Jacobian: Velocities and Static Forces
What is Jacobian?
• Jacobian estabilishes
– the relation between the joint velocities and
the velocity of the end-effector
– the relation between the joint torque and the
force acting at the end-effector
Linear Velocity
• Linear velocity of an object is the rate of
change of its position.

r : the position vector of the object
w.r.t a fixed reference frame
yo
z

r
The linear velocity of the object is given
 d 
v r
dt
x
The linear velocity is a Vector!!
zo
C
object
xo
O
y
Angular Velocity
• Angular velocity: Angular velocity of an object
describes the rate of change of its orientation.
 x 


   y 
 z 

 x : angular velocity about the x - axis
 y : angular velocity about the y - axis
 z : angular velocity about the z - axis
More on Angular Velocity
z
z

0
   0 


 

y
x
x
z

y
x
0

   
   
0
 
y

z

u : unit vector

y
x



  
  0
0
 



  u
More on Angular Velocity
When an object rotates about two axes simultaneously

z
b : unit vector

u : unit vector


y


  u  b

x
Angular velocity is the combination of the angular
velocities due to the two rotations
Joint Velocity and Velocity of End-Effector
• The positions of the joints of a manipulator are

represented by a n-dimensional vector  , where
n is the number of the joints.
• The joint velocity of the manipulator is given by 
• Denote the position
and orientation of the end
effector
 by x
–
x is 3 dimensional vector in planar cases (two
positions
+ one orientation)

– x is 6 dimensional vector in 3D space (3 positions + 3
orientation)

• The velocity of the end-effector is given by x
Jacobian
• Forward kinematics:

  
x  f ( ), where f is 3  1 or 6  1 vector function
• Differentiating the forward kinematics
 
 

 f ( ) 
f ( )

 , where  is the following m  n matrix :
x





 f1 ( ) f1 ( )

f
(
)

1


...
 1
 2 
 n  
   f ( ) f ( )


f
(
)

2
2
2
f ( ) 
...

  






n
1
2
Jacobian matrix of

...
...
...
...



 
the manipulator
 f m ( ) f m ( )
f m1 ( ) 
...








1

2n
J ( )
Jacobian (cont’)
• The Jacobian matrix relates the joint velocity
of a manipulator to the velocity of its endeffector
 

x  J ( )
• The Jacobian matrix establishes the relation
of differential motion between the joints and
the end-effector.
 2







x J ( )  x  J ( )
x: differential motion of the end - effector
 : differential motion of the joints

x
1
Example
Example: Find the Jacobian
matrix of a 3 DOF planar arm.
 1 
Joint angles :    2 
 
 3

(x,y)
l3

y
0
  x 
Position and orientation of the endpoint : x  y
 
 
x  l1c1  l 2 c12  l3c123
y  l1s1  l 2 s12  l3 s123
  1   2   3
 x

 1
  y
J ( )  
 1
 
 
 1
x
 2
y
 2

 2
x 

 3 
y    l1s1  l 2 s12  l3 s123
 l1c1  l 2 c12  l3c123
 3  
1

 
 3 
3
l2
2
l1
1
x0
 l 2 s12  l3 s123
l 2 c12  l3c123
1
 l3 s123 
l3c123 
1 
General Method for Jacobian Calculation
1.
Solve the forward kinematics of the robot manipulator
0

Tend effector
 R ( )

 013
 
p ( ) 
1 
2.
Define the linear and angular velocity of the end-effector
3.
The Jacobian matrix has the following form

  v 
x   
 
 
 p ( )  
 
 
x    
 B( ) 



How to Calculate B( ) ?
• Consider the rotation (angular
velocity) of the end-effector due
to motion of joint i.
• Case A: Joint is a prismatic joint
– When all the other joints do not
move, the motion of joint i does not
cause rotation of the end-effector.
– Therefore, the angular velocity due
to joint i is zero
i
end effector  0
i
end effector : angular velocity of the
end - effector due to joint i.
Joint i

How to Calculate B( ) ?
• Case B: Joint i is revolute
– When all the other joints do not
move, the motion of joint i will
cause rotation of the endeffector.
– Therefore, the angular velocity
due to joint i is as follows
i
0 

end effector  zi i

i
i
end effector : angular velocity of the
end - effector due to joint i.
zi
Joint i
Jacobian
• The angular velocity of the end-effector is
given by


n

i 1
0 
 i zi i
 i  0
  1
 i
If joint i is prismatic
If joint i is revolute
• Therefore, the Jacobian matrix is as follows:
 
