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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)

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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? - Let’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) : 𝑛 Γ— 𝑛 invertible matrices with matrix multiplication PRELIMINARY - Special Linear Group, SL(n) : GL(n) with determinant 1 - Orthogonal Group, O(n) : 𝑄 ∈ 𝐺𝐿 𝑛 𝑄 𝑇 𝑄 = 𝑄𝑄 𝑇 = 𝐼} 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) 𝑅 𝑇 𝑅 = 𝑅𝑅 𝑇 = 𝐼det 𝑅 = 1 β€’ Lie algebra of SO(3) : so(3) 𝑅 π‘Žπ‘ = [π‘₯ π‘Ž π‘¦π‘Ž 𝑧 π‘Ž] π‘₯ 𝑦 𝑧 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 ∢ π‘ π‘œ 3 β†’ 𝑆𝑂(3) exp ∢ 𝑠𝑒 3 β†’ 𝑆𝐸(3) exp ∢ 𝐿𝑖𝑒 π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Ž β†’ 𝐿𝑖𝑒 π‘”π‘Ÿπ‘œπ‘’π‘ 𝑅 π‘Žπ‘ 𝑣 𝑏 = 𝑣 π‘Ž
  8. 8. Terry Taewoong Um (terry.t.um@gmail.com) 8 SO(3) : ROTATION β€’ Exponential mapping (Cont.) e.g.) π‘…π‘œπ‘‘ 𝑧, πœƒ = 𝐼 + π‘ π‘–π‘›πœƒ 0 βˆ’1 0 1 0 0 0 0 0 + (1 βˆ’ π‘π‘œπ‘ πœƒ) 0 βˆ’1 0 1 0 0 0 0 0 0 βˆ’1 0 1 0 0 0 0 0 = 1 0 0 0 1 0 0 0 1 + 0 βˆ’π‘ π‘–π‘›πœƒ 0 π‘ π‘–π‘›πœƒ 0 0 0 0 0 + (1 βˆ’ π‘π‘œπ‘ πœƒ) βˆ’1 0 0 0 βˆ’1 0 0 0 0 = π‘π‘œπ‘ πœƒ βˆ’π‘ π‘–π‘›πœƒ 0 π‘ π‘–π‘›πœƒ π‘π‘œπ‘ πœƒ 0 0 0 1 β€’ Logarithm mapping log : 𝐿𝑖𝑒 π‘”π‘Ÿπ‘œπ‘’π‘ β†’ 𝐿𝑖𝑒 π‘Žπ‘™π‘”π‘’π‘π‘Ÿπ‘Ž
  9. 9. Terry Taewoong Um (terry.t.um@gmail.com) 9 SE(3) : ROTATION + TRANSLATION β€’ Special Euclidean group, SE(3) 𝑋 π‘Žπ‘ 𝑣 𝑏 = 𝑣 π‘Ž β€’ Exp & Log β€’ se(3) 𝑣 {𝑏} {π‘Ž}
  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‘Ÿπ‘Žπ‘‘) (0 for rev. joint) q = a point on the axis variableconstant c.f.) A seen from {0} 𝑅 π‘Žπ‘ 𝑣 𝑏 = 𝑣 π‘Ž π‘‡π‘Žπ‘ 𝑣 𝑏 = 𝑣 π‘Ž 𝐴𝑑 𝑇 π‘Žπ‘ [𝐴] 𝑏= [𝐴] π‘Ž 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 𝝎/𝒗 : 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œƒ = 𝐽𝑠 πœƒ
  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 𝑉 is required
  16. 16. Terry Taewoong Um (terry.t.um@gmail.com) 16 INVERSE DYNAMICS β€’ 𝑽 : β€’ 𝑽 : c.f.) β€’ 𝑭𝒐𝒓𝒄𝒆 ∢ 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œƒ = 𝐽𝑠 πœƒ
  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 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)
  23. 23. Terry Taewoong Um (terry.t.um@gmail.com) 23 Thank you

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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)

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