kadoms10

KADOMS KRiM, WIMIR, AGH Kraków 1

Katedra Robotyki i Mechatroniki

Akademia Górniczo-Hutnicza w Krakowie

Wojciech Lisowski

10

Manipulator Jacobian Matrix

Kinematics and Dynamics of Mechatronic Systems


Problems:

velocity and acceleration of a manipulator’s links

manipulator Jacobian matrix – definition and

interpretation

algorithm of algebraic determination of 0Je

algorithm of algebraic determination of eJe

pose singularity: definition, classification,

interpretation, necessary condition

transformation of the force vector from end-

effector to joints

Modelling of Robots

and Manipulators

KRiM, WIMIR, AGH Kraków 3

Determination of velocities and accelerations basing on homogeneous

transformation matrices.

Example of a 3 DOM manipulator

Vector of position of the 3rd link tip

Tppp zyxp 13

3

3

333

0 pTp

Velocity vector of the arm tip:

3

3

3332321313

3

3

3

32

2

31

1

33

333

0

3 pqUqUqUpqq

Tq

q

Tq

q

TpT

dt

dp

dt

dvp

Velocity vector of a j-th point of a j-th link (expressed by jpj vector with

respect to the j-th local frame)

j

jj

k

kjkpj pqUv

1

Modelling of Robots

and Manipulators


Example of a 3 DOM manipulator.

Vector of acceleration of the arm tip:

Vector of acceleration of a j-th link (expressed by jpj vector with respect

to the j-th local frame)

j

jj

k

j

m

j

k

kjkmkjkmpj pqUqqUv

1 1 1

3

3

33333333233213312322332323221321

13113313231213113

3

3332321313

pqUqqUqUqUqUqqUqUqU

qUqqUqUqUpqUqUqUdt

dvp

Modelling of Robots

and Manipulators


Cartesian angular speed

TzyxP

Position vector: Velocity vector:

Tzyxzyx vvvV

dt

Pdv

dt

dxvx

dt

dyv y

dt

dzvz

fdt

d?

dt

d?

dt

d?

dt

d

Modelling of Robots

and Manipulators


o

z

y

x

S

S

CCS

SCC

10

0

0

z

y

x

o

z

y

x

S

CSSSC

CCCS

SC

C

1

0

01

PS

vo

*

o

oS

IS

0

0*


Definition of the manipulator jacobian matrix:

PS

vo

*

qqf

Φ

Pq

qJqqfSΦ

PS

vqoo

**

qfSJ qo *

Differentiation of the trigonometric expressions present in the

geometrical model is inconvenient


qJv

ez

l

z

l

z

l

ey

l

y

l

y

l

ex

l

x

l

x

l

ez

l

z

l

z

l

ey

l

y

l

y

l

ex

l

x

l

x

l

m

l ddd

ddd

ddd

J

21

21

21

21

21

21Manipulator Jacobian Matrix

l – components of the velocity vectors are projected on axes of

coordinate frame No. l

m – Cartesian velocity vectors are determined for the origin of the

coordinate frame No. m


In practice there are two the most often used kinds of the manipulator

jacobian matrix:

0Je – velocity vectors are projected on axes of the reference frame (0).

This matrix allows to determine velocity of the end-effector (e) basing on

the joint velocities

eJe – velocity vectors are projected on axes of the reference frame (0)

with orientation changed to that corresponding to the actual (instant)

orientation of the local frame assigned to the end-effector (e).

This matrix is used to determine the joint velocities corresponding to the

end-effector motion with some velocity in some direction


Elements of the manipulator jacobian matrix depend on the joint

coordinates qi in a nonlinear way (trigonometric functions) and they take

different values in different parts of the workspace

rows of the manipulator jacobian matrix are coefficients of linear

combination of the joint velocities contributing to a particular component

of the Cartesian velocity vector (of some link or of the end-effector)

columns of the manipulator jacobian matrix are the components of the

translational as well as the rotational velocity vector corresponding to the

the motion of a single link with the unit velocity.

e

ez

l

z

l

z

l

ey

l

y

l

y

l

ex

l

x

l

x

l

ez

l

z

l

z

l

ey

l

y

l

y

l

ex

l

x

l

x

l

z

y

x

z

y

x

q

q

q

ddd

ddd

ddd

v

v

v

2

1

21

21

21

21

21

21


Determination of the eJe manipulator jacobian matrix with use of

algebraic operations basing on the differential kinematic relationships

Variation of an end-effector pose is a superposition of changes of a

relative position of each pair of adjacent links that take place in

consequent joints.

