Manipulator Jacobian

Manipulator Jacobian

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

2. Manipulator Jacobian
By:
AparnaParida
SonaliBehera
SubhasisMartha
AmarMajhi
SandeepMunda

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

4. RobotAnatomy for Human Arm

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

6. The Jacobian matrixin Robotics
•Weuse the Jacobian Matrix to find the velocity of an end
effector.
•We use a notation ᶓ(chi) to represent the velocity of the end
effectorin the base frame.
•The velocity of the end effectorat the base frame has 2
components:
•Linear velocity of the end effector(V
n
0
).
•Angular velocity of the end effector(ω
n
0
).
•The linear velocity has 3 components x,y,z.
•The angular velocity has 3 components rotation about x,rotation
about y,rotationabout z.

7. ᶓ=
We consider a parameter qas a generalisationof joint variation.
For each revolute jointq is ‘ϴ’ for the joint.
For a prismatic joint q is displacement of the joint ‘d’.
q’ is thetime derivative of each joint variable.
q’ is a vector with one columnand number of rowsequal to the number of
joints.
For an articulated manipulator, we have 3 revolute joints, so we have
q’ =
For a prismatic joint as 3
rd
joint of a manipulator we have
q’ =
V
n
0
ω
n
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 upperhalf 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