FORWARD KINEMATICS -...

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FORWARD KINEMATICS INTRODUCTION TO ROBOTICS: DISCUSSION 2 ROBERT PETER MATTHEW 20150908

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FORWARD KINEMATICS

INTRODUCTION TO ROBOTICS DISCUSSION 2ROBERT PETER MATTHEW

20150908

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ADMINISTRIVADISCUSSION 2 FORWARD KINEMATICS

bull Course text is available on bCoursesbull Course grading policybull Piazza questions

bull Discussion sections Tues 1000-1100 Wed 1100-1200bull Office hours Mon Thurs 1100-1200bull Homeworks Released Due and Graded by Thursdaybull DSP Students Letters of Accommodation required asap

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EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

119877119877 = 119877119877119885119885 1205791205793 119877119877119884119884minus1205871205872

119877119877119883119883 1205791205791

119877119877 =0 119904119904119904119904119904119904 1205791205791 minus 1205791205793 119888119888119888119888119904119904 1205791205791 minus 12057912057930 119888119888119888119888119904119904 1205791205791 minus 1205791205793 minus119904119904119904119904119904119904 1205791205791 minus 1205791205793minus1 0 0

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

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EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

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AXIS ANGLEDISCUSSION 1 ROTATIONS

119877119877 = 1201281201283 +120596120596120596120596 2

119904119904119904119904119904119904 120579120579 +1205961205962

120596120596 22 1 minus 119888119888119888119888119904119904 120579120579

Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula

(a derivation of this equation will be given next week)

120596120596 =120596120596112059612059621205961205963

=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

120596120596 2 = 119894119894=1

119899119899119909119909119894119894 2

1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992

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EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =100

=0 0 00 0 minus10 1 0

Consider a rotation about the x axis

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Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

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Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

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TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

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RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

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HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

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FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
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  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
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  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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Page 2: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

ADMINISTRIVADISCUSSION 2 FORWARD KINEMATICS

bull Course text is available on bCoursesbull Course grading policybull Piazza questions

bull Discussion sections Tues 1000-1100 Wed 1100-1200bull Office hours Mon Thurs 1100-1200bull Homeworks Released Due and Graded by Thursdaybull DSP Students Letters of Accommodation required asap

2104

rpmatthew ltatgt berkeleyedu

EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

119877119877 = 119877119877119885119885 1205791205793 119877119877119884119884minus1205871205872

119877119877119883119883 1205791205791

119877119877 =0 119904119904119904119904119904119904 1205791205791 minus 1205791205793 119888119888119888119888119904119904 1205791205791 minus 12057912057930 119888119888119888119888119904119904 1205791205791 minus 1205791205793 minus119904119904119904119904119904119904 1205791205791 minus 1205791205793minus1 0 0

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

3104

rpmatthew ltatgt berkeleyedu

EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

4104

rpmatthew ltatgt berkeleyedu

AXIS ANGLEDISCUSSION 1 ROTATIONS

119877119877 = 1201281201283 +120596120596120596120596 2

119904119904119904119904119904119904 120579120579 +1205961205962

120596120596 22 1 minus 119888119888119888119888119904119904 120579120579

Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula

(a derivation of this equation will be given next week)

120596120596 =120596120596112059612059621205961205963

=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

120596120596 2 = 119894119894=1

119899119899119909119909119894119894 2

1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992

5104

rpmatthew ltatgt berkeleyedu

EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =100

=0 0 00 0 minus10 1 0

Consider a rotation about the x axis

6104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

7104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
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  • Forward Kinematics
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  • Exponential Coordinates
  • Exponential Coordinates
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  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
Page 3: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

119877119877 = 119877119877119885119885 1205791205793 119877119877119884119884minus1205871205872

119877119877119883119883 1205791205791

119877119877 =0 119904119904119904119904119904119904 1205791205791 minus 1205791205793 119888119888119888119888119904119904 1205791205791 minus 12057912057930 119888119888119888119888119904119904 1205791205791 minus 1205791205793 minus119904119904119904119904119904119904 1205791205791 minus 1205791205793minus1 0 0

