Manipulator Jacobian

1. Manipulator Jacobian
Hitesh Mohapatra
https://www.linkedin.com/in/hiteshmohapatra/

2. Manipulator Jacobian
By:
Aparna Parida
Sonali Behera
Subhasis Martha
Amar Majhi
Sandeep Munda

3. •A robot manipulator is constructed using rigid links
connected by joints with one fixed end and one free end to
perform a given task (e.g., to move a box from one location
to the next).
•The joints to this robotic manipulator are the movable
components, which enables relative motion between the
adjoining links.
What is a Manipulator?

4. Robot Anatomy for Human Arm

5. Jacobian
•Jacobian is used in change of variables in
multiple integrals.
•Jacobian is basically a determinant.
•Jacobian is used to transform variables in one
coordinate frame to variables in another
coordinate frame.

6. The Jacobian matrix in Robotics
•We use the Jacobian Matrix to find the velocity of an end
effector.
•We use a notation ᶓ (chi) to represent the velocity of the end
effector in the base frame.
•The velocity of the end effector at the base frame has 2
components:
•Linear velocity of the end effector(Vn
0 ).
•Angular velocity of the end effector(ωn
0 ).
•The linear velocity has 3 components x,y,z.
•The angular velocity has 3 components rotation about x,rotation
about y,rotation about z.

7. ᶓ =
We consider a parameter q as a generalisation of joint variation.
For each revolute joint q is ‘ϴ’ for the joint.
For a prismatic joint q is displacement of the joint ‘d’.
q’ is the time derivative of each joint variable.
q’ is a vector with one column and number of rows equal to the number of
joints.
For an articulated manipulator, we have 3 revolute joints, so we have
q’ =
For a prismatic joint as 3rd joint of a manipulator we have
q’ =
Vn
0
ωn
0
ϴ
1’
ϴ
2’
ϴ
3’
ϴ
1’
ϴ
2’
d

8. We seek the expressions of the form:
•The above matrix is called the Manipulator Jacobian or Jacobian
for short.
• It is a 6 × n matrix where n is the number of links.

9. The upper half of the Jacobian is given as :

10. Now putting the upper and lower halves of the Jacobian together we have
shown that the Jacobian for an n-link manipulator is of the form

11. Let’s take an example-

13. Example-2