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Anna Yershova 1 , Steven M. LaValle 2 , and Julie C. Mitchell 3

Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration. Anna Yershova 1 , Steven M. LaValle 2 , and Julie C. Mitchell 3 1 Dept. of Computer Science, Duke University 2 Dept. of Computer Science, University of Illinois at Urbana-Champaign

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Anna Yershova 1 , Steven M. LaValle 2 , and Julie C. Mitchell 3

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  1. Generating Uniform Incremental Grids on SO(3) Using the HopfFibration Anna Yershova1, Steven M. LaValle2,and Julie C. Mitchell3 1Dept. of Computer Science, Duke University 2Dept. of Computer Science, University of Illinois at Urbana-Champaign 3Dept. of Mathematics, University of Wisconsin December 8, 2008 Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  2. Introduction Presentation Overview • Introduction • Motivation • Problem Formulation • Properties and Representations of the space of rotations, SO(3) • Literature Overview • Method Presentation • Conclusions and Discussion Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  3. Introduction Motivation Sampling SO(3) Occurs in: • Automotive Assembly • Computational Chemistryand Biology • Manipulation Planning • Medical applications • Computer Graphics(motions for digital actors) • Autonomous vehicles andspacecrafts Courtesy of Kineo CAM Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  4. Introduction Motivation Our Main Motivation: Motion Planning The graph over C-space should capture the “path connectivity” of the space Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  5. Introduction Motivation Our Main Motivation: Motion Planning • The quality of the undelying samples affect the quality of the graph • SO(3) is often the C-space Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  6. Introduction Problem Formulation Problem Formulation Desirable properties of samples over the SO(3): • uniform • deterministic • incremental • grid structure Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  7. Introduction Problem Formulation Problem Formulation Desirable properties of samples over the SO(3): Uniform: Dispersion: the radius of the largest empty ball Discrepancy: maximum volume estimation error • uniform • deterministic • incremental • grid structure Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  8. Introduction Problem Formulation Problem Formulation Desirable properties of samples over the SO(3): • uniform • deterministic • incremental • grid structure Deterministic: The uniformity measures can be deterministically computed Reason: resolution completeness Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  9. Introduction Problem Formulation Problem Formulation Desirable properties of samples over the SO(3): • uniform • deterministic • incremental • grid structure Incremental: The uniformity measures are optimized with every new point Reason: it is unknown how many points are needed to solve the problem in advance Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  10. Introduction Problem Formulation Problem Formulation Desirable properties of samples over the SO(3): • uniform • deterministic • incremental • grid structure Grid: Reason: Trivializes nearest neighbor computations Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  11. SO(3) Properties SO(3): Topology, Manifold Structure Fiber bundles • SO(3) is a Lie group • SO(3) is diffeomorphic to S3 with antipodal points identified • Haar measure on SO(3) corresponds to the surface measure on S3 • SO(3) has a fiber bundle structure • Fibers represent SO(3) as a product of S1 and S2. Locally it is a Cartesian product • Remark: sampling on spheres and SO(3) are related S3, SO(3) S1 S2 Mobius Band I S1 Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  12. SO(3) Properties SO(3) Parameterizations and Coordinates • Euler angles • Axis angle representation (topology) • Spherical coordinates (topology, Haar measure) • Quaternions (topology, Haar measure, group operation) • Hopf coordinates (topology, Haar measure, Hopf bundle) Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  13. Literature Overview Literature overview • Euclidean space, [0,1]d • Spheres, Sd • Special orthogonal group, SO(3) Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  14. Literature Overview Euclidean Spaces, [0,1]d + uniform + deterministic + incremental - grid structure + uniform + deterministic + incremental - grid structure + uniform - deterministic + incremental - grid structure Halton points Hammersley points Random sequence + uniform + deterministic - incremental + grid structure + uniform + deterministic - incremental + grid structure Sukharev grid A lattice Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  15. Literature Overview Euclidean Spaces, [0,1]d Layered Sukharev Grid Sequence [Lindemann, LaValle 2003] + uniform + deterministic + incremental + grid structure Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  16. Literature Overview Spheres, Sd, and SO(3) • Random sequences • subgroup method for random sequences SO(3) • almost optimal discrepancy random sequences for spheres [Beck, 84] [Diaconis, Shahshahani 87] [Wagner, 93] [Bourgain, Linderstrauss 93] • Deterministic point sets • optimal discrepancy point sets for Sd, SO(3) • uniform deterministic point sets for SO(3) [Lubotzky, Phillips, Sarnak 86] [Mitchell 07] • No deterministic sequences to our knowledge + uniform - deterministic + incremental - grid structure + uniform + deterministic - incremental - grid structure Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  17. Literature Overview Our previous approach: Spheres ~ uniform + deterministic + incremental + grid structure • Make a Layered Sukharev Grid sequence inside each face • Define the ordering across faces • Combine these two into a sequence on the cube • Project the faces of the cube outwards to form spherical tiling • Use barycentric coordinates to define the sequence on the sphere • [Yershova, LaValle, ICRA 2004] Ordering on faces + Ordering inside faces Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  18. Literature Overview 3 2 XY 1 4 XY Ordering on cells, Ordering inside cells Our previous approach: Cartesian Products • Make grid cells inside X and Y • Naturally extend the grid structure to XY • Define the cell ordering and the ordering inside each cell Y X • [Lindemann, Yershova, LaValle, WAFR 2004] Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  19. Method Presentation Our approach: SO(3) • Hopf coordinates preserve the fiber bundle structure of R P3 • Locally, R P3 is a product of S1 and S2 Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  20. Method Presentation Our approach: SO(3) • The method for Cartesian products can then be extended to R P3 • Need two grids, for S1 and S2 Grid on S2 Grid on S1 Healpix, [Gorski,05] Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  21. Method Presentation Our approach: SO(3) • The method for Cartesian products can then be extended to R P3 • Need two grids, for S1 and S2 Grid on S1 Grid on S2 Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  22. Method Presentation Our approach: SO(3) • The method for Cartesian products can then be extended to R P3 • Need two grids, for S1 and S2 • Ordering on cells, ordering on [0,1]3 + uniform + deterministic + incremental + grid structure Grid on S1 Grid on S2 Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  23. Method Presentation Propositions • The dispersion of the sequence T on SO(3) at the resolution level lis: in which is the dispersion of the sequence over S2. Note: The best bound so far to our knowledge. • The sequence T has the following properties: • The position of the i-th sample in the sequence T can be generated in O(logi) time. • For any i-th sample any of the 2d nearest grid neighbors from the same layer can be found in O((logi)/d) time. Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  24. Method Presentation Illustration on Motion Planning • Configuration space: SO(3) (a) (b) Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  25. Conclusions Conclusions 1. We have designed a sequence of samples over the SO(3) which are: • uniform • deterministic • incremental • grid structure 2. Main point: Hopf coordinates naturally preserve the grid structure on SO(3). (Subgroup aglorithm by Shoemake implicitly utilizes them) Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

  26. Conclusions Conclusions Thank you! Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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