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1Terry Taewoong Um ([email protected])
University of Waterloo
Department of Electrical & Computer Engineering
Terry Taewoong Um
MANIPULATOR JACOBIAN IN LIE GROUP FORMULATION
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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)
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Jacobian in Lie Group
From the chain rule,
Note that is forward kinematics solution
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Jacobian in Lie Group
The Lie algebra vector seen from {0} at (The Jacobian changes over time(pose))
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Example 1
The Lie algebra vector seen from {0} at (The Jacobian changes over time(pose))
[SCARA robot]
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Example 2
[Stanford arm]
prismatic
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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.