Lie Group Formulation for Robot Mechanics

Lie Group Formulation

for Robot Mechanics

Terry Taewoong Um

[email protected]

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])