Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSOutline:

• Introduction• Solvability • Manipulator subspace when n<6• Algebraic vs. Geometric• Example: Kinematics of PUMA Robot

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSIntroduction:

• Direct kinematics:Given the joint angles calculate the position and

orientation of the tool {T} relative to the station {S}• Inverse kinematics:

Given the desired position and orientation of the tool {T} relative to the station {S}, how to calculate the set of joint angles that give the desired result? – Find {W} from {T}; Find {B} from {S};

– Solve for joint angles

WTTBST

BWT

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSSolvability:

• Nonlinear problem: solve to find the values of

• Linear and nonlinear problems? • Ex.: Puma manipulator

01 1( , ,..., )B

W n nT T

1 1, ,..., n

11 12 13

21 22 230

31 32 33

0 0 0 1

x

yn

z

r r r P

r r r PT

r r r P

12 values find

12 equations and 6 unknowns, rotation part (r11 … r33), only 3 independent

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSSolvability:

• Nonlinear problem: solve to find the values of

• Linear and nonlinear problems? • Ex.: Puma manipulator

01 1( , ,..., )B

W n nT T

1 1, ,..., n

11 12 13

21 22 230

31 32 33

0 0 0 1

x

yn

z

r r r P

r r r PT

r r r P

12 values find

12 equations and 6 unknowns, rotation part (r11 … r33), only 3 independent

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSSolvability:

3 independent equations for the orientation and 3 independent equations for the position Nonlinear equations that are difficult to solve. – Note that these were for simple links α = 0, 90, -90, … and many

(d & a) = 0.– General case (α, d, & a) have other nonzero valuesMore complex case.

We must concern on – Existence of solution?– Multiple solutions?– Method of solution?

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSSolvability:

• Existence of solutionExistence or nonexistence of a solution defines the workspace of the manipulator.

• Workspace (W.S.): (Figure)– The volume of the space that the E.E. of the robot can reach.– A solution to exist the desired position & orientation (goal)

must lie in the W.S.

• Two types of workspaces:– Dextrous W.S.: volume of the W.S. in which the E.E. can reach

with all the orientations.– Reachable W.S.: volume of the W.S. in which the E.E. can reach in

at least one orientation. Dextrous W.S. is a subset of the reachable W.S.

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSSolvability:

• Example: if l1 = l2 determine the dextrous and reachable W.S.

Chapter 4: Inverse Kinematics

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Faculty of Engineering - Mechanical Engineering Department

ROBOTICSSolvability:

• Factors that affect the W.S. of a robot:– Limitations on joint anglesGenerally: min < θi < max. Most of the robots have limitations on their joint angles. The W.S. is reduced; this may affect the attained positions or the

number of orientations for the attained position. • Ex.: for the previous sketch 0 ≤ θi ≤ 180 sketch the W.S.

The same positions can be reached, however in only one orientation.

– Limitations on the number of DoF:A 3D-space goal is not attainable by a manipulator of DoF < 6.For the manipulator of the previous example can not attain a position of nonzero z-value.