VELOCITY ANALYSISJACOBIAN

University of Bridgeport

1

Introduction to ROBOTICS

KINEMATIC RELATIONS

6

5

4

3

2

1

zyx

X

Joint SpaceTask Space

θ =IK(X)

Location of the tool can be specified using a joint space or a cartesian space description

X=FK(θ)

VELOCITY RELATIONS Relation between joint velocity and cartesian

velocity. JACOBIAN matrix J(θ)

Joint Space Task Space

6

5

4

3

2

1

z

y

x

zyx

)(JX

XJ )(1

JACOBIAN Suppose a position and orientation vector of a

manipulator is a function of 6 joint variables: (from forward kinematics)

X = h(q)

zyx

X 16

6

5

4

3

2

1

)(

qqqqqq

h

166216

6215

6214

6213

6212

6211

),,,(),,,(),,,(),,,(),,,(),,,(

qqqhqqqhqqqhqqqhqqqhqqqh

JACOBIAN MATRIXForward kinematics

)( 116 nqhX

qdqqdh

dtdq

dqqdhqh

dtdX n )()()( 116

z

y

x

zyx

1

2

1

6

)(

nn

n

q

qq

dqqdh

1616 nnqJX

dqqdhJ )(

JACOBIAN MATRIX

z

y

x

zyx

1

2

1

6

)(

nn

n

q

qq

dqqdh

nn

n

n

n

qh

qh

qh

qh

qh

qh

qh

qh

qh

dqqdhJ

6

6

2

6

1

6

2

2

2

1

2

1

2

1

1

1

6

)(

Jacobian is a function of q, it is not a constant!

JACOBIAN MATRIX

1

2

1

1

nn

n

q

qq

q

Vz

yx

X

z

y

x

16 nnqJX

zyx

V

Linear velocity

z

y

x

Angular velocity

The Jacbian Equation

EXAMPLE 2-DOF planar robot arm

Given l1, l2 , Find: Jacobian2

1

(x , y)

l2

l1

),(),(

)sin(sin)cos(cos

212

211

21211

21211

hh

llll

yx

)cos()cos(cos)sin()sin(sin

21221211

21221211

2

2

1

2

2

1

1

1

llllll

hh

hh

J

2

1

Jyx

Y

SINGULARITIESThe inverse of the jacobian matrix cannot

be calculated whendet [J(θ)] = 0

Singular points are such values of θ that cause the determinant of the Jacobian to be zero

Find the singularity configuration of the 2-DOF planar robot arm

)cos()cos(cos)sin()sin(sin

21221211

21221211

llllll

J

2

1

Jyx

X

2

1

(x , y)

l2

l1

x

Y

=0

V

determinant(J)=0

02 Det(J)=0

JACOBIAN MATRIXPseudoinverse

Let A be an mxn matrix, and let be the pseudoinverse of A. If A is of full rank, then can be computed as:

AA

nmAAAnmAnmAAA

ATT

TT

1

1

1

][

][

JACOBIAN MATRIXExample: Find X s.t.

2

3011201x

2451

41

91

2115

0210

11][

11TT AAAA

1613

5

91bAx

Matlab Command: pinv(A) to calculate A+

JACOBIAN MATRIX Inverse Jacobian

Singularity

Jacobian is non-invertable Boundary Singularities: occur when the tool tip is on the

surface of the work envelop. Interior Singularities: occur inside the work envelope when two

or more of the axes of the robot form a straight line, i.e., collinear

666261

262221

161211

JJJ

JJJJJJ

qJX

6

5

4

3

2

1

qqqqqq

XJq 1

5q

1q

JACOBIAN MATRIX

• If

• Then the cross product

,x x

y y

z z

a bA a B b

a b

( )y z z y

x y z x z z x

x y z x y y x

i j k a b a bA B a a a a b a b

b b b a b a b

REMEMBER DH PARMETER

• The transformation matrix T

i i i i i i i

i i i i i i i

i i i

c -c s s s a cs c c -s c a s0 s c d0 0 0 1

iA

ii AAAT .....210

JACOBIAN MATRIX nJJJJ ....21