Lie Group Formulation for Robot Mechanics (http://terryum.io)

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)