Introduction to ROBOTICS
Velocity Analysis
Jacobian
University of Bridgeport
1
Kinematic relations
 1 
 

 2
 3 
  
 4 
 
5
 
 6 
X=FK(θ)
Joint Space
θ =IK(X)
x 
 
y
 
z 
X   
 
 
 
 
Task Space
Location of the tool can be specified using a joint space or a cartesian space
description
Velocity relations
• Relation between joint velocity and cartesian
velocity.
• JACOBIAN matrix J(θ)
 1  
X  J ( )
 
 2 
 
3
 

 4 
 
1
 5    J ( ) X
6 
Joint Space
 x 
 
y
 
 z 
 
 x 
 
y
 
 z 
Task Space
Jacobian
• Suppose a position and orientation vector of a
manipulator is a function of 6 joint variables: (from
forward kinematics)
X = h(q)
 q1 
x 
 
 
q
y
 2
 
q3 
z 
X     h (   ) 6 1
q4 
 
q 
 
5


 
q

 6 
 
 h1 ( q 1 , q 2 ,  , q 6 ) 


h 2 ( q1 , q 2 ,  , q 6 )


 h3 ( q1 , q 2 ,  , q 6 ) 


h
(
q
,
q
,

,
q
)
4
1
2
6


 h (q , q , , q ) 
5
1
2
6


 h 6 ( q 1 , q 2 ,  , q 6 )  6 1
Jacobian Matrix
Forward kinematics
X 6 1  h ( q n 1 )
d
dh ( q ) dq
dh ( q )

X 6 1 
h ( q n 1 ) 

q
dt
dq dt
dq
 x 
 
y
 
 z 
 
 x 
 
y
 
 z 
 q 1 
 
 dh ( q )   q 2 

  
dq

 6 n 
 
 q n  n 1
X 6 1  J 6  n q n 1
J 
dh ( q )
dq
Jacobian Matrix
 x 
 
y
 q 1 
 
 z   dh ( q )   q 2 
 
  
  
 x   dq  6  n 
 
 
y
 q n  n 1
 
 z 
 dh ( q ) 

J  
 dq  6  n
Jacobian is a function of
q, it is not a constant!
  h1
 q
 1
  h2
  q
1
 
 h
 6
  q 1
 h1
q 2
 h2
q 2

 h6
q 2




 h1 
q n 

 h2 
q n 
 
 h6 

 q n  6  n
Jacobian Matrix
 x 
 
y
 
 z   V 

X     
 x    
 
y
 
  z 
Linear velocity
 x 
 
V  y
 
 z 
The Jacbian Equation
X  J 6  n q n 1
Angular velocity
  x   



   y   
   
 z

q n 1
 q 1 
 
q 2



  
 
 q n  n 1
Example
• 2-DOF planar robot arm
– Given l1, l2 , Find: Jacobian
(x , y)
2
 x   l1 cos  1  l 2 cos(  1   2 )   h1 ( 1 ,  2 ) 
 

  
l
sin


l
sin(



)
h
(

,

)
y
   1
1
2
1
2 
 2 1 2 
l2
 1 
 x 

Y     J 

 y 
 2 
1 l1
  h1
 
J  1
  h2
   1
 l 2 sin(  1   2 ) 

l 2 cos(  1   2 ) 
 h1 
  2    l1 sin  1  l 2 sin(  1   2 )
 
 h2 
 l1 cos  1  l 2 cos(  1   2 )
  2 
Singularities
• The inverse of the jacobian matrix cannot be
calculated when
det [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
determinant(J)=0
Not full rank
 1 
x 

X     J  

 y 
 2 
  l1 sin  1  l 2 sin(  1   2 )
J  
 l1 cos  1  l 2 cos(  1   2 )
V
 l 2 sin(  1   2 ) 

l 2 cos(  1   2 ) 
l2
Y
2 =0
Det(J)=0
2  0
1 l1
x
(x , y)
Jacobian Matrix
• Pseudoinverse
– Let A be an mxn matrix, and let A  be the pseudoinverse
of A. If A is of full rank, then A  can be computed as:
 A T [ AA T ]  1
 1

A  A
[ A T A ] 1 A T

mn
m n
mn
Jacobian Matrix
– Example: Find X s.t.
1

1

0
1
2
 3 
x  

0

2


A  A [ AA ]
T
T
1
1

 0

 2
1 
 5
1 
 1

0 
1

2
1
1
1

1

9
 4
Matlab Command: pinv(A) to calculate A+
 5
1


x A b
13

9
 16 
4 

5

 2 
Jacobian Matrix
• Inverse Jacobian
 J 11

J 21


X  J q 
 

 J 61
q  J
1
X
J 12

J 22



J 62

J 16 

J 26

 