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

3104

rpmatthew ltatgt berkeleyedu

EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

4104

rpmatthew ltatgt berkeleyedu

AXIS ANGLEDISCUSSION 1 ROTATIONS

119877119877 = 1201281201283 +120596120596120596120596 2

119904119904119904119904119904119904 120579120579 +1205961205962

120596120596 22 1 minus 119888119888119888119888119904119904 120579120579

Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula

(a derivation of this equation will be given next week)

120596120596 =120596120596112059612059621205961205963

=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

120596120596 2 = 119894119894=1

119899119899119909119909119894119894 2

1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992

5104

rpmatthew ltatgt berkeleyedu

EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =100

=0 0 00 0 minus10 1 0

Consider a rotation about the x axis

6104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

7104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 4: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXAMPLE 2 EULER ANGLESDISCUSSION 1 ROTATIONS

Find the net rotation described by (21205871205873

minus1205871205872

minus1205871205873

) in Z-Y-X form

4104

rpmatthew ltatgt berkeleyedu

AXIS ANGLEDISCUSSION 1 ROTATIONS

119877119877 = 1201281201283 +120596120596120596120596 2

119904119904119904119904119904119904 120579120579 +1205961205962

120596120596 22 1 minus 119888119888119888119888119904119904 120579120579

Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula

(a derivation of this equation will be given next week)

120596120596 =120596120596112059612059621205961205963

=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

120596120596 2 = 119894119894=1

119899119899119909119909119894119894 2

1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992

5104

rpmatthew ltatgt berkeleyedu

EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =100

=0 0 00 0 minus10 1 0

Consider a rotation about the x axis

6104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

7104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 5: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

AXIS ANGLEDISCUSSION 1 ROTATIONS

119877119877 = 1201281201283 +120596120596120596120596 2

119904119904119904119904119904119904 120579120579 +1205961205962

120596120596 22 1 minus 119888119888119888119888119904119904 120579120579

Given a rotational axis 120596120596 and angle 120579120579 the corresponding rotation matrix 119877119877 is given by the Rodrigues formula

(a derivation of this equation will be given next week)

120596120596 =120596120596112059612059621205961205963

=0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

120596120596 2 = 119894119894=1

119899119899119909119909119894119894 2

1 2= 11990911990912 + 11990911990922 + ⋯1199091199091198991198992

5104

rpmatthew ltatgt berkeleyedu

EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =100

=0 0 00 0 minus10 1 0

Consider a rotation about the x axis

6104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

7104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 6: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =100

=0 0 00 0 minus10 1 0

Consider a rotation about the x axis

6104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

7104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 7: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =110

=0 0 10 0 minus1minus1 1 0

7104

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 8: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

Consider a rotation about the x axisEXAMPLE AXIS ANGLEDISCUSSION 1 ROTATIONS

120596120596 =111

=0 minus1 11 0 minus1minus1 1 0

8104

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 9: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

TERMINOLOGYDISCUSSION 2 FORWARD KINEMATICS

JOINT POSITION TORQUE

END EFFECTOR POSITION TORQUE

Forward Kinematicsndash Given joint positions find end

effector coordinatesInverse Kinematics

ndash Given end effector coordinates find required joint positions

Forward Dynamicsndash Given joint torques find end

effector forcestorquesInverse Dynamics

ndash Given a desired end effector forcetorque find required joint torques

9104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 10: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

10104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 11: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

11104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 12: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Translation of Origin Relative Rotation

12104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 13: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

13104

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 14: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Homogeneous Coordinates

Note all configurations are RELATIVE

14104

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 15: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

HOMOGENEOUS COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Points Vectors

15104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 16: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodies

that make up the manipulator

Forward Kinematics allows the configuration (position and orientation) of the end effector to be determined given a description of the joints

and the joint angles

16104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 17: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

17104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
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  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 18: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

18104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
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  • Terminology
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  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
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  • Exponential Rotations
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  • Exponential Rigid Body Motion
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  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 19: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

19104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 20: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

20104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 21: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120788120788120782120782120782120782

21104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 22: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

22104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 23: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

23104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 24: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

24104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
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  • Forward Kinematics
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  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
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  • Forward Kinematics
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  • Forward Kinematics
Page 25: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

25104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 26: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119912119912 =120785120785120782120782120782120782

26104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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Page 27: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

27104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 28: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

28104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
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  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
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  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 29: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

29104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 30: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

