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Anna Yershova Thesis Defense Dept. of Computer Science, University of Illinois August 5, 2008

Sampling and Searching Methods for Practical Motion Planning Algorithms. Anna Yershova Thesis Defense Dept. of Computer Science, University of Illinois August 5, 2008. Anna Yershova. Thesis Defense. Introduction. Presentation Overview. Introduction Motion Planning

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Anna Yershova Thesis Defense Dept. of Computer Science, University of Illinois August 5, 2008

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  1. Sampling and Searching Methodsfor Practical Motion Planning Algorithms Anna Yershova Thesis Defense Dept. of Computer Science, University of Illinois August 5, 2008 Anna Yershova Thesis Defense

  2. Introduction Presentation Overview • Introduction • Motion Planning • Incremental Sampling and Searching (ISS) Framework • Thesis Overview • Technical Contributions • Nearest Neighbor Searching • Uniform Deterministic Sampling • Guided Sampling • Conclusions and Discussion Anna Yershova Thesis Defense

  3. Introduction Motion Planning The Motion Planning Problem • Given: • , , • Initial and goal configurations • Extensions: Task: • Compute a collision free path that connects initial and goal configurations [J. Cortes] Anna Yershova Thesis Defense

  4. Introduction Motion Planning The Motion Planning Problem • Given: • , , • Initial and goal configurations • Extensions: Task: • Compute a collision free path that connects initial and goal configurations xgoal xinit Anna Yershova Thesis Defense

  5. Introduction Motion Planning The Motion Planning Problem • Conceptually simple, but in reality… • obstacles in C-spaces are not explicitly defined • they are described by an astronomical number of geometric primitives • free C-spaces have complicated topologies • feasible configurations may lie on lower dimensional algebraic varieties, which are also not explicitly defined Anna Yershova Thesis Defense

  6. Introduction Motion Planning Applications • Automotive Assembly [Yershova, et. al., 2005] Courtesy of Kineo CAM The solution path traverses a narrow passage in SE(3) Anna Yershova Thesis Defense

  7. Introduction Motion Planning Applications • Automotive Assembly • Computational Chemistryand Biology [Yershova, et. al., 2005] Courtesy of LAAS 330 dimensional C-space Anna Yershova Thesis Defense

  8. Introduction Motion Planning Applications • Automotive Assembly • Computational Chemistryand Biology • Manipulation Planning • Medical applications • Computer Graphics(motions for digital actors) • Autonomous vehicles andspacecrafts courtesy of Volvo Cars and FCC Anna Yershova Thesis Defense

  9. Introduction Motion Planning History • Grid Sampling, AI Search (beginning of time-1977) • Experimental mobile robotics, etc. • Problem Formalization (1977-1983) • Configuration space (Lozano-Perez, 1978-1981) • PSPACE-hardness (Reif, 1979) • Combinatorial Solutions (1983-1988) • Cylindrical algebraic decomposition (Schwartz, Sharir, 1983) • Stratifications, roadmap (Canny, 1987) • Sampling-based Planning (1988-present) • Randomized potential fields (Barraquand, Latombe, 1989) • Ariadne's clew algorithm (Ahuactzin, Mazer, 1992) • Probabilistic Roadmaps (PRMs) (Kavraki, Svestka, Latombe, Overmars, 1994) • Rapidly-exploring Random Trees (RRTs) (LaValle, Kuffner, 1998) Collision detection is used as a “black box” Anna Yershova Thesis Defense

  10. Introduction ISS Framework Incremental Sampling and Searching Framework Build a graph over the configuration space that connects initial and goal configurations: • Graph is embedded in C-space • Every vertex is a configuration • Every edge is a path xgoal xinit Anna Yershova Thesis Defense

  11. Introduction ISS Framework Typical Architecture Uniform Sampling Guided Sampling Input geometry Solution path Nearest Neighbor Search Collision Detection yes no Path Exists ? Anna Yershova Thesis Defense

