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Deterministic Sampling Methods for Spheres and SO (3)

Deterministic Sampling Methods for Spheres and SO (3). Anna Yershova Steven M. LaValle Dept. of Computer Science University of Illinois Urbana, IL, USA. Motivation. Sampling over spheres arises in sampling-based algorithms for solving:. Motion planning problems

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Deterministic Sampling Methods for Spheres and SO (3)

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  1. Deterministic Sampling Methods for Spheres and SO(3) Anna Yershova Steven M. LaValle Dept. of Computer Science University of Illinois Urbana, IL, USA

  2. Motivation Sampling over spheres arises in sampling-based algorithms for solving: • Motion planning problems • Optimization problems Target applications are: • Robotics • Computer graphics • Control theory • Computational biology One important special case and our main motivation: • Problem of motion planning for a rigid body in .

  3. Motion planning for 3D rigid body Given: • Geometric models of a robot and obstacles in 3D world • Configuration space • Initial and goal configurations Task: • Compute a collision free path that connects initial and goal configurations

  4. Motion planning for 3D rigid body Existing techniques: • Sampling-based motion planning algorithms based on random sequences [Amato, Wu, 96; Bohlin, Kavraki, 00;Kavraki, Svestka, Latombe,Overmars, 96;LaValle, Kuffner, 01;Simeon, Laumond, Nissoux, 00;Yu, Gupta, 98] Drawbacks: • Would we like to exchange probabilistic completeness for resolution completeness? • In some applications resolution completeness is crucial (e.g. verification problems)

  5. The Goal Deterministic sequences for have been shown to perform well in practice (sometimes even with the improvement in the performance over random sequences) [Lindemann, LaValle, 2003], [Branicky, LaValle, Olsen, Yang 2001] [LaValle, Branicky, Lindemann] [Matousek 99] [Niederreiter 92] Problem: • Uniformity measure is induced by the metric, and therefore, partially by the topology of the space • Cannot be applied to configuration spaces with different topology The Goal: • Extend deterministic sequences to spheres and SO(3) [Arvo 95][Blumlinger 91], [Rote,Tichy 95] [Shoemake 85, 92] [Kuffner 04] [Mitchell 04]

  6. Parameterization of SO(3) • Uniformity depends on the parameterization. • Haar measure defines the volumes of the sets in the space, so that they are invariant up to a rotation • The parameterization of SO(3) with quaternions respects the unique (up to scalar multiple) Haar measure for SO(3) • Quaternions can be viewed as all the points lying on S 3 with the antipodal points identified Close relationship between sampling on spheres and SO(3)

  7. Uniformity Criteria on Spheres and SO(3) Discrepancy of a point set: • The largest empty volume that can fit in between the points Dispersion of a point set: • The radius of the largest empty ball

  8. The Outline of the Rest of the Talk • Provide general approach for sampling over spheres • Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which: • is deterministic • achieves low dispersion and low discrepancy • is incremental • has lattice structure • can be efficiently generated • Properties and experimental evaluation of this sequence on the problems of motion planning

  9. The Outline of the Rest of the Talk • Provide general approach for sampling over spheres • Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which: • achieves low dispersion and low discrepancy • is deterministic • is incremental • has lattice structure • can be efficiently generated • Properties and experimental evaluation of this sequence on the problems of motion planning

  10. … Platonic Solids Regular polygons in R2: Regular polyhedra in R3: Regular polytopes in R4: Regular polytopes in Rd , d > 4: Properties of the vertices of Platonic solids in R(d+ 1): • Form a distribution on S d • Provide uniform coverage of S d • Provide lattice structure, natural for building roadmaps for planning simplex, cube, cross polytope,24-cell, 120-cell, 600-cell simplex, cube, cross polytope

  11. Platonic Solids Problem: • In higher dimensions there are only few regular polytopes • How to obtain evenly distributed points for n points in Rd • Is it possible to avoid distortions? General idea: • Borrow the structure of the regular polytopes and transform generated points on the surface of the sphere

  12. General Approach forDistributions on Spheres • Take a good distribution of points on the surface of a polytope • Project the faces of the polytope outward to form spherical tiling • Use the same baricentric coordinates on spherical faces as they are on polytope faces

  13. Example. Sukharev Grid on S2 • Take a cube in R3 • Place Sukharev grid on each face • Project the faces of the cube outwards to form spherical tiling • Place a Sukharev grid on each spherical face Important note: similar procedure applies for any Sd

  14. The Outline of the Rest of the Talk • Provide general approach for sampling over spheres • Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which: • achieves low dispersion and low discrepancy • is deterministic • is incremental • has lattice structure • can be efficiently generated • Properties and experimental evaluation of this sequence on the problems of motion planning

  15. Layered Sukharev Grid Sequencein [0, 1]d • Places Sukharev grids one resolution at a time • Achieves low dispersion and low discrepancy at each resolution • Performs well in practice • Can be easily adapted forspheres and SO(3) [Lindemann, LaValle 2003]

  16. Layered Sukharev Grid Sequence for Spheres • Take a Layered Sukharev Grid sequence inside each face • Define the ordering on faces • Combine these two into a sequence on the sphere Ordering on faces + Ordering inside faces

  17. The Outline of the Rest of the Talk • Provide general approach for sampling over spheres • Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which: • achieves low dispersion and low discrepancy • is deterministic • is incremental • has lattice structure • can be efficiently generated • Properties and experimental evaluation of this sequence on the problems of motion planning

  18. Properties • The dispersion of the sequence Ts at the resolution level l containing points is: • The relationship between the discrepancy of the sequence T at the resolution level l taken over d-dimensional spherical canonical rectangles and the discrepancy of the optimal sequence, To, is: • 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.

  19. ExperimentsPRM method • SO(3) configuration space • Averaged over 50 trials

  20. ExperimentsPRM method • SO(3) configuration space • Averaged over 50 trials

  21. Conclusion • We have proposed a general framework for uniform sampling over spheres and SO(3) • We have developed and implemented a particular sequence which extends the layered Sukharev grid sequence designed for a unit cube • We have tested the performance of this sequence in a PRM-like motion planning algorithm • We have demonstrated that the sequence is a useful alternative to random sampling, in addition to the advantages that it has Future Work • Reduce the amount of distortion introduced with more dimensions and with the size of polytope’s faces • Design deterministic sequences for more general topological spaces

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