Deterministic Sampling Methods for Spheres and SO (3)

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: • Resolution completeness is crucial (manufacturing, verification problems) • If the planner does not return an answer, what are the guarantees on the existence of a solution?

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) [LaValle, Branicky, Lindemann 03] [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 subsets of locally compact topological groups, so that they are invariant up to a rotation • The parameterization of SO(3) with quaternions respects the bi-invariant Haar measure on 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 for Spheres and SO(3) • Let R (range space) denote a collection of subsets of a sphere • Discrepancy: “maximum volume estimation error over all boxes” R

8. Uniformity Criteria for Spheres and SO(3) • Let  denote metric on a sphere • Dispersion: “radius of the largest empty ball”

9. The Outline of the Rest of the Talk • Literature on sampling over spheres • General approach for sampling over spheres • 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 • Extension of this sequence to cross product spaces and SE(3) • Properties and experimental evaluation of this sequence on the problems of motion planning

10. The Outline of the Rest of the Talk • Literature on sampling over spheres • General approach for sampling over spheres • 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 • Extension of this sequence to cross product spaces and SE(3) • Properties and experimental evaluation of this sequence on the problems of motion planning

11. Literature on sampling spheres 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 SO(3) • uniform deterministic point sets for SO(3) [Lubotzky, Phillips, Sarnak 86] [Mitchell 04] • No deterministic sequences to our knowledge

12. The Outline of the Rest of the Talk • Literature on sampling over spheres • General approach for sampling over spheres • A particular sequence (Layered Sukharev grid sequence) which: • is deterministic • achieves low dispersion and low discrepancy • is incremental • has lattice structure • can be efficiently generated • Extension of this sequence to cross product spaces and SE(3) • Properties and experimental evaluation of these sequences on the problems of motion planning

13. … 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

14. 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

15. 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

16. Example. Sukharev Grid on S1 • Take a square in R2 • Place Sukharev grid on each edge • Project the edges of the square outwards to form circle tiling • Place a Sukharev grid on each circular edge Important note: similar procedure applies for any Sd

17. 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

18. Properties of Spherical Sukharev Grids Advantages: • distortions are easy to calculate • lattice structure is beneficial for motion planning • calculations are efficient • easily extendable to sequences Disadvantages: • distortions grow with dimension

19. The Outline of the Rest of the Talk • Literature on sampling over spheres • General approach for sampling over spheres • A particular sequence (Layered Sukharev grid sequence) which: • is deterministic • achieves low dispersion and low discrepancy • is incremental • has lattice structure • can be efficiently generated • Extension of this sequence to cross product spaces and SE(3) • Properties and experimental evaluation of these sequences on the problems of motion planning

20. 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]

21. 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

22. The Outline of the Rest of the Talk • Literature on sampling over spheres • General approach for sampling over spheres • A particular sequence (Layered Sukharev grid sequence) which: • is deterministic • achieves low dispersion and low discrepancy • is incremental • has lattice structure • can be efficiently generated • Extension of this sequence to cross product spaces and SE(3) • Properties and experimental evaluation of these sequences on the problems of motion planning

23. 3 2 XY 1 4 Ordering on cells + Ordering inside cells XY Layered Sukharev Grid Sequence for XY • Take cell structure in X and Y • Define a cell structure in XY • Determine the cell ordering and the ordering inside each cell Y X

24. Layered Sukharev Grid Sequence for SE(3) • SE(3) = SO(3)  R3 • The measure can be defined as SO(3) R3 • This measure corresponds to the left-invariant Haar measure on SE(3) That is, defined construction will respect this Haar measure on SE(3)

25. The Outline of the Rest of the Talk • Literature on sampling over spheres • General approach for sampling over spheres • A particular sequence (Layered Sukharev grid sequence) which: • is deterministic • achieves low dispersion and low discrepancy • is incremental • has lattice structure • can be efficiently generated • Extension of this sequence to cross product spaces and SE(3) • Properties and experimental evaluation of these sequences on the problems of motion planning

26. 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.

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

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

29. ExperimentsPRM method • SE(3) configuration space • Averaged over 50 trials

30. ExperimentsPRM method • SE(3) configuration space • Averaged over 50 trials

31. Conclusion • We have proposed a general framework for uniform sampling over spheres, SO(3), and cross product spaces • 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 other configuration spaces