30104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

31104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
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  • Terminology
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  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
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  • Exponential Rigid Body Motion
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Page 32: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

32104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 33: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

33104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 34: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

34104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
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  • Terminology
  • Rigid Body Motion
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  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
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  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 35: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

35104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
Page 36: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119936119936

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119912119912 =120785120785120782120782120782120782

36104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 37: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 11199541199541198601198601

1199541199541198601198601 = 119929119929119860119860119862119862 119953119953119860119860119862119862

120782120782 11199541199541198621198621

The Kinematics of a robotic manipulator describes the relationship between the motion of the joints and the motion of the rigid bodiesthat make up the manipulator

Note all configurations are RELATIVE

37104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 38: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

38104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 39: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

39104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
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  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 40: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

40104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 41: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

41104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Coordinates
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  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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Page 42: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

42104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

43104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
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  • Terminology
  • Rigid Body Motion
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  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Rotations
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  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 44: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120782120782

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119937119937

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

44104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
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  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
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  • Homogeneous Coordinates
  • Forward Kinematics
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  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Exponential Rigid Body Motion
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Page 45: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

45104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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  • Forward Kinematics
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Page 46: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

46104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
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Page 47: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

47104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
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  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
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  • Exponential Rigid Body Motion
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  • Forward Kinematics
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Page 48: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

48104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 49: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

49104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 50: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

50104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 51: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119929119929119912119912119912119912 = 119929119929119937119937

119953119953119912119912119912119912 =120782120782120782120782120785120785

119954119954119914119914 =120785120785120782120782120782120782

119929119929119912119912119914119914 = 119929119929119936119936

119953119953119912119912119914119914 =120785120785120782120782120782120782

1199541199541198601198601 = 119929119929119860119860119860119860 119953119953119860119860119860119860

120782120782 1119929119929119860119860119862119862 119953119953119860119860119862119862120782120782 1

1199541199541198621198621

51104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 52: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

coordinate transforms

52104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 53: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Rigid body motion as

solutions to differential equationscoordinate transforms

53104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 54: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL COORDINATESDISCUSSION 2 FORWARD KINEMATICS

Twists and screws describe the same concept

Any rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that axis

-Chasles Theorem

These rotations and translations can be found using twists andor screw theory

54104

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 55: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

TWISTS AND SCREWSDISCUSSION 2 FORWARD KINEMATICS

Twistsbull Rotational velocity lsquo120596120596rsquobull Linear velocity lsquo119907119907rsquo bull A magnitude lsquo120579120579rsquo

Screwsbull An axis of rotation lsquo119897119897rsquobull A Pitch lsquoℎrsquobull A magnitude lsquo119872119872rsquo

55104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 56: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

56104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
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Page 57: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

57104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 58: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL ROTATIONSDISCUSSION 2 FORWARD KINEMATICS

Given the differential equation 119902 = 120596120596 times 119902119902 119905119905

We know that the solution is given by 119902119902 120579120579 = 119890119890 120596120596120579120579119902119902 0

Looking at its structure we see that 119890119890 120596120596120579120579 acts as a rotation ieR 120596120596120579120579 = 119890119890 120596120596120579120579

where

120596120596 =0 minus1205961205963 12059612059621205961205963 0 minus1205961205961minus1205961205962 1205961205961 0

58104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 59: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Using similar analysis we can derive

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

where

120585120585 = 120596120596 119907119907120782120782 0

bull 120596120596 is the rotation velocity vectorbull 119907119907 is the linear velocity vectorbull 119901119901 0 is the zero configuration and is given in ABSOLUTE

coordinates

59104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 60: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

There are two operators for the exponential RBM 119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

bull lsquoWedgersquo 120585120585^ = 119907119907120596120596

^= 120596120596 119907119907

120782120782 0 = 120585120585

bull lsquoVeersquo 120585120585or = 120596120596 119907119907120782120782 0

or= 119907119907

120596120596 = 120585120585

bull Note the ^ symbol refers to two different operations

60104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0

Compute the matrix exponent 119890119890120585120585120579120579

bull Analytically

119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579120782120782 1

bull Computationally expm

61104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 62: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

119901119901 120579120579 = 119890119890120585120585120579120579119901119901 0120585120585 = 119907119907