J 66 
 q 1 
 
q
 2
 q 3 
 
 q 4 
 q 
5
 
 q 6 
q5
q1
• Singularity
– rank(J)<n : Jacobian Matrix is less than full rank
– 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
Singularity
• At Singularities:
– the manipulator end effector cant move in
certain directions.
– Bounded End-Effector velocities may
correspond to unbounded joint velocities.
– Bounded joint torques may correspond to
unbounded End-Effector forces and torques.
Jacobian Matrix
• If
ax 
 bx 
 
 
A  a y , B  by
 
 
 a z 
 b z 
• Then the cross product
i
j
A  B  ax
ay
bx
by
 a y bz  a z b y 


a z    ( a x bz  a z bx ) 
 a b a b 
bz
y x
 x y

k
Remember DH parmeter
 c i

s i

Ai 
 0

 0
-c  i s  i
s is i
c ic i
-s  i c  i
s i
c i
0
0
a ic i 

a is i

di 

1 
• The transformation matrix T
T 0  A1 A 2 ..... Ai
i
Jacobian Matrix
J  J 1
J2
....
Jn
Jacobian Matrix
2-DOF planar robot arm
Given l1, l2 , Find: Jacobian
• Here, n=2,
(x , y)
2
l2
1 l1
 c i

s i
Ai  
 0

 0
-c  i s  i
s is i
c ic i
-s  i c  i
s i
c i
0
0
a ic i 

a is i

di 

1 
Where (θ1 + θ2 ) denoted by θ12 and
0 
 
Z 0  Z1  0
 
 1 
0 
 a1 cos  1 
 a1 cos  1  a 2 cos( 1   2 ) 
 




O 0  0 , O1  a1 sin  1 , O 2  a1 sin  1  a 2 sin( 1   2 )
 




 0 




0
0
19
Jacobian Matrix
2-DOF planar robot arm
Given l1, l2 , Find: Jacobian
• Here, n=2
 z 0  (o2  o0 ) 
 z1  ( o 2  o1 ) 
J1  
 , J2  

z
z
0

1



(x , y)
2
l2
1 l1
Jacobian Matrix
 z0  (o2  o0 ) 
J1  

z
0


 0   a1 cos  1  a 2 cos( 1   2 ) 
  

Z 0  ( o 2  o 0 )  0  a1 sin  1  a 2 sin( 1   2 )
  

 1  

0
i



0

 a1 cos  1  a 2 cos( 1   2 )
  a1 sin  1  a 2 sin( 1   2 ) 


 a1 cos  1  a 2 cos( 1   2 )




0
j
0
a1 sin  1  a 2 sin( 1   2 )
k

1

0 
Jacobian Matrix
 z 1  ( o 2  o1 ) 
J2  

z

1

 0   a 2 co s( 1   2 ) 
  

Z 1  ( o 2  o1 )  0  a 2 sin ( 1   2 )
  

 1  

0
i



0

 a 2 co s( 1   2 )
  a 2 sin ( 1   2 ) 


 a 2 co s( 1   2 )




0
j
0
a 2 sin ( 1   2 )
k

1

0 
Jacobian Matrix
  a1 sin  1  a 2 sin( 1   2 ) 


a1 cos  1  a 2 cos( 1   2 )




0
J1  

0




0


1


  a 2 sin ( 1   2 ) 


a 2 co s( 1   2 )




0
J2  

0




0


1


The required Jacobian matrix J
J 
 J1
J2

Stanford Manipulator
The DH parameters are:
 c i

s i
Ai  
 0

 0
-c  i s  i
s is i
c ic i
-s  i c  i
s i
c i
0
0
a ic i 

a is i

di 

1 
Stanford Manipulator
T 0  A1
1
2
T0
 c1 c 2

s1 c 2
 A1 A2  
  s2

 0
 c1 c 2

s1 c 2
3

T 0  A1 A2 A3 
  s2

 0
 s1
c1 s 2
c1
s1 s 2
0
c2
0
0
 s1
c1 s 2
c1
s1 s 2
0
c2
0
0
 d 2 s1 

d 2 c1

0 

1 
0 
 c1 s 2 
  s1 
 c1 s 2 
 


z 0  0 z   c  z   s s  z 3  s1 s 2


  1  1  2  1 2
 c 2 
 1 
 0 
 c 2 
  d 2 s1 
 d 3 c1 s 2  d 2 s1 
0 






O

d
c
O

d
s
s

d
c
d 3 c1 s 2  d 2 s1  O 0  O1  0 2  2 1  3  3 1 2
2 1
 

 0 


d 3c2
d 3 s1 s 2  d 2 c 1
 0 


d 3c2

1

Stanford Manipulator
T4 =
T0  A1 A2 A3 A4
4
T0  A1 A2 A3 A4 A5
5
T0  A1 A2 A3 A4 A5 A6
6
[ c1c2c4-s1s4,
-c1s2, -c1c2s4-s1*c4, c1s2d3-sin1d2]
[ s1c2c4+c1s4, -s1s2, -s1c2s4+c1c4, s1s2d3+c1*d2]
[-s2c4, -c2, s2s4, c2*d3]
[ 0, 0, 0, 1]
  c1 c 2 s 4  s 1 c 4 


