velocity propagation between robot links...

Velocity Propagation Between Robot Links 3/4

Instructor: Jacob Rosen

Advanced Robotic - MAE 263D - Department of Mechanical & Aerospace Engineering - UCLA

Introduction – Velocity Propagation



Jacobian Matrix - Introduction

• In the field of robotics the Jacobian matrix

describe the relationship between the joint

angle rates ( ) and the translation and

rotation velocities of the end effector ( ).

This relationship is given by:

• In addition to the velocity relationship, we are

also interested in developing a relationship

between the robot joint torques ( ) and the

forces and moments ( ) at the robot end

effector (Static Conditions). This

relationship is given by:

N

x

Jx

F

F

FJT



Jacobian Matrix - Calculation Methods

Jacobian Matrix

Differentiation the

Forward Kinematics Eqs. Iterative Propagation

(Velocities or Forces / Torques)



Summary – Changing Frame of Representation

• Linear and Rotational Velocity

– Vector Form

– Matrix Form

• Angular Velocity

– Vector Form

– Matrix Form

Q

BA

BB

A

Q

BA

BBORG

A

Q

A PRVRVV

B

A

Q

BA

B

A

BQ

BA

BBORG

A

Q

A PRRVRVV

Q

BP

TA

B

B

C

A

B

A

B

A

C RRRRR

C

BA

BB

A

C

A R



Frame - Velocity

• As with any vector, a velocity vector may be described in terms of any frame,

and this frame of reference is noted with a leading superscript.

• A velocity vector computed in frame {B} and represented in frame {A} would be

written

Q

BA

Q

BA Pdt

dV )(

Q

Computed

(Measured)

Represented

(Reference Frame)

0BORG

AV

0RA

B



Position Propagation

• The homogeneous transform matrix provides a complete description of the

linear and angular position relationship between adjacent links.

• These descriptions may be combined together to describe the position of a link

relative to the robot base frame {0}.

• A similar description of the linear and angular velocities between adjacent links

as well as the base frame would also be useful.

TTTT i

i

oo

i

11

21



Position Propagation

TTTTTTTT TT

65

6

4

5

3

4

2

3

1

2

0

1

0



Motion of the Link of a Robot

• In considering the motion of a robot link we will always use link frame {0} as the

reference frame (Computed AND Represented). However any frame can be used

as the reference (represented) frame including the link’s own frame (i)

Where: - is the linear velocity of the origin of link frame (i) with respect to

frame {0} (Computed AND Represented)

- is the angular velocity of the origin of link frame (i) with respect to

frame {0} (Computed AND Represented)

• Expressing the velocity of a frame {i} (associated with link i ) relative to the robot

base (frame {0}) using our previous notation is defined as follows:

iv

i



iii 000

iii VVv 000

Velocities - Frame & Notation

• The velocities differentiate (computed) relative to the base frame {0} are often

represented relative to other frames {k}. The following notation is used for this

conditions

i

k

i

k

i

k

i

k vRVRVv 0

0

0

0

i

k

i

k

i

k

i

k RR 0

0

0

0



Velocity Propagation

• Given: A manipulator - A chain of

rigid bodies each one capable of

moving relative to its neighbor

• Problem: Calculate the linear and

angular velocities of the link of a

robot

• Solution (Concept): Due to the

robot structure (serial mechanism)

we can compute the velocities of

each link in order starting from the

base.

The velocity of link i+1 will be that of

link i , plus whatever new velocity

components were added by joint i+1



Velocity of Adjacent Links - Angular Velocity 0/5




• From the relationship developed previously

• we can re-assign link names to calculate the velocity of any link i relative to the

base frame {0}

• By pre-multiplying both sides of the equation by ,we can convert the frame

of reference for the base {0} to frame {i+1}

C

BA

BB

A

C

A R

1

0

iC

iB

A

1

00

1

0

i

i

iii R

Ri 1

0




• Using the recently defined notation, we have

- Angular velocity of frame {i+1} measured relative to the robot base, and

expressed in frame {i+1} - Recall the car example

- Angular velocity of frame {i} measured relative to the robot base, and

expressed in frame {i+1}

- Angular velocity of frame {i+1} measured relative to frame {i} and


1

01

0

01

01

01

0

i

i

i

i

i

i

i

i RRRR

1

11

1

1

i

ii

ii

i

i

i R

1

1

i

i

i

i 1

1

1

i

ii

i R

c

c

c

wcvV




• Angular velocity of frame {i} measured relative to the robot base, expressed in

