Manipulator Jacobian for robotics in Lie group formuation

1Terry Taewoong Um (terry.t.um@gmail.com)

University of Waterloo

Department of Electrical & Computer Engineering

Terry Taewoong Um

MANIPULATOR JACOBIAN IN LIE GROUP FORMULATION

Jacobian in Lie Group

space Jacobian

from {0} to {tool}

space vel.(There exist body vel. & body Jacobian as well)

[6x1] [6x6] [6x1]

[4x4]

• Space Jacobian

• Space Velocity

[4x4] [4x4]

(Note that velocity of a point on the body is as follows due to coordinate transformation)

Jacobian in Lie Group

From the chain rule,

Note that is forward kinematics solution

Jacobian in Lie Group

The Lie algebra vector seen from {0} at (The Jacobian changes over time(pose))

Example 1

The Lie algebra vector seen from {0} at (The Jacobian changes over time(pose))

[SCARA robot]

Example 2

[Stanford arm]

prismatic

Body Jacobian

• In robotics, body Jacobian is preferred because the body Jacobian is independent of the choice of base frame

coordinate change

Reference

• R. Murray, A Mathematical Introduction to Robotics (pp 115-123), CRC press, 1994.