Lie Group Formulation for Robot Mechanics

Terry Taewoong Um ([email protected])

University of Waterloo

Department of Electrical & Computer Engineering

Terry Taewoong Um

LIE GROUP FORMULATION

FOR ROBOT MECHANICS

1


CONTENTS

1. Motion and Lie Group

2. Kinematics and Dynamics

3. Summary + Q&A

2


CONTENTS

3

1. Motion and Lie Group


MOTIVATION

4

• Coordinate-free approach

http://arxiv.org/pdf/1404.1100.pdf

- Which coordinate should we choose?

- Let’s remove the dependency on the choice of reference frames!

→ Use the right representation for motion → Lie group & Lie algebra

[Newton-Euler formulation]

- Geodesic : a shortest path b/w two points

- Euler angle-based trajectory is not a geodesic!


PRELIMINARY

5

• Differential Manifolds

Implicit representation

Explicit representation

Local coordinate

n-dim manifold is a set that locally resembles n-dim Euclidean space

- Each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

Local coordinate : vector space! Riemannian metric

Minimal geodesics

distortion

Terry Taewoong Um ([email protected])6

- General Linear Group, GL(n)

: 𝑛 × 𝑛 invertible matrices with matrix multiplication

PRELIMINARY

- Special Linear Group, SL(n) : GL(n) with determinant 1

- Orthogonal Group, O(n) : 𝑄 ∈ 𝐺𝐿 𝑛 𝑄𝑇𝑄 = 𝑄𝑄𝑇 = 𝐼}

• Lie Group : a group that is also a differentiable manifold

e.g.)

• Lie Algebra : the tangent space at the identity of Lie group

a vector space with Lie bracket operation [x, y]

- Lie bracket

Non-commutativeLie group

Lie algebra


SO(3) : ROTATION

• Special Orthogonal group, SO(3)

𝑅𝑇𝑅 = 𝑅𝑅𝑇 = 𝐼det 𝑅 = 1

• Lie algebra of SO(3) : so(3)

𝑅𝑎𝑏 = [𝑥𝑎 𝑦𝑎 𝑧𝑎]

𝑥 𝑦

𝑧

𝑥 of {b} w.r.t. {a}

- You can express SO(3) with the rotation axis & angle!

http://goo.gl/uqilDV

so(3) : skew-symm. matrices

• Exponential mapping

exp ∶ 𝑠𝑜 3 → 𝑆𝑂(3) exp ∶ 𝑠𝑒 3 → 𝑆𝐸(3)

exp ∶ 𝐿𝑖𝑒 𝑎𝑙𝑔𝑒𝑏𝑟𝑎 → 𝐿𝑖𝑒 𝑔𝑟𝑜𝑢𝑝

𝑅𝑎𝑏𝑣𝑏 = 𝑣𝑎


SO(3) : ROTATION

• Exponential mapping (Cont.)

e.g.) 𝑅𝑜𝑡 𝑧, 𝜃 = 𝐼 + 𝑠𝑖𝑛𝜃0 −1 01 0 00 0 0

+ (1 − 𝑐𝑜𝑠𝜃)0 −1 01 0 00 0 0

0 −1 01 0 00 0 0

=1 0 00 1 00 0 1

+0 −𝑠𝑖𝑛𝜃 0

𝑠𝑖𝑛𝜃 0 00 0 0

+ (1 − 𝑐𝑜𝑠𝜃)−1 0 00 −1 00 0 0

=𝑐𝑜𝑠𝜃 −𝑠𝑖𝑛𝜃 0𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 00 0 1

• Logarithm mapping log : 𝐿𝑖𝑒 𝑔𝑟𝑜𝑢𝑝 → 𝐿𝑖𝑒 𝑎𝑙𝑔𝑒𝑏𝑟𝑎


SE(3) : ROTATION + TRANSLATION

• Special Euclidean group, SE(3)

𝑋𝑎𝑏𝑣𝑏 = 𝑣𝑎

• Exp & Log

• se(3)

𝑣{𝑏}

{𝑎}


ADJOINT MAPPING


a vector space with Lie bracket operation [x, y]

• Small adjoint mapping

• Large adjoint mapping

cross product

For so(3),

For se(3),

For so(3),

For se(3),

coordinate change


CONTENTS

11

2. Kinematics & Dynamics


FORWARD KINEMATICS

• Product of Exponential (POE) Formula

- D-H Convention

- POE formula from robot configuration

h = pitch (m/𝑟𝑎𝑑) (0 for rev. joint)

q = a point on the axis

variableconstant

c.f.)

A seen from {0}

𝑅𝑎𝑏𝑣𝑏 = 𝑣𝑎

𝑇𝑎𝑏𝑣𝑏 = 𝑣𝑎

𝐴𝑑𝑇𝑎𝑏[𝐴]𝑏= [𝐴]𝑎

Coord. change

SE(3) from {0} to {n} at home position


FORWARD KINEMATICS


DIFFERENTIAL KINEMATICS

• Angular velocity by rotational motionfrom space(fixed frame) to body

c.f.)

body velocity

𝝎/𝒗 : angular/linear velocity of the {body} attached to the body relative to the {space} but expressed @{body}

• Spatial velocity by screw motion

• Jacobian

From

𝜃 = 𝐽𝑠 𝜃


PRELIMINARY FOR DYNAMICS

• Coordinate transformation rules

for velocity-like se(3) for force-like se(3)

generalized momentum

dual map

c

• Time derivatives

: :

c.f.)

wholederivative

component-wisederivative

𝑉 is required


INVERSE DYNAMICS

• 𝑽 :

• 𝑽 : c.f.)

• 𝑭𝒐𝒓𝒄𝒆 ∶

propagated forces


INVERSE DYNAMICS


CONTENTS

18

3. Summary + Q&A


SUMMARY

• Lie Group : a group that is also a differentiable manifold


• SO(3), so(3), SE(3), se(3), exp, log, Ad, adcoord. trans.

for se(3)cross product

for se(3)

• Forward Kinematics

• Lie algebra is vector space! (easier to apply pdf)

• Inverse Dynamics

• Differential Kinematics 𝜃 = 𝐽𝑠 𝜃


Q & A

• What are the benefits/drawbacks of using Lie group for rigid body dynamics?

• What are the key differences between Lie groups and other 6D formulations (e.g., Featherstone's spatial notation)?

[Featherstone's cross operation]

skew-symmetric

Lie bracket


Q & A

[From Featherstone's book]


Q & A

• Can you do a high-level overview of the mathematical details of the Wang’s paper (for those of us who got lost in the math)?

? - Convolution for Lie group (Chirikjian, 1998)

- Error propagation – 1st order (Wang and Chirikjian, 2006)

- Error propagation – 2nd order (Wang and Chirikjian, 2008)


Thank you