Fundamentals__Robotics_Jacobian_part1.ppt

robotics lecture

1. Velocity Analysis
Jacobian
University of Bridgeport
1
Introduction to ROBOTICS

2. Kinematic relations

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














6
5
4
3
2
1






 



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


















z
y
x
X
Joint Space
Task Space
θ =IK(X)
Location of the tool can be specified using a joint space or a cartesian space
description
X=FK(θ)

3. Velocity relations
Joint Space
Task Space
•Relation between joint velocity and cartesian
velocity.
•JACOBIAN matrix J(θ)



















6
5
4
3
2
1











 



















z
y
x
z
y
x





 

)(JX XJ

)(
1




4. Jacobian
•Suppose a position and orientation vector of a
manipulator is a function of 6 joint variables: (from
forward kinematics)
X = h(q)























z
y
x
X 16
6
5
4
3
2
1
)(






















q
q
q
q
q
q
h 166216
6215
6214
6213
6212
6211
),,,(
),,,(
),,,(
),,,(
),,,(
),,,(






















qqqh
qqqh
qqqh
qqqh
qqqh
qqqh







5. Jacobian Matrix
Forward kinematics)(
116 
nqhX q
dq
qdh
dt
dq
dq
qdh
qh
dt
d
X
n

)()(
)(
116 
 



















z
y
x
z
y
x





 1
2
1
6
)(





















nn
n
q
q
q
dq
qdh



 1616 
nnqJX  dq
qdh
J
)(


6. Jacobian Matrix



















z
y
x
z
y
x





 1
2
1
6
)(





















nn
n
q
q
q
dq
qdh



 nn
n
n
n
q
h
q
h
q
h
q
h
q
h
q
h
q
h
q
h
q
h
dq
qdh
J


















































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!

7. Jacobian Matrix




























Vz
y
x
X
z
y
x






 16 
nnqJX  










z
y
x
V



Linear velocity



















z
y
x Angular velocity
The Jacbian Equation1
2
1
1















nn
n
q
q
q
q






8. Example
•2-DOF planar robot arm
–Given l
1, l
2 ,Find: Jacobian

2

1
(x , y)
l
2
l
1





















),(
),(
)sin(sin
)cos(cos
212
211
21211
21211




h
h
ll
ll
y
x 





























)cos()cos(cos
)sin()sin(sin
21221211
21221211
2
2
1
2
2
1
1
1




lll
lll
hh
hh
J 













2
1







J
y
x
Y

9. 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

10. •Find the singularity configuration of the 2-DOF
planar robot arm








)cos()cos(cos
)sin()sin(sin
21221211
21221211


lll
lll
J 













2
1







J
y
x
X

2

1
(x , y)
l
2
l
1
x
Y
=0
V
determinant(J)=0 Not full rank0
2
Det(J)=0

11. Jacobian Matrix
•Pseudoinverse
–Let A be an mxn matrix, and let be the pseudoinverse
of A. If A is of full rank, then can be computed as: 
A 
A 












nmAAA
nmA
nmAAA
A
TT
TT
1
1
1
][
][

12. Jacobian Matrix
–Example: Find X s.t.













 2
3
011
201
x 






























24
51
41
9
1
21
15
02
10
11
][
1
1TT
AAAA 











16
13
5
9
1
bAx
Matlab Command: pinv(A)to calculate A
+

13. Jacobian Matrix
•Inverse Jacobian
•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












666261
262221
161211
JJJ
JJJ
JJJ
qJX




 



















6
5
4
3
2
1
q
q
q
q
q
q





 XJq

1
 5
q
1
q

14. 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.

15. Jacobian Matrix
•If
•Then the cross product,
xx
yy
zz
ab
A a B b
ab
   
   

   
   
    ()
y z z y
x y z x z z x
x y z x y y x
i j k a b a b
A B a a a a b a b
b b b a b a b


    




16. Remember DH parmeter
•The transformation matrix Ti i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
i
A
     
     






 i
i
AAAT .....
210

17. Jacobian Matrix 
nJJJJ ....
21

18. Jacobian Matrix
2-DOF planar robot arm
Given l1, l2 ,Find: Jacobian
•Here, n=2,

2

1
(x , y)
l
2
l
1

19. 19
Where (θ
1 +θ
2 ) denoted by θ
12 andi i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
i
A
     
     






 01
0
0
1
ZZ





 1 1 1 1 2 1 2
0 1 1 1 2 1 1 2 1 2
0 cos cos cos( )
0 , sin , sin sin( )
0 0 0
a a a
O O a O a a
   
   
     
     
    
     
     
     

20. Jacobian Matrix
2-DOF planar robot arm
Given l1, l2 ,Find: Jacobian
•Here, n=20 2 0 1 2 1
12
0 1
() ()
,
z o o z o o
JJ
z z
  

 


2

1
(x , y)
l
2
l
1

21. Jacobian Matrix0 2 0
1
0
()z o o
J
z



 1 1 2 1 2
0 2 0 1 1 2 1 2
1 1 2 1 2 1 1 2 1 2
1 1 2 1 2
1 1 2 1 2
0 cos cos( )
( ) 0 sin sin( )
10
0 0 1
cos cos( ) sin sin( ) 0
sin sin( )
cos cos( )
0
aa
Z o o a a
i j k
a a a a
aa
aa
  
  
     
  
  
   
   
     
   
   
   




    

  

  




22. Jacobian Matrix1 2 1
2
1
()z o o
J
z



 2 1 2
1 2 1 2 1 2
2 1 2 2 1 2
2 1 2
2 1 2
0 cos( )
( ) 0 sin( )
10
0 0 1
cos( ) sin( ) 0
sin( )
cos( )
0
a
Z o o a
i j k
aa
a
a


   


   
   
    
   
   
   




 








23. Jacobian Matrix2 1 2
2 1 2
2
sin( )
cos( )
0
0
0
1
a
a
J











 1 1 2 1 2
1 1 2 1 2
1
sin sin( )
cos cos( )
0
0
0
1
aa
aa
J
  
  
  








  
12
J J J
The required Jacobian matrix J

24. Inverse Velocity
–The relation between the joint and end-effector velocities:
where j (m×n). If J is a square matrix (m=n), the joint
velocities:
–If m<n, let pseudoinverse J
+
where()X J q q 1
()q J q X

 1
[]
TT
J J JJ

 ()q J q X



25. Acceleration
–The relation between the joint and end-effector velocities:
–Differentiating this equation yields an expression for the
acceleration:
–Given of the end-effector acceleration, the joint
acceleration ()X J q q ( ) [ ( )]
d
X J q q J q q
dt
 X q ( ) [ ( )]
d
J q q X J q q
dt
 1
( )[ ( )] ]
d
q J q X J q q
dt