120596120596

Special cases

bull Pure Rotation120585120585 = minus120596120596 times 119902119902

120596120596

bull Pure Translation120585120585 = 119907119907

0

62104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 63: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

63104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 64: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

64104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 65: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Rotational velocity

120596120596 =001

65104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 66: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Linear velocity119907119907 = minus120596120596 times 119902119902

119902119902 is ANY point on the rotational axis

119907119907 = minus001

times011989711989710

=119897119897100

66104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 67: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Twist

119907119907 =119897119897100

120596120596 =001

120585120585 = 119907119907120596120596 =

119897119897100001

67104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 68: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Exponential

119890119890120585120585120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

68104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 69: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0119890119890120585120585120579120579 = 119890119890 120596120596120579120579 119868119868 minus 119890119890 120596120596120579120579 120596120596 times 119907119907 + 120596120596120596120596119879119879119907119907120579120579

120782120782 1

bull Initial Configuration119892119892 0 = 1198771198770 1198791198790

120782120782 1

119892119892 0 = 120128120128011989711989710

120782120782 1

69104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 70: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

70104

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 71: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

EXPONENTIAL RIGID BODY MOTIONDISCUSSION 2 FORWARD KINEMATICS

Example Single rotation about the 119885119885 axis

bull RBM119892119892 120579120579 = 119890119890120585120585120579120579119892119892 0

119892119892 120579120579 =cos 120579120579 minus119904119904119904119904119904119904 120579120579 0119904119904119904119904119904119904 120579120579 119888119888119888119888119904119904 120579120579 0

0 0 1

1198971198971119904119904119904119904119904119904 1205791205791198971198971 1 minus 119888119888119888119888119904119904 120579120579

0120782120782 1

120128120128011989711989710

120782120782 1

119892119892 120579120579 =cos θ minussin θ 0sin θ cos θ 0

0 0 1

0l10

120782120782 1

71104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 72: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

72104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 73: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

73104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 74: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

74104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 75: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

75104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

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Page 76: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283600

120782120782 1

76104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 77: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

77104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 78: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

78104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 79: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

79104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 80: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

80104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 81: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283300

120782120782 1

81104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 82: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

82104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 83: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

83104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 84: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

84104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 85: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

85104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 86: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

000001

119892119892(0) = 1201281201283303

120782120782 1

86104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 87: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

87104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 88: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

88104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
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Page 89: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

89104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 90: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

90104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 91: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 11989011989012058512058511205791205791119892119892 0

1205851205851 = minus120596120596 times 119902119902120596120596 =

minus300010

119892119892(0) = 1201281201283303

120782120782 1

91104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 92: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

The effect of multiple RBMs can be found via the composition of multiple matrix exponents

For any reference frame at a zero configuration we can write

Note all configurations are in ABSOLUTE coordinates

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

92104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 93: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

93104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 94: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

94104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 95: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

95104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 96: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

96104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 97: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

97104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 98: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283600

120782120782 1

1205851205851 =

000001

1205851205852 =

0minus30001

98104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 99: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

99104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 100: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

100104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 101: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

101104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 102: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

102104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
Page 103: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

103104

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
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Page 104: FORWARD KINEMATICS - robertpetermatthew.comrobertpetermatthew.com/wp-content/uploads/2018/01/Discussion02... · Kinematics. of a robotic manipulator describes the relationship between

rpmatthew ltatgt berkeleyedu

FORWARD KINEMATICSDISCUSSION 2 FORWARD KINEMATICS

119892119892 120579120579 = 1198901198901205851205851120579120579111989011989012058512058521205791205792119892119892 0

119892119892(0) = 1201281201283603

120782120782 1

1205851205851 =

000001

1205851205852 =

minus303010

104104

  • Forward Kinematics
  • Administriva
  • Example 2 Euler Angles
  • Example 2 Euler Angles
  • Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Example Axis Angle
  • Terminology
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Rigid Body Motion
  • Homogeneous Coordinates
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
  • Exponential Coordinates
  • Exponential Coordinates
  • Exponential Coordinates
  • Twists and Screws
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rotations
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Exponential Rigid Body Motion
  • Forward Kinematics
  • Forward Kinematics
  • Forward Kinematics
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  • Forward Kinematics
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