lie group formulation for robot mechanics
TRANSCRIPT
Terry Taewoong Um ([email protected])
University of Waterloo
Department of Electrical & Computer Engineering
Terry Taewoong Um
LIE GROUP FORMULATION
FOR ROBOT MECHANICS
1
Terry Taewoong Um ([email protected])
CONTENTS
1. Motion and Lie Group
2. Kinematics and Dynamics
3. Summary + Q&A
2
Terry Taewoong Um ([email protected])
MOTIVATION
4
β’ Coordinate-free approach
http://arxiv.org/pdf/1404.1100.pdf
- Which coordinate should we choose?
- Letβs remove the dependency on the choice of reference frames!
β Use the right representation for motion β Lie group & Lie algebra
[Newton-Euler formulation]
- Geodesic : a shortest path b/w two points
- Euler angle-based trajectory is not a geodesic!
Terry Taewoong Um ([email protected])
PRELIMINARY
5
β’ 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)
: π Γ π invertible matrices with matrix multiplication
PRELIMINARY
- Special Linear Group, SL(n) : GL(n) with determinant 1
- Orthogonal Group, O(n) : π β πΊπΏ π πππ = πππ = πΌ}
β’ Lie Group : a group that is also a differentiable manifold
e.g.)
β’ 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
Terry Taewoong Um ([email protected])7
SO(3) : ROTATION
β’ Special Orthogonal group, SO(3)
π ππ = π π π = πΌdet π = 1
β’ Lie algebra of SO(3) : so(3)
π ππ = [π₯π π¦π π§π]
π₯ π¦
π§
π₯ 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
β’ Exponential mapping
exp βΆ π π 3 β ππ(3) exp βΆ π π 3 β ππΈ(3)
exp βΆ πΏππ πππππππ β πΏππ ππππ’π
π πππ£π = π£π
Terry Taewoong Um ([email protected])8
SO(3) : ROTATION
β’ Exponential mapping (Cont.)
e.g.) π ππ‘ π§, π = πΌ + π πππ0 β1 01 0 00 0 0
+ (1 β πππ π)0 β1 01 0 00 0 0
0 β1 01 0 00 0 0
=1 0 00 1 00 0 1
+0 βπ πππ 0
π πππ 0 00 0 0
+ (1 β πππ π)β1 0 00 β1 00 0 0
=πππ π βπ πππ 0π πππ πππ π 00 0 1
β’ Logarithm mapping log : πΏππ ππππ’π β πΏππ πππππππ
Terry Taewoong Um ([email protected])9
SE(3) : ROTATION + TRANSLATION
β’ Special Euclidean group, SE(3)
ππππ£π = π£π
β’ Exp & Log
β’ se(3)
π£{π}
{π}
Terry Taewoong Um ([email protected])10
ADJOINT MAPPING
β’ Lie Algebra : the tangent space at the identity of Lie group
a vector space with Lie bracket operation [x, y]
β’ Small adjoint mapping
β’ Large adjoint mapping
cross product
For so(3),
For se(3),
For so(3),
For se(3),
coordinate change
Terry Taewoong Um ([email protected])12
FORWARD KINEMATICS
β’ Product of Exponential (POE) Formula
- D-H Convention
- POE formula from robot configuration
h = pitch (m/πππ) (0 for rev. joint)
q = a point on the axis
variableconstant
c.f.)
A seen from {0}
π πππ£π = π£π
ππππ£π = π£π
π΄ππππ[π΄]π= [π΄]π
Coord. change
SE(3) from {0} to {n} at home position
Terry Taewoong Um ([email protected])14
DIFFERENTIAL KINEMATICS
β’ Angular velocity by rotational motionfrom space(fixed frame) to body
c.f.)
body velocity
π/π : angular/linear velocity of the {body} attached to the body relative to the {space} but expressed @{body}
β’ Spatial velocity by screw motion
β’ Jacobian
From
π = π½π π
Terry Taewoong Um ([email protected])15
PRELIMINARY FOR DYNAMICS
β’ Coordinate transformation rules
for velocity-like se(3) for force-like se(3)
generalized momentum
dual map
c
β’ Time derivatives
: :
c.f.)
wholederivative
component-wisederivative
π is required
Terry Taewoong Um ([email protected])16
INVERSE DYNAMICS
β’ π½ :
β’ π½ : c.f.)
β’ πππππ βΆ
propagated forces
Terry Taewoong Um ([email protected])19
SUMMARY
β’ Lie Group : a group that is also a differentiable manifold
β’ Lie Algebra : the tangent space at the identity of Lie group
β’ SO(3), so(3), SE(3), se(3), exp, log, Ad, adcoord. trans.
for se(3)cross product
for se(3)
β’ Forward Kinematics
β’ Lie algebra is vector space! (easier to apply pdf)
β’ Inverse Dynamics
β’ Differential Kinematics π = π½π π
Terry Taewoong Um ([email protected])20
Q & A
β’ What are the benefits/drawbacks of using Lie group for rigid body dynamics?
β’ 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
Terry Taewoong Um ([email protected])22
Q & A
β’ Can you do a high-level overview of the mathematical details of the Wangβs paper (for those of us who got lost in the math)?
? - Convolution for Lie group (Chirikjian, 1998)
- Error propagation β 1st order (Wang and Chirikjian, 2006)
- Error propagation β 2nd order (Wang and Chirikjian, 2008)
