Introduction Robotics lecture - day 2 out of 7.ppt · Introduction Robotics Introduction Robotics,...
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1/47 Introduction Robotics Introduction Robotics, lecture 2 of 7 dr Dragan Kostić WTB Dynamics and Control September - October 2009
Transcript of Introduction Robotics lecture - day 2 out of 7.ppt · Introduction Robotics Introduction Robotics,...
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Introduction Robotics
Introduction Robotics, lecture 2 of 7
dr Dragan Kostić
WTB Dynamics and Control
September - October 2009
Introduction Robotics
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New Plan for the Lecture Rooms
• Every Tuesday, 3e+4e hours
• Week 37+40+42+43 in AUD.10
Introduction Robotics, lecture 2 of 7
• Week 37+40+42+43 in AUD.10
• Week 38+39 in AUD.14
• Week 41 in AUD.15
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• Recapitulation
• Parameterization of rotations
Outline
Introduction Robotics, lecture 2 of 7
• Rigid motions
• Homogenous transformations
• Robotic manipulators
• Forward kinematics problem
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Recapitulation
Introduction Robotics, lecture 2 of 7
Recapitulation
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Representing rotations in 3D
Each axis of the frame o1x1y1z1 is projected onto o0x0y0z0:
Introduction Robotics, lecture 2 of 7
R10 ∈ SO(3)
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Basic rotation matrices
Introduction Robotics, lecture 2 of 7
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Introduction Robotics, lecture 2 of 7
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Introduction Robotics, lecture 2 of 7
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Parameterization of rotations
Rigid motions
Introduction Robotics, lecture 2 of 7
Rigid motions
Homogenous transformations
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Basic homogenous transformations
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Robotic Manipulators
Introduction Robotics, lecture 2 of 7
Robotic Manipulators
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Robot manipulators
• Kinematic chain is series
of links and joints.SCARAgeometry
Introduction Robotics, lecture 2 of 7
• Types of joints:
– rotary (revolute, θ)
– prismatic (translational, d).
geometry
schematic representations of robot joints
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Anthropomorphic geometry (RRR)
• Resembles
geometry of
human arm.
• Features at least
Introduction Robotics, lecture 2 of 7
Upper arm• Features at least
3 rotary joints.
• Also known as
articulated
kinematic
scheme.
Staubli
(ex Unimation)
PUMA robot
http://www.ifr.org/pictureGallery/robType.htm
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Spherical geometry (RRP)
• Robot axes form a polar coordinate system.
Introduction Robotics, lecture 2 of 7
Stanford Arm - 1969
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SCARA geometry (RRP)
• Robot having 2 parallel rotary joints to provide compliance in x-y
direction (suitable for assembly tasks); sufficiently rigid in z-direction.
• SCARA stands for Selective Compliant Assembly Robot Arm or
Selective Compliant Articulated Robot Arm.
Introduction Robotics, lecture 2 of 7
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Cylindrical geometry (RPP)
• Robot axes form a cylindrical
coordinate system.
Introduction Robotics, lecture 2 of 7
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Cartesian geometry (PPP)
• Robot having 3 prismatic joints, whose axes are coincident with
a Cartesian coordinate frame.
Introduction Robotics, lecture 2 of 7
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Parallel geometry
• Robot whose arms have
concurrent prismatic or
revolute joints.
• Some subset of robot links
Introduction Robotics, lecture 2 of 7
• Some subset of robot links
form a closed kinematic
chain.
• There are at least two open
kinematic chains
connecting the base to the
end-effector.
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Workspace per robot geometry (1/2)
Introduction Robotics, lecture 2 of 7
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Workspace per robot geometry (2/2)
Introduction Robotics, lecture 2 of 7
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Forward Kinematics
Introduction Robotics, lecture 2 of 7
Forward Kinematics
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Scope
• Formulate problem of Forward Kinematics (FK) in robotics.
• Introduce Denavits-Hartenberg (DH) formalism to assign
Introduction Robotics, lecture 2 of 7
coordinate frames to robot joints.
• Solve FK problem using matrix manipulation
of homogenous transformations.
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Forward kinematics problem
• Determine position and orientation of the end-effector as
function of displacements in robot joints.
Introduction Robotics, lecture 2 of 7
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Illustration: planar manipulator with 2 joints
• Position of end-effector:
Introduction Robotics, lecture 2 of 7
• Orientation of end-effector:
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Conventions (1/2)1. there are n joints and hence
n + 1 links; joints 1, 2, …, n;
links 0, 1, …, n,
2. joint i connects link i − 1
to link i,
Introduction Robotics, lecture 2 of 7
to link i,
3. actuation of joint i
causes link i to move,
4. link 0 (the base) is fixed
and does not move,
5. each joint has a single
degree-of-freedom (dof):
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Conventions (2/2)
6. frame oixiyizi is attached to
link i; regardless of motion of
the robot, coordinates of each
point on link i are constant
Introduction Robotics, lecture 2 of 7
when expressed in frame
oixiyizi,
7. when joint i is actuated, link i
and its attached frame oixiyizi
experience resulting motion.
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DH convention for homogenous transformations (1/2)
• An arbitrary homogeneous transformation is based on 6
independent variables: 3 for rotation + 3 for translation.
• DH convention reduces 6 to 4, by specific choice of
the coordinate frames.
Introduction Robotics, lecture 2 of 7
the coordinate frames.
• In DH convention, each homogeneous transformation has the form:
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DH convention for homogenous transformations (2/2)
• Position and orientation of coordinate frame i with respect to
frame i-1 is specified by homogenous transformation matrix:
qi
q0
qi+1
xn
z0
zn
Introduction Robotics, lecture 2 of 7
ai
qi
qi
x0
xi-1
xi
zi
zi-1
y0 yn
zn
di
αi‘0’ ‘ ’n
where
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Mathematical rational (1/3)
• There exists unique
homogenous transformation matrix A1 that expresses
coordinates from frame 1
relative to frame 0.
