This document presents an overview of lie group formulations for robot mechanics. It discusses rigid body motion using SO(3) and SE(3) transformations. It also covers generalized velocity and force, dynamics of open chain systems, hybrid dynamics, and recursive inverse dynamics algorithms. The document is authored by Terry Taewoong Um from the University of Waterloo and is intended to summarize concepts from another source on lie group dynamics.

1. Lie Group Formulation
for Robot Mechanics
Terry Taewoong Um
terry.t.um@gmail.com
Adaptive Systems Laboratory
Electrical and Computer Engineering
University of Waterloo

2. These slides are made based
on Junnggon Kim’s note
http://www.cs.cmu.edu/~junggon/tools/liegroupdynamics.pdf
made by Terry. T. Um (terry.t.um@gmail.com)

3. Dynamics of a Rigid Body
made by Terry. T. Um (terry.t.um@gmail.com)

4. Rigid Body Motion
• SO(3) & SE(3)
ab : cord. {B} w.r.t cord. {A}
• se(3) : Lie algebra of SE(3)
4x4
4x4
skew symmetric matrix
• Adjoint mapping
4x4
made by Terry. T. Um (terry.t.um@gmail.com)
or
6x6
or
dse(3) mapping

5. Generalized Velocity & Force
• Notation @{body} : w.r.t the frame attached to the (moving) body
@{space} : w.r.t. the frame attached to the (fixed) reference frame
• Generalized Velocity & Force
4x4
• Coordinate Transformation Rules
made by Terry. T. Um (terry.t.um@gmail.com)
or
6x6
흎 / 풗 : angular / linear velocity of the {body} attached to the body relative
relative to the {space} but expressed @{body}
푭 : a moment and force action on the body viewed @{body}
Let {A}, {B} be two different coord. frames attached to the same body but diff. pos.
(recall )
푭 ∈ dse(3)

6. Generalized Inertial & Momentum
• Kinematic Energy
: generalized momentum @{body}
• Coordinate Transformation Rules
: generalized
inertia @{body}
6x6
Let {A}, {B} be two different coord. frames attached to the same body but diff. pos.
made by Terry. T. Um (terry.t.um@gmail.com)
3x3 inertia matrix @{body}
= 0 if the origin is
located on the CoM
if the origin @CoM
like

7. Time Derivative and Force
• Time derivative of a 3-dim vector
• Time derivative of se(3) & dse(3)
made by Terry. T. Um (terry.t.um@gmail.com)
• Generalized Force
component-wise
time derivative
whole derivative component-wise
time derivative

8. Dynamics of Open Chain Systems
made by Terry. T. Um (terry.t.um@gmail.com)

9. Hybrid Dynamics
• Hybrid Dynamics : Mixture of Forward & Inverse Dynamics
made by Terry. T. Um (terry.t.um@gmail.com)
u : inverse dynamics, i.e.
v : forward dynamics, i.e.
thus,
• Notation
: inertial frame (stationary)
: the frame of the ith body
: the frame of the parent
of the ith body

10. Recursive Inverse Dynamics
• Generalized Velocity of the ith frame
relative velocity w.r.t. its parent
: Jacobin of the joint i connecting with it parents
• To build the dynamics equations for each body, 푽 is required
Force of a rigid body : : 푉 is requiraed
made by Terry. T. Um (terry.t.um@gmail.com)

11. Recursive Inverse Dynamics
• Time derivative of the generalized velocity, 푽
made by Terry. T. Um (terry.t.um@gmail.com)
recall
• Force of the i th body, 푭풊
propagated forces
external force acting
on the ith body
recall
reaction

12. Recursive Inverse Dynamics
• Recursive Inverse Dynamics Algorithm
made by Terry. T. Um (terry.t.um@gmail.com)

13. Recursive Inverse Dynamics
made by Terry. T. Um (terry.t.um@gmail.com)

14. Recursive Inverse Dynamics (Comparison)
made by Terry. T. Um (terry.t.um@gmail.com)

15. Recursive Inverse Dynamics (Comparison)
made by Terry. T. Um (terry.t.um@gmail.com)