In case the motion in the i -th joint

e

ii

e

ieTDTD

1

ei

i

ei

eAADAAD 1

1

1

Tz

i

y

i

x

i

z

i

y

i

x

ii

dddD ,,,,,

The equivalent variation of the end-effector pose can be expressed as:


Rotation around zi axis by i+1 angle: Ti

d 000 Ti 100

0

100

ˆˆˆ

x

y

zyx

i

p

p

ppp

kji

p

a

o

n

i

z

l

i

y

l

i

x

l

z

T

z

e

z

T

y

e

z

T

x

e

xyyx

T

xyz

e

xyyx

T

xyy

e

xyyx

T

xyx

e

aa

oo

nn

papappad

popoppod

pnpnppnd

100

100

100

0

0

0

j

n p n p

o p o p

a p a p

n

o

a

i

x y y x

x y y x

x y y x

z

z

z

1

dpad

dpod

dpnd

ii

z

l

ii

y

l

ii

x

l


a

o

n

i

z

l

i

y

l

i

x

l

Ti

d 100 Ti 000

Translation along zi axis by di+1:

0000

0000

0000

100

100

100

T

z

e

T

y

e

T

x

e

z

T

z

e

z

T

y

e

z

T

x

e

a

o

n

aad

ood

nnd

j n o ai z z z

T

1 0 0 0, , , , ,

dpad

dpod

dpnd

ii

z

l

ii

y

l

ii

x

l


a

o

n

i

z

l

i

y

l

i

x

l

Ti

d 001 Ti 000

Translation along xi axis by ai+1:

0000

0000

0000

001

001

001

T

z

e

T

y

e

T

x

e

x

T

z

e

x

T

y

e

x

T

x

e

a

o

n

aad

ood

nnd

j n o ai x x x

T

1 0 0 0, , , , ,

dpad

dpod

dpnd

ii

z

l

ii

y

l

ii

x

l


Rotation around xi axis by i+1 angle: Ti

d 000 Ti 001

y

z

zyx

i

p

p

ppp

kji

p

0

001

ˆˆˆ

a

o

n

i

z

l

i

y

l

i

x

l

x

T

z

e

x

T

y

e

x

T

x

e

yzzy

T

yzz

e

yzzy

T

yzy

e

yzzy

T

yzx

e

aa

oo

nn

papappad

popoppod

pnpnppnd

001

001

001

0

0

0

j

n p n p

o p o p

a p a p

n

o

a

i

y z z y

y z z y

y z z y

x

x

x

1

dpad

dpod

dpnd

ii

z

l

ii

y

l

ii

x

l


Link No.

d

a

Motion range

1

1 v

0

a1

0

-120o120o

2

2 v

0

a2

0o150o

3

0

d3 v

0

0

0.1 m 0.3 m

4

4 v

0

0

0

-180o180o

x 0

y0

z0

x 1

y 1

z 1

x 2 y 2

z 2

x 3 y 3

z 3

x 4 y 4

z 4

An example –SCARA manipulator of RRPR kinematic structure


The first rotation 1:

j

n p n p

o p o p

a p a p

n

o

a

i

x y y x

x y y x

x y y x

z

z

z

1

0

4

1 2 4 1 2 4 1 1 2 12

1 2 4 1 2 4 1 1 2 12

3

0

0

0 0 1

0 0 0 1

T

a C a C

a S a S

d

cos( ) sin( )

sin( ) cos( )

j

a S a S a C a C

a S a S a C a C

1

1 2 4 1 1 2 12 1 2 4 1 1 2 12

1 2 4 1 1 2 12 1 2 4 1 1 2 12

0

0

0

1

cos( ) sin( )

sin( ) cos( )