frame {i+1}

1

11

1

1

i

ii

ii

i

i

i R

i

ii

ii

i R 11




• Angular velocity of frame {i+1} measured (differentiate) in frame {i} and

represented (expressed) in frame {i+1}

• Assuming that a joint has only 1 DOF. The joint configuration can be either

revolute joint (angular velocity) or prismatic joint (Linear velocity).

• Based on the frame attachment convention in which we assign the Z axis

pointing along the i+1 joint axis such that the two are coincide (rotations of a link

is preformed only along its Z- axis) we can rewrite this term as follows:

1

1

1 0

0

i

i

ii

i R

1

2

3

i+1

i

1

11

1

1

i

ii

ii

i

i

i R




• The result is a recursive equation that shows the angular velocity of one link in

terms of the angular velocity of the previous link plus the relative motion of the

two links.

• Since the term depends on all previous links through this recursion, the

angular velocity is said to propagate from the base to subsequent links.

1

1

1

1 0

0

i

i

ii

ii

i R

1

1

i

i



Velocity of Adjacent Links - Linear Velocity 0/6




• Simultaneous Linear and Rotational Velocity

• The derivative of a vector in a moving frame (linear and rotation velocities) as

seen from a stationary frame

• Vector Form

• Matrix Form

Q

BA

BB

A

Q

BA

BBORG

A

Q

A PRVRVV

B

A

Q

BA

B

A

BQ

BA

BBORG

A

Q

A PRRVRVV

Q

BP




• From the relationship developed previously (matrix form)

• we re-assign link frames for adjacent links (i and i +1) with the velocity

computed relative to the robot base frame {0}

• By pre-multiplying both sides of the equation by ,we can convert the

frame of reference for the left side to frame {i+1}

Q

BA

B

A

BQ

BA

BBORG

A

Q

A PRRVRVV

1

0

iC

iB

A

1

00

1

00

1

0

i

i

iii

i

iii VRVPRRV

Ri 1

0




• Which simplifies to

• Factoring out from the left side of the first two terms

1

01

0

01

01

001

01

01

0

i

i

i

i

i

i

i

i

ii

i

i

i VRRVRPRRRVR

1

101

01

001

01

01

0

i

ii

ii

i

i

i

ii

i

i

i VRVRPRRRVR

Ri

i

1

1

10

01

00

0

1

1

01

0

i

ii

ii

i

i

i

ii

ii

ii

i VRVRPRRRRVR




- Linear velocity of frame {i+1} measured relative to frame {i} and


• Assuming that a joint has only 1 DOF. The joint configuration can be either

revolute joint (angular velocity) or prismatic joint (Linear velocity).

• Based on the frame attachment convention in which we assign the Z axis

pointing along the i+1 joint axis such that the two are coincide (translation of a

link is preformed only along its Z- axis) we can rewrite this term as follows:

1

10

01

00

0

1

1

01

0

i

ii

ii

i

i

i

ii

ii

ii

i VRVRPRRRRVR

1

1

1 0

0

i

i

ii

i

d

VR

1

1

i

ii

i VR




1

0

01

00

0

1

1

01

0 0

0

i

i

i

i

i

ii

ii

ii

i

d

VRPRRRRVR

i

i

i

i

i

iTi

i

i

ii

i RRRRRRRR 0

0

00

0

0

00

0

Multiply by Matrix

Definition

i

i

i

i vVR 0

0

Definition

1

1

1

01

0

i

i

i

i vVR

Definition




• The result is a recursive equation that shows the linear velocity of one link in

terms of the previous link plus the relative motion of the two links.

• Since the term depends on all previous links through this recursion, the

angular velocity is said to propagate from the base to subsequent links.