Introduction Robotics, lecture 2 of 7
relative to frame 0.
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Mathematical rational (2/3)
0; 0101
010101
010101
010101
0
1 =⋅⇒⊥
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
= zxzx
zzzyzx
xzxyyx
xzxyxx
R
• Each column and row of R10 has unique length, implying
Introduction Robotics, lecture 2 of 7
1)()(and1)()( 2
01
2
01
2
01
2
01 =⋅+⋅=⋅+⋅ zzzyyxxx
• Each column and row of R1 has unique length, implying
and in turn there exist unique θ and α such that
)cos,(sin),(
)sin,(cos),(
0101
0101
αα
θθ
=⋅⋅
=⋅⋅
zzzy
yxxx
• Other elements of R10 ensure R1
0∈SO(3).
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Mathematical rational (3/3)
0
1
0
0
0
0
0
1 =++= axdzoo
• Displacement between the origins of frames 1 and 0 is linear
combination of vectors z0 and x1:
Introduction Robotics, lecture 2 of 7
.sin
cos
0
sin
cos
1
0
0
0
0
0
1001
=
=
+
+
=
=++=
d
a
a
ad
axdzoo
θ
θ
θ
θ
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Physical meaning of DH parameters• Link length ai is distance from zi-1 to
zi measured along xi.
• Link twist αi is angle between zi-1
and zi measured in plane normal
to x (right-hand rule). ai
qi
q0
qi+1
xx
zi
xn
y0 yn
z0
zn
αi‘0’ ‘ ’n
Introduction Robotics, lecture 2 of 7
to xi (right-hand rule).
• Link offset di is distance from origin
of frame i-1 to the intersection xi
with zi-1, measured along zi-1.
• Joint angle θi is angle from xi-1 to xi
measured in plane normal to zi-1
(right-hand rule).
ai
qi
x0
xi-1
xi
zi-1di
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DH convention to assign coordinate frames1. Assign zi to be the axis of actuation for joint i+1 (unless otherwise stated zn
coincides with zn-1).
2. Choose x0 and y0 so that the base frame is right-handed.
3. Iterative procedure for choosing oixiyizi depending on oi-1xi-1yi-1zi-1 (i=1, 2, …, n-1):
a) zi−1 and zi are not coplanar; there is an unique shortest line segment from zi−1 to
zi, perpendicular to both; this line segment defines xi and the point where the
Introduction Robotics, lecture 2 of 7
zi, perpendicular to both; this line segment defines xi and the point where the
line intersects zi is the origin oi; choose yi to form a right-handed frame,
b) zi−1 is parallel to zi; there are infinitely many common normals; choose xi as
the normal passes through oi−1; choose oi as the point at which this normal
intersects zi; choose yi to form a right-handed frame,
c) zi−1 intersects zi; axis xi is chosen normal to the plane formed by zi and zi−1;
it’s positive direction is arbitrary; the most natural choice of oi is the
intersection of zi and zi−1, however, any point along the zi suffices;
choose yi to form a right-handed frame.
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Forward kinematics (1/2)
• Homogenous transformation matrix relating the frame oixiyizi to
oi-1xi-1yi-1zi-1:
Introduction Robotics, lecture 2 of 7
Ai specifies position and orientation of oixiyizi w.r.t. oi-1xi-1yi-1zi-1.
• Homogenous transformation matrix Tji expresses position and
orientation of ojxjyjzj with respect to oixiyizi:
jjiiij AAAAT 121 −++= K
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• Forward kinematics of a serial manipulator with n joints can be
represented by homogenous transformation matrix Hn0 which
defines position and orientation of the end-effector’s (tip)
frame o x y z relative to the base coordinate frame o x y z :
Forward kinematics (2/2)
Introduction Robotics, lecture 2 of 7
frame onxnynzn relative to the base coordinate frame o0x0y0z0:
=
⋅⋅==
×1
)()()(
),()()()(
13
00
0
11
00
0
qqq
nn
n
nnnn
xRH
qAqATH K
[ ];00013 =×
0
[ ]Tnqq L1=q
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Case-study: RRR robot manipulator
-q3
x
x2
x3
y1
y2
y3
d2a3
d3
z2
z3
elbow
Introduction Robotics, lecture 2 of 7 x0
q1
q2
x1
y0
y1
z0
α1
d1
a2
z1
z2
waist
shoulder
elbow
α1 - twist angle
ai - link lenghts
di - link offsets
qi - displacements
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DH parameters of RRR robot manipulator
Introduction Robotics, lecture 2 of 7
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Forward kinematics of RRR robot manipulator (1/2)
• Coordinate frame o3x3y3z3 is related with the base frame o0x0y0z0 via
homogenous transformation matrix:
== 32103 (q)(q)A(q)AA(q)T
Introduction Robotics, lecture 2 of 7
=
× 131
03
03
0
(q)x(q)R
whereT
qqq ][ 321=qT
zyx ][)(03 =qx
]000[31 =×0
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Forward kinematics of RRR robot manipulator (2/2)
,
• Position of end-effector:
[ ] 132223231 )sin(cos)cos(cos qddqaqqaqx ++++=
[ ] 132223231 )cos(cos)cos(sin qddqaqqaqy +−++=
Introduction Robotics, lecture 2 of 7
, 122323 sin)sin( dqaqqaz +++=
• Orientation of end-effector:
++
−+−+
+−+
=
0)cos()sin(
cos)sin(sin)cos(sin
sin)sin(cos)cos(cos
3232
1321321
132132103
qqqq
qqqqqqq
qqqqqqq
R