1

0

0

0

cos

)sin(

1

0

0

0

)cos()cos(

)sin()sin(

42421

42421

21421214211

21421214211

1

Caa

Saa

aa

aa

j


The second rotation 2:

j

n p n p

o p o p

a p a p

n

o

a

i

x y y x

x y y x

x y y x

z

z

z

1

1

4

2 4 2 4 2 2

2 4 2 4 2 2

3

0

0

0 0 1

0 0 0 1

T

a C

a S

d

cos( ) sin( )

sin( ) cos( )

1

0

0

0

1

0

0

0

cos

sin

1

0

0

0

cossin

sincos

42

42

2422

2422

2422422

2422422

2

Ca

Sa

a

a

CSa

CSa

j


2

4

4 4

4 4

3

0 0

0 0

0 0 1

0 0 0 1

T

C S

S C

d

The third motion – translation by d3 along z2 axis

Tj 0001003

j n o ai z z z

T

1 0 0 0, , , , ,

The fourth motion – rotation by 4 around z3 axis

3

4 4

4 4

4 4

0 0

0 0

0 0 1 0

0 0 0 1

T A

C S

S C

Tj 1000004

j

n p n p

o p o p

a p a p

n

o

a

i

x y y x

x y y x

x y y x

z

z

z

1


SCARA manipulator jacobian matrix

4

4

1 2 4 2 4 2 4

1 2 4 2 4 2 4

0 0

0 0

0 0 1 0

0 0 0 0

0 0 0 0

1 1 0 1

J

a a S a S

a a C a C

sin( )

cos( )

x 0

y 0 z 0

x 1

y 1 z 1

x 2 y 2

z 2

x 3 y 3

z 3

x 4 y 4

z 4


Formulas for determination of columns of the manipulator

jacobian matrix 0Je

Vectors of differential displacement (velocity) are projected on the axes

parallel to the reference frame (0) axes.

Translation along zi axis by di+1:

i

z

y

x

i

a

a

a

j

0

0

0

0

1

0

Translation along xi axis by ai+1:

i

z

y

x

i

n

n

n

j

0

0

0

0

1

0


Rotation around xi axis by

i+1 angle:

Rotation around zi axis by i+1

angle:

z

y

x

x

i

zy

i

z

x

i

yy

i

y

x

i

xy

i

x

i

a

a

a

popn

popn

popn

j 1

0

z

y

x

y

i

zz

i

z

y

i

yz

i

y

y

i

xz

i

x

i

n

n

n

papo

papo

papo

j 1

0

𝑇𝑖0 𝑇𝑒

𝑖 n, o, a p


Example manipulator

of PPPRR kinematic structure

x0

y0

z0

x1

z1

y1

x2

z2

y2

x3

z3

y3

x4

z4

y4 x5

z5

y5

Link

No.