z 4   s1 c 2 s 4  c 1 c 4




s2 s4
 d 3 c1 s 2  d 2 s 1 


O 4  d 3 s1 s 2  d 2 c1




d 3c2
Stanford Manipulator
T5 =
[ (c1c2c4-s1s4)c5-c1s2s5, c1c2s4+s1c4, (c1c2c4-s1s4)s5+c1s2c5,
c1s2d3-s1d2]
[ (s1c2c4+c1s4)c5-s1s2s5, s1c2s4-c1c4, (s1c2c4+c1s4)s5+s1s2c5,
s1s2d3+c1d2]
[ -s2c4c5-c2s5, -s2s4, -s2c4s5+c2c5, c2d3]
[ 0, 0, 0, 1]
 c1 c 2 c 4 s 5  s1 s 4 s 5  c1 s 2 c 5 


z 5  s1 c 2 c 4 s 5  c1 s 4 s 5  s1 s 2 c 5




 s 2 c 4 s5  c 2 c5
 c1 c 2 c 4 s 5  s1 s 4 s 5  c1 s 2 c 5 


z 5  s1 c 2 c 4 s 5  c1 s 4 s 5  s1 s 2 c 5




 s 2 c 4 s5  c 2 c5
Stanford Manipulator
T5 =
[ (c1c2c4-s1s4)c5-c1s2s5, c1c2s4+s1c4, (c1c2c4-s1s4)s5+c1s2c5,
c1s2d3-s1d2]
[ (s1c2c4+c1s4)c5-s1s2s5, s1c2s4-c1c4, (s1c2c4+c1s4)s5+s1s2c5,
s1s2d3+c1d2]
[ -s2c4c5-c2s5, -s2s4, -s2c4s5+c2c5, c2d3]
[ 0, 0, 0, 1]
 c1 c 2 c 4 s 5  s1 s 4 s 5  c1 s 2 c 5 


z 5  s1 c 2 c 4 s 5  c1 s 4 s 5  s1 s 2 c 5




 s 2 c 4 s5  c 2 c5
 d 3 c1 s 2  d 2 s1 


O 5  d 3 s1 s 2  d 2 c1




d 3c2
Stanford Manipulator
T6 = [ c6c5c1c2c4-c6c5s1s4-c6c1s2s5+s6c1c2s4+s6s1c4, c5c1c2c4+s6c5s1s4+s6c1s2s5+c6c1c2s4+c6s1c4, s5c1c2c4-s5s1s4+c1s2c5,
d6s5c1c2c4-d6s5s1s4+d6c1s2c5+c1s2d3-s1d2]
[ c6c5s1c2c4+c6c5c1s4-c6s1s2s5+s6s1c2s4-s6c1c4, -s6c5s1c2c4s6c5c1s4+s6s1s2s5+c6s1c2s4-c6c1c4, s5s1c2c4+s5c1s4+s1s2c5,
d6s5s1c2c4+d6s5c1s4+d6s1s2c5+s1s2d3+c1d2]
[ -c6s2c4c5-c6c2s5-s2s4s6, s6s2c4c5+s6c2s5-s2s4c6, -s2c4s5+c2c5, d6s2c4s5+d6c2c5+c2d3]
[ 0, 0, 0, 1]
 d6s5c1c2c4  d6s5s1s4  d6c1s2c5  c1s2d3  s1d2 


O 6  d6s5s1c2c4  d6s5c1s4  d6s1s2c5  s1s2d3  c1d2




 d6s2c4s5  d6c2c5  c2d3
Stanford Manipulator
 z 0  (o6  o0 ) 
 z1  ( o 6  o1 ) 
J1  
 , J2  

z
z
0

1



 z2 
J3   
0
Joints 1,2 are revolute
Joint 3 is prismatic
 z 3  ( o6  o3 ) 
 z 5  (o6  o5 ) 
 z 4  (o6  o4 ) 
J4  
 , J5  

 , J6  
z
z
z
3
5

4





The required Jacobian matrix J
J 
 J1
J2J3J4J5J6

Inverse Velocity
– The relation between the joint and end-effector velocities:
X  J (q )q
where j (m×n). If J is a square matrix (m=n), the joint
velocities:
q J
1
(q ) X
– If m<n, let pseudoinverse J+ where J   J T [ JJ T ]  1

q  J (q ) X
Acceleration
– The relation between the joint and end-effector velocities:
X  J (q )q
– Differentiating this equation yields an expression for the
acceleration:
X  J (q )q  [
d
J ( q )]q
dt
– Given X of the end-effector acceleration, the joint
acceleration q
J (q )q  X  [
q J
1
( q )[ X 
d
dt
d
dt
J ( q )] q
J ( q )] q ]