1

1

1

1

1 0

0

i

i

i

i

i

i

ii

ii

i

d

vPRv

1

1

i

i v



Velocity of Adjacent Links - Summary

• Angular Velocity

• Linear Velocity

1

1

1

1

1 0

0

i

i

i

i

iii

ii

i

d

vPRv

1

1

1

1 0

0

i

i

ii

ii

i R

0 - Prismatic Joint

0 - Revolute Joint



Angular and Linear Velocities - 3R Robot - Example

• For the manipulator shown in the figure, compute the angular and linear velocity

of the “tool” frame relative to the base frame expressed in the “tool” frame (that

is, calculate and ).

4

44

4v




• Frame attachment




• DH Parameters

i 1i 1ia id i

1 0 0 0 1

2 90 L1 0 2

3 0 L2 0 3

4 0 L3 0 0




• From the DH parameter table, we can specify the homogeneous transform

matrix for each adjacent link pair:

1000

0

1111

1111

1

1

dccscss

dsscccs

asc

Tiiiiii

iiiiiii

iii

i

i

1000

0100

0011

0011

0

1

cs

sc

T

1000

0022

0100

1022

1

2cs

Lsc

T

1000

0100

0033

2033

2

3

cs

Lsc

T

1000

0100

0010

3001

3

4

L

T




• Compute the angular velocity of the end effector frame relative to the base

frame expressed at the end effector frame.

• For i=0

1

1

1

1 0

0

i

i

ii

ii

i R

111

0

01

01

1 0

0

0

0

0

0

0

100

011

011

0

0

cs

sc

R




• For i=1

• For i=2

• For i=3

• Note

2

1

1

212

1

12

12

2 2

2

0

0

0

0

010

202

202

0

0

c

s

cs

sc

R

32

1

1

32

1

1

3

2

23

23

3 23

23

0

0

2

2

100

033

033

0

0

c

s

c

s

cs

sc

R

32

1

1

32

1

1

3

34

34

4 23

23

0

0

0

23

23

100

010

001

0

0

0

c

s

c

s

R

4

4

3

3




• Compute the linear velocity of the end effector frame relative to the base frame

expressed at the end effector frame.

•

• Note that the term involving the prismatic joint has been dropped from the

equation (it is equal to zero).

1

1

1

1

1 0

0

i

i

i

i

iii

ii

i

d

vPRv

0




• For i=0

• For i=1

0

0

0

0

0

0

0

0

0

0

0

0

100

011

011

0

0

1

0

0

01

01

1 cs

sc

vPRv

111

1

1

2

1

1

12

12

2 0

0

0

0

0

0

0

1

0

0

010

202

202

L

L

cs

sc

vPRv




• For i=3

1

2

2

11

1

12

1

1

2

2

3

2

2

23

23

3

)221(

32

32

122

2

0

100

033

033

1

0

0

0

0

2

2

2

100

0303

033

cLL

cL

sL

LcL

Lcs

sc

L

L

c

s

cs

sc

vPRv




• For i=4

1

32

2

1

2

2

32

1

1

3

3

4

3

3

34

34

4

)233221(

3)332(

32

)221(

32

32

0

0

3

23

23

100

010

001

cLcLL

LLcL

sL

cLL

cL

sLL

c

s

vPRv




• Note that the linear and angular velocities ( ) of the end effector where

differentiate (measured) in frame {0} however represented (expressed) in frame

{4}

• In the car example: Observer sitting in the “Car”

Observer sitting in the “World”

Solve for and by multiply both side of the questions from the left by

i

k

i

k

i

k

i

k vRVRVv 0

0

0

0

i

k

i

k

i

k

i

k RR 0

0

0

0

4

4

04

4 vRv

4

4

04

4 R

4

4

4

4 , v

C

WCV

C

WWV

4v4

14

0

R




• Multiply both sides of the equation by the inverse transformation matrix, we

finally get the linear and angular velocities expressed and measured in the

stationary frame {0}

4

40

44

44

04

414

04 vRvRvRv T

4

40

44

44

04

414

04 RRR T

TTTTT 3

4

2

3

1

2

0

1

0

4




1

1

1 0

0




2

1

1

2

2 2

2

c

s




32

1

1

3

3 23

23

c

s




4

4

3

3




0

0

0

1

1v




11

2

2 0

0

L

v




1

2

2

3

3

)221(

32

32

cLL

cL

sL

v




1

32

2

4

4

)233221(

3)332(

32

cLcLL

LLcL

sL

v



velocity propagation between robot links...

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