Differential Kinematics and Statics

Ref: 理论力学，洪嘉振，杨长俊，高等教育出版社， 2001

Incremental MotionIncremental Motion

What small (incremental) motions at the end-effector (x, y, z) result from small motions of the joints (1, 2, …, n )?

Alternatively, what velocities at the end-effector (vx, vy, vz) result from velocities at the joints (1, 2, … n)?

ii

t

vx

tv

y

tv

z

txn

yn

zn, , ,

Some DefinitionsSome Definitions

Linear Velocity: The instantaneous rate-of-change in linear position of a point relative to some frame.

v=(vx, vy, vz)T

Angular Velocity: The instantaneous rate-of-change in the orientation of one frame relative to another.– Angular Velocity depends on the way to represent orientat

ion (Euler Angles, Rotation Matrix, etc.)– Angular Velocity Vector and the Angular Velocity Matrix.

Some DefinitionsSome Definitions

Angular Velocity Vector: A vector whose direction is the instantaneous axis of rotation of one frame relative to another and whose magnitude is the rate of rotation about that axis.

Tzx y )( ＝

x

y

z

Free VectorFree Vector

Linear velocity are insensitive to shifts in origin but are sensitive to orientation.

{D}

DBA vvv

x

x

Free VectorFree Vector

Angular velocity are insensitive to shifts in origin but are sensitive to orientation.

DBA

{A}

{B}

{D}x

x

xx

Velocity FramesVelocity Frames

frame of reference: this is the frame used to measure the object’s velocity

frame of representation.: this is the frame in which the velocity is expressed.

X0

Y0

x0

y0

0

Y1X1

0

x2

a1

v

vv

v

R

a2

y2

Figure 2.13: Two-Link Planar Robot

X0

Y0

x0

y0

0

0 v

vv

v

End-effector velocity for 1

r0n

n01 r

X0

Y0

x0

y0

0

0 v

vv

v

End-effector velocity for 2

r1n

n12 r

Two-Link Planar RobotTwo-Link Planar Robot

Direct kinematics equation Direct kinematics equation

x x c y s a c a c

y x s y c a s a s

2

0 2

0 2 12 12 2 12 1 1

2 12 12 2 12 1 1

Incremental MotionIncremental Motion taking derivatives of the position taking derivatives of the position

equation w.r.t. time we have equation w.r.t. time we have

note thatnote that

v x a y c a s

v x a ) c y s a c

x 2

y 2

( )( ) ( )

( ( ) ( )

2 2 1 2 12 1 2 12 1 1 1

2 2 1 2 12 1 2 12 1 1 1

s

d(c

dt)s

d(s

dt)c

12

1 2 1212

1 2 12

)( ,

)(

Incremental MotionIncremental Motion

written in the more common matrix written in the more common matrix form, form,

or in terms of incremental motion,or in terms of incremental motion,

v

v =

x a y c a s x a y c

x a )c y s a c x a )c y sx

y

2 2

2 2

( ) ( )

( (2 2 12 12 1 1 2 2 12 12

2 2 12 12 1 2 2 12 12

1

2

s s

x

y =

x a y c a s x a y c

x a )c y s a c x a )c y s2 2

2 2

( ) ( )

( (2 2 12 12 1 1 2 2 12 12

2 2 12 12 1 2 2 12 12

1

2

s s

Differential KinematicsDifferential Kinematics

Find the relationship between the joint velocities and the end-effector linear and angular velocities.

Linear velocity

Angular velocity

i

ii d

q for a revolute joint

for a prismatic joint

Differential KinematicsDifferential Kinematics

Differential kinematics equation

Geometric Jacobian 6Rv

Relationship with T(q)Relationship with T(q)

Direct kinematics equation

Linear velocity

Angular velocity?

)(qpp

？)(qRq

qpJ P

)(

Vector (Cross) ProductVector (Cross) Product

Vector product of x and y

Skew-symmetric matrix

321

321

yyy

xxx

kji

yx

0SS T

Vector (Cross) ProductVector (Cross) Product

Skew-symmetric matrix

Derivative of a Rotation MatrixDerivative of a Rotation Matrix

define

S(t) is skew-symmetric

Interpretation of S(t)Interpretation of S(t)

Interpretation of S(t)Interpretation of S(t)

Given R(t)

Example 3.1: Rotation about ZExample 3.1: Rotation about Z