d

a

1

0

0

a1 v

/2

2

0

d2 v

0

/2

3

0

d3 v

0

-/2

4

4 v

0

0

/2

5

5 v

d5

0

0


The first translation by a1

i

z

y

x

i

n

n

n

j

0

0

0

0

1

0

IT 0

0 Tj 0000011

0

The second translation by d2

i

z

y

x

i

a

a

a

j

0

0

0

0

1

0

1000

0010

0100

001 1

11

a

AT Tj 0000102

0


The third translation by d3

i

z

y

x

i

a

a

a

j

0

0

0

0

1

0

1000

0100

010

001

2

1

212

d

a

AAT

Tj 0001003

0


The fourth rotation by 4

z

y

x

x

i

zy

i

z

x

i

yy

i

y

x

i

xy

i

x

i

a

a

a

popn

popn

popn

j 1

0

1000

010

100

001

3

2

1

3213d

d

a

AAAT

1000

0055

5445454

5445454

545

3

CS

dCCSSCS

dSSSCCC

AAT

0

1

0

0

0

1

0

1

0

0

0

0

1

54

54

5454

4

0 dS

dC

dSdC

j


The fifth rotation by 5

z

y

x

x

i

zy

i

z

x

i

yy

i

y

x

i

xy

i

x

i

a

a

a

popn

popn

popn

j 1

0

1000

0

010

0

344

2

144

43214dCS

d

aSC

AAAAT

1000

100

00

00

5

55

55

55

4

d

CS

SC

AT

4

4

4

4

4

4

5

0

0

0

0

0

0

0

0

1

0

00

C

S

C

S

S

C

j


4

4

54

54

5

0

0000

01000

0000

0100

00010

0001

C

S

dS

dC

J

x0

y0

z0

x1

z1

y1

x2

z2

y2

x3

z3

y3

x4

z4

y4 x5

z5

y5

PPPRR manipulator jacobian matrix


Invertion of the manipulator Jacobian matrix

DOM=DOF=6

Inversion of the square manipulator Jacobian matrix is possible, except

for the local singular poses.

qJv

D e

l

le

lele

le

le

e

lle

e

l vJDJq

11

DOM>DOF=1,2,...

Redundant manipulators

The pseudoinverse matrix is determined (the best – optimal solution

according to the criterion of minimum least squares error):

DJJJDJql

T

e

l

e

lT

e

ll

I

e

l 1

Modelling of Robots

and Manipulators


DOM=DOF<6

In case of non-redundant manipulators:

- formulate the equations of kinematic constraints

- find the conditions of kinematic constraint by solving the constraint

equations

- remove the dependent rows from the Jacobian matrix

- invert the square matrix

The questions arise – which components of the velocity vectors are

dependent? and how do they depend on each other?

The conditions of kinematic constraints are the algebraic

relationships between components of the velocity vectors v and .

MMiR Analiza osobliwości 32

RRP manipulator

x 0

y 0 z 0

x 1

y 1 z 1

x 2 y 2

z 2

x 3 y 3

z 3

011

000

000

100

0

00

221

21

3

3

aCa

Sa

J

Kinematic constraint

equation

yxz vvf333

,

2 zero rows

22

2

3

2

3

3

Sa

Cv

a

vxy

z

100

01

001

2221

221

21

1

3

3

aSaa

aCa

Sa

J

Inverse jacobian

matrix of the

manipulator:


Singularity of an end-efector pose:

Necessary condition:

determinant of a manipulator Jacobian matrix is equal to zero

Properties of a singular pose:

loss of mobility – the end-effector cannot be moved in some direction

loss of controllability – the end-effector velocity in this direction is

zero for any values of the joint variables

solution of the inverse kinematic problem is indefinite – infinite

number of solutions.

Types of pose singularities:

inside the workspace – more difficult to be determined, makes

the tracking difficult

on the border of the workspace – easier to be determined, rarely

attained


Condition of zero value of the Jacobian matrix:

543 dSa Corresponds to the singular pose:

4

d5

a3

1000

000

000

000

00

00

4

554

554

44

555454

555454

5435

5

C

CSS

SCS

SC

dSSCSS

dCCCCS

dSaJ

Rows No. 1 and 2 are mutually dependent, what

results in the translational end-effector velocity

component parallel to z3 axis equal to zero for any

values of the joint velocities.

For any value of 1 joint angle within the motion range the end-effector

possesses the same position – the one on the axis of the first motion.


Examples of RPY wrist singularity

x4

y4 z4

x3

y3

z3

x7

y7

z7

x4

y4 z4

x5

y5

z5

x6

y6

z6

Values of 4 and 6 angles cannot be determined unambiguously.

Loss of mobility in this case consists in no possibility of rotation of

the end-effector around the x4 axis


In the static equilibrium:

0 QFDFWTqeTe

QFQJFTq

e

eTe

Transformation of the force vector from an end-effector to joints:

Vector of driving forces: Tn

q

FFFF 21

Vector of joint virtual displacements: TnqqqQ 21

For a eF force vector sometimes there does not exist the corresponding qF vector. There are directions in which the end-effector cannot be

moved - the translational/angular velocities are zero.

The manipulator joint forces/torques cannot counterbalance the external

forces/torques acting on the end-effector in these directions. These

forces/torques are balanced by the manipulator structural elements (joints

and links). Correspondingly the end-effector cannot exert forces/torques

in these directions.

FJFeT

e

eq

kadoms10

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