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5HC99 lecture 1
Kinematic Modelling in Robotics
dr Dragan Kostić
WTB Dynamics and Control
October 22th, 2010
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Outline
• Representing rotations and rotational transformations
• Parameterization of rotations
• Rigid motions and homogenous transformations
• DH convention for modeling of robot kinematics
• Forward kinematics
• Case-study: kinematics of RRR-arm
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Representing rotations in coordinate frame 0
• Rotation matrix
• xi and yi are the unit vectors in oixiyi
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Representing rotations in coordinate frame 1
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Representing rotations in 3D (1/4)
Each axis of the frame o1x1y1z1 is projected onto o0x0y0z0:
R10 SO(3)
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Representing rotations in 3D (2/4)
Example: Frame o1x1y1z1 is obtained from frame o0x0y0z0
by rotation through an angle about z0 axis.
all other dot products are zero
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Representing rotations in 3D (3/4)
Basic rotation matrix about z-axis
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Representing rotations in 3D (4/4)
Similarly, basic rotation matrices about x- and y-axes:
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Rotational transformations
pi: coordinates of p in oixiyizi
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Parameterization of rotations (1/2)
Euler angles
ZYZEuler angle transformation:
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Parameterization of rotations (2/2)
Roll, pitch, yaw angles
XYZyaw-pitch-roll angle transformation:
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Rigid motions
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Homogenous transformations (1/2)
• We have
• Note that
• Consequently, rigid motion (d, R) can be described by matrix
representing homogenous transformation:
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Homogenous transformations (2/2)
• Since R is orthogonal, we have
• We augment vectors p0 and p1 to get their homogenous
representations
and achieve matrix representation of coordinate transformation
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Basic homogenous transformations
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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,
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
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
Position and orientation of coordinate frame i with respect to
frame i-1 is specified by homogenous transformation matrix:
ai
qi
q0
qi
qi+1
x0
xi-1
xi
zi
zi-1
xn
y0 yn
z0zn
di
i‘0’ ‘ ’n
where
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Physical meaning of DH parameters• Link length ai is distance from zi-1 to zi
measured along xi.
• Link twist i is angle between zi-1 and zi
measured in plane normal
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 i is angle from xi-1 to xi
measured in plane normal to zi-1 (right-
hand rule).
ai
qi
q0
qi
qi+1
x0
xi-1
xi
zi
zi-1
xn
y0 yn
z0zn
di
i‘0’ ‘ ’n
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DH convention to assign coordinate frames
1. 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 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)
Ai specifies position and orientation of oixiyizi w.r.t. oi-1xi-1yi-1zi-1.
Homogenous transformation matrix relating the frame oixiyizi to
oi-1xi-1yi-1zi-1:
Homogenous transformation matrix Tji expresses position and
orientation of ojxjyjzj with respect to oixiyizi:
jjiiij AAAAT 121
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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 onxnynzn relative to the base coordinate frame o0x0y0z0:
Forward kinematics (2/2)
1
)()()(
),()()()(
31
000
1100
0
qqq
nnn
nnnn
xRH
qAqATH
;00031 0
Tnqq 1q
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Case-study: RRR robot manipulator
x 0
q 1
q 2
-q 3
x 1
x 2
x 3
y 0
y 1
y 2
y 3
z 0
1
d 1
d 2
a 2
a 3
d 3
z 1
z 2
z 3
w ais t
sh o u ld e r
e lb o w
1 - tw is t an g le
a i - lin k len g h ts
d i - lin k o ffse tsq i - d isp lacem en ts
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DH parameters of RRR robot manipulator
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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:
131
03
03
32103
0
(q)x(q)R
(q)(q)A(q)AA(q)T
whereTqqq ][ 321q Tzyx ][)(0
3 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
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