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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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5HC99 lecture 1

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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5HC99 lecture 1

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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5HC99 lecture 1

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

qq

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