Successfully reported this slideshow.
Upcoming SlideShare
×

# Lie Group Formulation for Robot Mechanics

A brief summary of Lie group formulation for robot mechanics. For more details, please refer to the book, "A first course in robot mechanics" written by Frank C. Park from the follow link.

http://robotics.snu.ac.kr/fcp/files/_pdf_files_publications/a_first_coruse_in_robot_mechanics.pdf

(http://terryum.io)

See all

### Related Audiobooks

#### Free with a 30 day trial from Scribd

See all
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

### Lie Group Formulation for Robot Mechanics

1. 1. Terry Taewoong Um (terry.t.um@gmail.com) University of Waterloo Department of Electrical & Computer Engineering Terry Taewoong Um LIE GROUP FORMULATION FOR ROBOT MECHANICS 1
2. 2. Terry Taewoong Um (terry.t.um@gmail.com) CONTENTS 1. Motion and Lie Group 2. Kinematics and Dynamics 3. Summary + Q&A 2
3. 3. Terry Taewoong Um (terry.t.um@gmail.com) CONTENTS 3 1. Motion and Lie Group
4. 4. Terry Taewoong Um (terry.t.um@gmail.com) MOTIVATION 4 β’ Coordinate-free approach http://arxiv.org/pdf/1404.1100.pdf - Which coordinate should we choose? - Letβs 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!
5. 5. Terry Taewoong Um (terry.t.um@gmail.com) 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
6. 6. Terry Taewoong Um (terry.t.um@gmail.com) 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 πΊπΏ π π π π = ππ π = πΌ} 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-commutative Lie group Lie algebra
7. 7. Terry Taewoong Um (terry.t.um@gmail.com) 7 SO(3) : ROTATION β’ Special Orthogonal group, SO(3) π π π = ππ π = πΌdet π = 1 β’ Lie algebra of SO(3) : so(3) π ππ = [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 βΆ π π 3 β ππ(3) exp βΆ π π 3 β ππΈ(3) exp βΆ πΏππ πππππππ β πΏππ ππππ’π π ππ π£ π = π£ π
8. 8. Terry Taewoong Um (terry.t.um@gmail.com) 8 SO(3) : ROTATION β’ Exponential mapping (Cont.) e.g.) πππ‘ π§, π = πΌ + π πππ 0 β1 0 1 0 0 0 0 0 + (1 β πππ π) 0 β1 0 1 0 0 0 0 0 0 β1 0 1 0 0 0 0 0 = 1 0 0 0 1 0 0 0 1 + 0 βπ πππ 0 π πππ 0 0 0 0 0 + (1 β πππ π) β1 0 0 0 β1 0 0 0 0 = πππ π βπ πππ 0 π πππ πππ π 0 0 0 1 β’ Logarithm mapping log : πΏππ ππππ’π β πΏππ πππππππ
9. 9. Terry Taewoong Um (terry.t.um@gmail.com) 9 SE(3) : ROTATION + TRANSLATION β’ Special Euclidean group, SE(3) π ππ π£ π = π£ π β’ Exp & Log β’ se(3) π£ {π} {π}
10. 10. Terry Taewoong Um (terry.t.um@gmail.com) 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
11. 11. Terry Taewoong Um (terry.t.um@gmail.com) CONTENTS 11 2. Kinematics & Dynamics
12. 12. Terry Taewoong Um (terry.t.um@gmail.com) 12 FORWARD KINEMATICS β’ Product of Exponential (POE) Formula - D-H Convention - POE formula from robot configuration h = pitch (m/πππ) (0 for rev. joint) q = a point on the axis variableconstant c.f.) A seen from {0} π ππ π£ π = π£ π πππ π£ π = π£ π π΄π π ππ [π΄] π= [π΄] π Coord. change SE(3) from {0} to {n} at home position
13. 13. Terry Taewoong Um (terry.t.um@gmail.com) 13 FORWARD KINEMATICS
14. 14. Terry Taewoong Um (terry.t.um@gmail.com) 14 DIFFERENTIAL KINEMATICS β’ Angular velocity by rotational motion from 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 = π½π  π
15. 15. Terry Taewoong Um (terry.t.um@gmail.com) 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.) whole derivative component-wise derivative π is required
16. 16. Terry Taewoong Um (terry.t.um@gmail.com) 16 INVERSE DYNAMICS β’ π½ : β’ π½ : c.f.) β’ π­ππππ βΆ propagated forces
17. 17. Terry Taewoong Um (terry.t.um@gmail.com) 17 INVERSE DYNAMICS
18. 18. Terry Taewoong Um (terry.t.um@gmail.com) CONTENTS 18 3. Summary + Q&A
19. 19. Terry Taewoong Um (terry.t.um@gmail.com) 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, ad coord. 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 = π½π  π
20. 20. Terry Taewoong Um (terry.t.um@gmail.com) 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
21. 21. Terry Taewoong Um (terry.t.um@gmail.com) 21 Q & A [From Featherstone's book]
22. 22. Terry Taewoong Um (terry.t.um@gmail.com) 22 Q & A β’ Can you do a high-level overview of the mathematical details of the Wangβs 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)
23. 23. Terry Taewoong Um (terry.t.um@gmail.com) 23 Thank you

### Be the first to comment

• #### neosarchizo

Jul. 23, 2015
• #### doomaridev

Sep. 14, 2015
• #### hirokkun

Oct. 29, 2015
• #### utotch

Jan. 12, 2016
• #### yyukiyoshikawa

Jan. 20, 2016
• #### JaeminCho6

Feb. 26, 2017

Aug. 1, 2017
• #### ssuserb3a202

Oct. 13, 2017
• #### AkihiroI

Feb. 16, 2018
• #### ssuser5fb55e

May. 30, 2018

Jul. 6, 2018
• #### BaaGiiShark

Feb. 16, 2019
• #### LiuYajun1

Jul. 13, 2021

A brief summary of Lie group formulation for robot mechanics. For more details, please refer to the book, "A first course in robot mechanics" written by Frank C. Park from the follow link. http://robotics.snu.ac.kr/fcp/files/_pdf_files_publications/a_first_coruse_in_robot_mechanics.pdf (http://terryum.io)

Total views

41,217

On Slideshare

0

From embeds

0

Number of embeds

33,805

120

Shares

0