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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 3 Dept. of Mathematics, University of Wisconsin December 8, 2008 Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershova, et. al. Uniform Incremental Grids on SO(3) 1

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

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

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

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

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

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

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

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Problem Formulation uniform deterministic incremental grid structure 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 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 9 Desirable properties of samples over the SO(3):IntroductionProblem Formulation Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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

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SO(3): Topology, Manifold Structure SO(3) is a Lie group SO(3) is diffeomorphic to S 3 with antipodal points identified Haar measure on SO(3) corresponds to the surface measure on S 3 SO(3) has a fiber bundle structure Fibers represent SO(3) as a product of S 1 and S 2. Locally it is a Cartesian product Remark: sampling on spheres and SO(3) are related SO(3) is a Lie group SO(3) is diffeomorphic to S 3 with antipodal points identified Haar measure on SO(3) corresponds to the surface measure on S 3 SO(3) has a fiber bundle structure Fibers represent SO(3) as a product of S 1 and S 2. Locally it is a Cartesian product Remark: sampling on spheres and SO(3) are related 11 SO(3) Properties Anna Yershova, et. al. Uniform Incremental Grids on SO(3) S 3, SO(3) S 1 S 2 S 3, SO(3) S 1 S 2 Fiber bundles Mobius Band I S 1 Mobius Band I S 1

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

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

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

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

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Spheres, S d, 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 S d, 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 + uniform deterministic incremental grid structure + uniform deterministic incremental grid structure 16 Literature Overview Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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Our previous approach: Spheres ~ uniform deterministic + incremental grid structure ~ uniform deterministic + incremental grid structure Ordering on faces + Ordering inside faces 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] 17 Literature Overview Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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Our previous approach: Cartesian Products X YX Y 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 X Y X YX Y Ordering on cells, Ordering inside cells Literature Overview Anna Yershova, et. al. Uniform Incremental Grids on SO(3) [Lindemann, Yershova, LaValle, WAFR 2004]

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Our approach: SO(3) Method Presentation Hopf coordinates preserve the fiber bundle structure of R P 3 Locally, R P 3 is a product of S 1 and S 2 19 Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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The method for Cartesian products can then be extended to R P 3 Need two grids, for S 1 and S 2 Healpix, [Gorski,05] Grid on S 2 Grid on S 1 20 Our approach: SO(3) Method Presentation Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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The method for Cartesian products can then be extended to R P 3 Need two grids, for S 1 and S 2 Grid on S 2 Grid on S 1 21 Our approach: SO(3) Method Presentation Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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The method for Cartesian products can then be extended to R P 3 Need two grids, for S 1 and S 2 Ordering on cells, ordering on [0,1] 3 Grid on S 2 Grid on S 1 + uniform deterministic + incremental grid structure + uniform deterministic + incremental grid structure 22 Our approach: SO(3) Method Presentation Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

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

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

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ConclusionsConclusions 25Conclusions Anna Yershova, et. al. Uniform Incremental Grids on SO(3) 1. We have designed a sequence of samples over the SO(3) which are: uniform deterministic incremental grid structure 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)

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

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