Lie Group Formulation
for Robot Mechanics
Terry Taewoong Um
Adaptive Systems LaboratoryElectrical and Computer Engineering
University of Waterloo
These slides are made based
on Junnggon Kim’s note
http://www.cs.cmu.edu/~junggon/tools/liegroupdynamics.pdf
made by Terry. T. Um ([email protected])
Dynamics of a Rigid Body
made by Terry. T. Um ([email protected])
Rigid Body Motion
ab : cord. {B} w.r.t cord. {A}
• se(3) : Lie algebra of SE(3)
4x4
4x4
skew symmetric matrix
• Adjoint mapping
• SO(3) & SE(3)
4x4
made by Terry. T. Um ([email protected])
or6x6
or
dse(3) mapping
Generalized Velocity & Force
• Coordinate Transformation Rules
4x4
• Generalized Velocity & Force
made by Terry. T. Um ([email protected])
or6x6
𝝎 / 𝒗 : angular / linear velocity of the {body} attached to the body relativerelative to the {space} but expressed @{body}
𝑭 : 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 )
𝑭 ∈ dse(3)
• 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 Inertial & Momentum
• Coordinate Transformation Rules
• Kinematic Energy
made by Terry. T. Um ([email protected])
Let {A}, {B} be two different coord. frames attached to the same body but diff. pos.
: generalized inertia @{body}
6x6
3x3 inertia matrix @{body}
= 0 if the origin islocated on the CoM
if the origin @CoM
: generalized momentum @{body}
like
Time Derivative and Force
• Time derivative of se(3) & dse(3)
• Time derivative of a 3-dim vector
made by Terry. T. Um ([email protected])
• Generalized Force
component-wisetime derivative
whole derivative component-wisetime derivative
Dynamics of Open Chain Systems
made by Terry. T. Um ([email protected])
Hybrid Dynamics
• Hybrid Dynamics : Mixture of Forward & Inverse Dynamics
made by Terry. T. Um ([email protected])
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
Recursive Inverse Dynamics
• Generalized Velocity of the ith frame
made by Terry. T. Um ([email protected])
relative velocity w.r.t. its parent
: Jacobin of the joint i connecting with it parents
• To build the dynamics equations for each body, 𝑽 is required
: 𝑉 is requiraedForce of a rigid body :
Recursive Inverse Dynamics
• Time derivative of the generalized velocity, 𝑽
made by Terry. T. Um ([email protected])
recall
• Force of the i th body, 𝑭𝒊
propagated forcesexternal force acting
on the ith bodyrecall
reaction
Recursive Inverse Dynamics
• Recursive Inverse Dynamics Algorithm
made by Terry. T. Um ([email protected])
Recursive Inverse Dynamics
made by Terry. T. Um ([email protected])
Recursive Inverse Dynamics (Comparison)
made by Terry. T. Um ([email protected])
Recursive Inverse Dynamics (Comparison)
made by Terry. T. Um ([email protected])