Jacobian robotics

7/30/2019 Jacobian robotics

1/48

ME444: Module 3

Velocities, Jacobians and StaticForces

1


2/48

Differentiation of position vectors

ttQttQQ

dtdV

BB

t

BQB

)()(lim

0

We are calculating the derivative of Qrelative to frame B.

Derivative of a vector:

2


3/48

Differentiation of position vectors

Qdtd

VB

A

Q

BA

A velocity vector may be described interms of any frame:

.QBA

BQ

BA

VRV

We may write it:

VV CORGUC

Special case: Velocity of the origin of a frame relative tosome understood universe reference frame

3


4/48

Example

A fi xed universal f rame

30 mph

100 mph

.70)(

.100)100()(

.

30

1

1

XRRVRRVRV

XRXRRvvV

XvVPdt

d

U

T

U

CCORG

TU

T

C

UCORG

TC

TCORG

TC

U

C

C

UT

C

UT

C

TORG

UC

CCORG

U

CORG

U

U

Both vehiclesare heeding in Xdirection of U

4


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Angular velocity vector:

Linear velocity attribute of a point

Angular velocity attribute of a body

Since we always attach a frame

to a body we can considerangular velocity as describingrational motion of a frame.

5


6/48

Angular velocity vector:

describes the rotation of frame {B} relative to {A}

B

A

B

AMagni tude of

indicates speed of rotation

B

A

C

U

C

I n the case which there is an

understood reference frame:

direction of

indicates instantaneous axis

of rotation

6


7/48

Linear velocity of a rigid body

.QBA

BBORG

A

Q

A

VRVV

We wish to describe motion

of {B} relative to frame {A}

RA

BI f rotation

is not changing with time: 7

0RAB


8/48

Rotational velocity of a rigid body

Two frames with coincident origins

The orientation of B withrespect to A is changing

in time.

Lets consider that vectorQ is constant as viewed

from B.

B

A

QB

{A}{B}

0QBV

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Is perpendicularto and


QV

tQ

A

B

A

Q

A

B

AA

Q

)|)(|sin|(|||

B

A

Magnitude of differentialchange is:

QA

|| Q

Vector cross product

9


10/48


.QRVRVBA

BB

A

Q

BA

BQ

A

In general case: QVV ABA

Q

BA

Q

A )(

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11/48

.QRVRVV

BA

BB

A

Q

BA

BBORG

A

Q

A

Simultaneous linear and rotationalvelocity

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Motion of the Links of a Robot

At any instant, each l ink of a robot in motion has some linear and

angular velocity.

Written in frame i

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Velocity of a Link

Remember that linear velocity is

associated with a point and angular

velocity is associated with a body. Thus:

The velocity of a link means the linear

velocity of the origin of the link frameand the rotational velocity of the link

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14/48

Velocity Propagation From Link to Link

We can compute the velocities of each

link in order starting from the base.

The velocity of linki+1 will be that of linki, plus whatever new velocity component

added by joint i+1.

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15/48

Rotational Velocity

Rotational velocities may be added when

both w vectors are written with respect to

the same frame.Therefore the angular velocity of linki+1

is the same as that of linkiplus a new

component caused by rotational velocityat joint i+1.

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Velocity Vectors of Neighboring Links

.1

1

111

i

i

i

i

ii

i

i

i ZR 16


17/48

By premultiplying both sides of previousequation to:

.

.

1

1

1

1

1

1

1

1

11

11

1

1

i

i

ii

ii

ii

i

i

i

i

i

i

i

ii

ii

ii

ii

i

ZR

ZRRRR


1

1

1

1

1 0

0

i

i

i

i

i Z

Note that:

Ri i1

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Linear Velocity

The linear velocity of the origin of frame

{i+1} is the same as that of the origin of

frame {i} plus a new component causedby rotational velocity of linki.

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).(

).(

1

1

1

1

1

1

1

1

i

i

i

i

i

ii

ii

i

i

i

i

i

i

ii

ii

ii

i

PvRv

PvRvR

Linear Velocity

.QRVRVV BABBA

Q

BA

BBORG

A

Q

A

Simultaneous linear and rotationalvelocity:

.11 ii

i

i

i

i

i

i Pvv By premultiplying both sides of previousequation to: Ri i

1

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.)(,

1

1

11

1

1

1

1

1

1

i

i

ii

i

i

i

i

ii

ii

i

i

ii

ii

i

ZdPvRvR

For the case that joint i+1 is prismatic:

Prismatic Joints Link

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Applying those previous equations

successfully from link to link, we can

compute the rotational and linearvelocities of the last link.

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Example

A 2-li nk manipulator with rotational joints

Calculate the velocityof the tip of the armas a function of joint

rates?

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Example

Frame assignments for the two linkmanipulator

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.

1000

0100

0010001

,

1000

0100

000

,

1000

0100

0000 2

2

3

22

122

1

2

11

11

0

1

l

Tcslsc

Tcssc

T

Example

We compute link transformations:

. 11

1

1

1

1

i

i

ii

ii

ii

i ZR

.)(1

1

11

1

1

1

i

i

ii

i

i

i

i

ii

ii

i ZdPvRv 24


25/48

.0

0

)(,

,

00

0

100

0

0

,0

0

,

0

0

0

,0

0

2

1

2221

21

212121

121

3

3

2

2

3

3

121

121

1122

22

2

2

21

2

2

1

1

1

1

1

llcl

sllcl

sl

v

cl

sl

lcs

sc

v

v

Example

Link to link transformation

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26/48

.

0

)()(

.

