Fundamentals__Robotics_Jacobian_part2.ppt

Jacobian lecture

1. Velocity Analysis
Jacobian
Stanford Manipulator
University of Bridgeport
1
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)
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












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)
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

















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
•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


    




6. 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

7. Stanford Manipulator i 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
     
     







The DH parameters are:

8. Stanford Manipulator 1
01
TA 1 2 1 1 2 2 1
1 2 1 1 2 2 12
0 1 2
22
00
0 0 0 1
c c s c s d s
s c c s s d c
T A A
sc





 0
0
0
1
z





 1
11
0
s
zc





 12
2 1 2
2
cs
z s s
c





 1 2 1 1 2 3 1 2 2 1
1 2 1 1 2 3 1 2 2 13
0 1 2 3
2 2 3 2
0
0 0 0 1
c c s c s d c s d s
s c c s s d s s d c
T A A A
s c d c






 12
3 1 2
2
cs
z s s
c





 3 1 2 2 1
3 3 1 2 2 1
32
d c s d s
O d s s d c
dc





 01
0
0
0
OO





 21
2 2 1
0
ds
O d c







9. Stanford Manipulator 4
0 1 2 3 4
T A A A A 5
0 1 2 3 4 5
T A A A A A 6
0 1 2 3 4 5 6
T A A A A A A 1 2 4 1 4
4 1 2 4 1 4
24
c c s s c
z s c s c c
ss


  



T4 =
[ 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]3 1 2 2 1
4 3 1 2 2 1
32
d c s d s
O d s s d c
dc







10. 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]1 2 4 5 1 4 5 1 2 5
5 1 2 4 5 1 4 5 1 2 5
2 4 5 2 5
c c c s s s s c s c
z s c c s c s s s s c
s c s c c


  


 1 2 4 5 1 4 5 1 2 5
5 1 2 4 5 1 4 5 1 2 5
2 4 5 2 5
c c c s s s s c s c
z s c c s c s s s s c
s c s c c


  




11. 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]1 2 4 5 1 4 5 1 2 5
5 1 2 4 5 1 4 5 1 2 5
2 4 5 2 5
c c c s s s s c s c
z s c c s c s s s s c
s c s c c


  


 3 1 2 2 1
5 3 1 2 2 1
32
d c s d s
O d s s d c
dc







12. Stanford Manipulator 6
d6s5c1c2c4 d6s5s1s4 d6c1s2c5 c1s2d3 s1d2
d6s5s1c2c4 d6s5c1s4 d6s1s2c5 s1s2d3 c1d2
d6s2c4s5 d6c2c5 c2d3
O
   

    

   

T6 = [ c6c5c1c2c4-c6c5s1s4-c6c1s2s5+s6c1c2s4+s6s1c4, -
c5c1c2c4+s6c5s1s4+s6c1s2s5+c6c1c2s4+c6s1c4, s5c1c2c4-s5s1s4+c1s2c5,
d6s5c1c2c4-d6s5s1s4+d6c1s2c5+c1s2d3-s1d2]
[ c6c5s1c2c4+c6c5c1s4-c6s1s2s5+s6s1c2s4-s6c1c4, -s6c5s1c2c4-
s6c5c1s4+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]

13. Stanford Manipulator 0 6 0 1 6 1
12
0 1
() ()
,
z o o z o o
JJ
z z
  

 

Joints 1,2 are revolute2
3
0
z
J




Joint 3 is prismatic3 6 3 5 6 5 4 6 4
4 5 6
35 4
( ) ( ) ()
,,
z o o z o o z o o
J J J
zz z
        
  
    
   
The required Jacobian matrix J 
1 2 3 4 5 6
J J J J J J J