 
 
  p ( ) p( ) ... p( ) 
J ( )   1
 2
 n 
0
0 
  0 z
 1 1  2 z 2 ...  n z n 
Example
z1
• A 3-DOF arm
 i 1
i
1
2
3
4
0
90 
0
0
 c1  s1
s
c1
0
T1   1
0
0
0
0

 c3
s
2
T3   3
0
 0

 s3
c3
0
0
0
0
1
0
0
0
1
0
ai-1
0
0
0
0
0
0
0
1 
0
0
0
1 
di
0
d2
0
l3
y1
i
1
y2
End-effector frame
x1 x2
z2
0
0
1
0
3
T4  
0
0

xe
ze
y3 x
3
3
1
0
1
T2  
0
0

ye
0
0
1
0
0
1
0
0
0
1
0
0
0
0
1
0
z3
0 
 d2 
0 
1 
0
0
l3 
1 
z0
y0
x0
l3: distance between x3 and x4
Example
• Forward kinematics
0
z1
 c1  s1

c1
0
T1   s1
0
0
0
0

0
0
1
0
0
0
0
1 
0
z2
s1 d 2 s1 
 c1 0


s
c
d
c


0
0
0 1
1
1
2
1
T2  T1 T2  

0 1
0 0
 0 0

0 1


 c1c3  c1s3
s c  s s
0
0
2
1 3
T3  T2 T3   1 3
s
c3
 0 3
0

s1 d 2 s1 
 c1  d 2 c1 

0 0

0 1

 c1c3  c1s3
s c  s s
0
0 3
1 3
Tend effector  T3 T4   1 3
c3
s
 0 3
0

0
z3

position p
s1
(d 2  l3 ) s1 
 c1  (d 2  l3 )c1 

0
0

0
1

Example
• Then, the Jacobian is as follows:

 p x ( )

 1


p y ( )


J ( )   1


 p z ( )

 1
  0 z
 1 1

p x ( )
d 2

p y ( )
d 2

p z ( )
d 2
0
 2 z2

p x ( ) 
  (l  d )c
s1 0 
3
2 1
 3  

   (l3  d 2 ) s1  c1 0 
p y ( )  

0
0
0

 3   
s1 
0
0
  

p z ( )  
c
0
0

1

 3  

1
0
0


0

 3 z3 
Singularities of Jacobian
• Velocity relationship
 

x  J ( )
n: the number of joints
m: the number of degrees of freedom
of the end-effector
• Given a joint velocity, it is always possible to
calculate the velocity of the end-effector.
• Question: Given a velocity of the end-effector, can
we always calculate the joint velocity?
– When n=m


 
1
If J ( ) exists,   J ( ) x


1
If J ( ) does not exist,  cannot be calculated
1
Singular configurations of a manipulator are those configurations
where the inverse of its Jacobain does not exist
Example of Singularities
• For a 2 DOF planar arm
l2
l2
2
l1
l1
1
1
x0
Jacobian
 l s l s
J ( )   1 1 2 12
 l1c1  l 2 c12
 l 2 s12 
l 2 c12 
2  0
x0
When  2  0
   (l  l ) s
J ( )   1 2 1
 (l1  l 2 )c1
singular
 l 2 s1 
l 2 c1 
The solution of the joint velocity
is not unique
Redundant Manipulators
• When n>m: There are more joints than the DOF of the
end-effector. In this case, the manipulator is a redundant
manipulator
 


From x  J ( ) , we cannot solve the  uniquely

 
 

  J ( ) x  ( I  J ( ) J ( ))k
where 

 T  1

T
J ( )  J ( )( J ( ) J ( ))
k is an arbitrary
vector





( I  J ( ) J ( ))k : vector in the null space of J ( )



Null space of J ( ) :



 
k : J ( )k  0
Equivalence of Forces
• Consider equivalence of forces acting on the
end-effector and torque at the joints.
– Equivalence: If two set of forces (torques) cause the
same motion to the manipulator, they are equivalent.
• Question: Suppose that a force acting on the
end-effector. What is the equivalent torque
acting at the joints of the force?
l2
f
2
l1
1
l2
2
1
l1
x0
x0
Equivalence of Forces
• Equivalence of forces  cause the same motion 
the works done by the equivalent forces must be equal
• Consider works done by the force and torque under
differential motion.


For a differential motion
  at the joints,
the different motion
x at the end - effector :  2



x  J ( )

The workdone byf :  
T 
f x  f T J ( )

T


The work done by joint
toru
qe



,...,

:

1
2
n

T
 
x
1
Equivalence of Forces
• Since the works done must be equal
T   T  
   f J ( )

T
T
T
    ( J ( ) f ) 
  

T
  J ( ) f
This represents the relation between the force acting the
end-effector and the torque acting at the joints