100

0

0

2

1

12212211

12212211

21122111

21122111

3

30

33

0

1212

1212

2

3

1

2

0

1

0

3

clclcl

slslslclclslsl

vRv

cs

sc

RRRR

Example

Velocities with respect to nonmoving base

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Derivative of a Vector Function

If we have a vector function r whichrepresents a particles position as a

function of time t:

dt

dr

dt

dr

dt

dr

dt

d

rrr

zyx

zyx

r

r

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Vector Derivatives

Weve seen how to take a derivative ofa vector vs. A scalar

What about the derivative of a vectorvs. A vector?

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Jacobian

A Jacobian is a vector derivative with respectto another vector

If we have f(x), the Jacobian is a matrix ofpartial derivatives- one partial derivative foreach combination of components of thevectors

The Jacobian is usually written as j(f,x), butyou can really just think of it as df/dx

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30/48

Jacobian

N

MM

N

x

f

x

f

x

f

x

f

x

f

x

f

x

f

J

......

............

......

...

,

1

2

2

1

2

1

2

1

1

1

xf

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31/48

Partial Derivatives

The use of the symbol instead of d forpartial derivatives just implies that it is

a single component in a vectorderivative.

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33/48

.)(

.)(

0

0

0

0

Jv

XXJY

V

In the field of robotics, we generally speak ofJacobians which relatejoint velocities toCartesian velocities of the tip of the arm.

Jacobian

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Jacobian

For a 6 joint robot the Jacobian is 6x6, .is a 6x1 and v is 6x1.

The number of rows in Jacobian is equal

to number of degrees of freedom in

Cartesian space and the number of

columns is equal to the number of joints.

)(00 JV

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35/48

Jacobian

.

0

)(

)(

2

1

12212211

1221221121122111

21122111

330

330

clclclslslslclcl

slsl

vRv

In the earlier example we had:

12212211

122122110

)( clclcl

slslsl

J

Thus:

2221

2130

)(llcl

slJ

And also:

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36/48

Jacobian

Jacobian might be found by directly

differentiating the kinematic equations of

the mechanism for linear velocity,

however there is no 3x1 orientation vectorwhose derivative is rotational velocity.

Thus we get Jacobian using successive

application of:

1

1

1

1

1

1

i

i

ii

ii

ii

i ZR

)( 11

1

1

i

i

i

i

i

ii

ii

i PvRv

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Given a transformation relating joint velocity toCartesian velocity then

Is this matrix invertible? ( Is it non singular)

Singularities

gularitynonJ

ingularityJ

sin:0]det[

s:0]det[

v

J)(

1

)(00 JV

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Singularities

Singularities are categorized into two class:

Workspace boundary singularities:

Occur when the manipulator is fully starched or

folded back on itself.

Workspace interior singularities:

Are away from workspace boundary and are

caused by two or more joint axes lining up.

All manipulators have singularity at boundaries of theirworkspace. In a singular configuration one or more degree offreedom is lost. ( movement is impossible )

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Example

180,0

.00

|)(|)]([

2

221

2221

21

sll

llcl

slJJDET

Workspace boundary singulari ties

2221

2130

)(llcl

slJ

In the earlier example we had:

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Example

.

,

.1

)(

21

12

22

12

21

121

1221112211

122122

221

10

sl

c

sl

c

slc

slslclcl

slcl

sllJ

As the arm stretches out toward 2=0 both

joint rates go to infinity 40


41/48

Static Forces in Manipulators

i

i

n

f force exerted on link i by link i -1

torque exerted on link i by link i-1

Force and moments propagation

To solve for joint

torques in static

equilibrium

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Solve for the joint torques which must be acting

to keep the system in static equi l ibrium.


0111 ii

i

i

i

i

i

i fPnn

01 ii

i

i ff

Summing the force and

setting them equal to zero

Summing the torques aboutthe or igin of frame i

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43/48

.

,

.

,

11

1

1

1

1

1

111

1

i

i

i

i

i

ii

ii

i

i

ii

ii

i

i

i

i

i

i

i

i

i

i

i

i

i

fPnRn

fRf

fPnn

ff

Working down from last l ink to the base we

formulate the force moment expressions

Static force propagation

fr om link to link:


Important question:What torques

are needed at the joint to balance

reaction forces and moments acting

on the links?

.

.

i

iT

i

i

i

i

iT

i

i

i

Zf

Zn

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44/48

Jacobians in the Force Domain

XF

Work is the dot product of a vector force or torque and a

vector displacement

jX

I t can be written as:

.

.

.

00F

F

FF

T

T

TTTT

J

J

JJ

TT XF

The defini tion of jacobian is

So we have

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Cartesian Transformation ofVelocities and Static Forces

vV

General velocityof a body

General forceof a body

6x 6 transformations map these quantities from one frame

to another.

3 x1 linear velocity

3 x1 angular velocity

N

FF

3 x1 force vector

3 x1 moment vector

45


46/48

Since two frames are rigidly connected

Cartesian Transformation of Velocitiesand Static Forces

Where the cross product isthe matrix operator

01

i

. 11

11

11

i

iii

iiii

i ZR (5.45)

A

A

AA

B

A

BORGABABA

B

B

BB

w

v

R

PRR

w

v

0

0

00

xy

xz

yz

pp

pppp

P

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47/48

A

A

AA

B

A

BORGAB

ABA

B

B

BB v

R

PRRv

0

We use the termvelocity transformation

Description of velocity in terms of A when given the quantities in B


A

A

v

A

BB

BvTv

B

B

B

B

A

B

A

BBORG

AA

B

A

A

A

A v

R

RPRv

0

B

B

v

A

BA

A

vTv 47


48/48

A force-moment transformation

With similari ty to Jacobians


B

B

B

B

A

B

A

BBORG

A

A

B

A

A

A

A

N

F

RRP

R

N

F 0

B

B

f

A

BA

A FTF

T

v

A

Bf

A

B TT

Documents

Jacobian robotics