  12. Introduction Thesis Overview Central Theme The performance of motion planning algorithms can be significantly improved by identifying and addressing the key issues in sampling and searching framework. • ISSUES ADDRESSED: • efficient nearest-neighbor computations • uniform deterministic sampling over configuration spaces • guided sampling for efficient exploration Thesis Overview: Chapter 1: Introduction Chapter 2: ISS Framework Chapter 3: NearestNeighbor Search Chapter 4: Uniform Sampling Chapter 5: Guided Sampling Anna Yershova Thesis Defense

  13. Technical Approach Nearest Neighbor Search Presentation Overview • Introduction • Motion Planning • ISS Framework • Thesis Overview • Technical Contributions • Nearest Neighbor Searching • Uniform Deterministic Sampling • Guided Sampling • Conclusions and Discussion Anna Yershova Thesis Defense

  14. Technical Approach Nearest Neighbor Search Motivation ISS methods often compute the nearest vertex in the graph q Anna Yershova Thesis Defense

  15. Technical Approach Nearest Neighbor Search Problem Formulation Given: a d-dimensional manifold, T, and a set of data points in T Goal: preprocess these points so that, for any query point q inT, the nearest data point to q can be found quickly Manifolds of interest: • Euclidean space, [0,1]d • Spheres, Sd • Projective space, R P3 • Cartesian products of the above Anna Yershova Thesis Defense

  16. Technical Approach Nearest Neighbor Search Problem Formulation Given: a d-dimensional manifold, T, and a set of data points in T Goal: preprocess these points so that, for any query point q inT, the nearest data point to q can be found quickly Manifolds of interest: Hypercube embedded in R d with Euclidean metric • Euclidean space, [0,1]d • Hyperspheres, Sd • Projective space, R P3 • Cartesian products of the above Anna Yershova Thesis Defense

  17. Technical Approach Nearest Neighbor Search Problem Formulation Given: a d-dimensional manifold, T, and a set of data points in T Goal: preprocess these points so that, for any query point q inT, the nearest data point to q can be found quickly Manifolds of interest: • Euclidean space, [0,1]d • Hyperspheres, Sd • Projective space, R P3 • Cartesian products of the above d-sphere embedded in R d+1 with induced metric Anna Yershova Thesis Defense

  18. Technical Approach Nearest Neighbor Search Problem Formulation Given: a d-dimensional manifold, T, and a set of data points in T Goal: preprocess these points so that, for any query point q inT, the nearest data point to q can be found quickly Manifolds of interest: • Euclidean space, [0,1]d • Hyperspheres, Sd • Projective space, R P3 • Cartesian products of the above , metric compatible with Haar measure Anna Yershova Thesis Defense

  19. Technical Approach Nearest Neighbor Search Problem Formulation Given: a d-dimensional manifold, T, and a set of data points in T Goal: preprocess these points so that, for any query point q inT, the nearest data point to q can be found quickly Manifolds of interest: • Euclidean space, [0,1]d • Hyperspheres, Sd • Projective space, R P3 • Cartesian products of the above weighed metric Anna Yershova Thesis Defense

  20. Technical Approach Nearest Neighbor Search 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 5 5 5 5 5 5 5 5 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 11 11 11 11 11 11 11 11 Example: Torus, S1xS1 4 6 7 q 8 5 9 10 3 2 1 11 Universal cover of torus allows visualization of the nearest neighbor search Anna Yershova Thesis Defense

  21. Technical Approach Nearest Neighbor Search Literature review Euclidean spaces: [Friedman, 77] [Sproull, 91] [Arya, 93] [Agarwal, 02] [indyk, 04] The most successful method used in practice is based on kd-trees [Arya 93] General metric spaces: Consider metric as a “black box” [Clarkson, 03,05] [Beygelzimer, 04][Krauthgamer, 04] [Hjaltason, 03] The spaces we consider are manifolds, i.e. locally Euclidean, with identifications on the boundary.This allows extension of kd-trees. Anna Yershova Thesis Defense

  22. Technical Approach Nearest Neighbor Search 4 6 l1 l9 7 l5 l6 8 l3 l2 5 9 10 3 l10 l8 l7 2 5 4 11 8 2 1 l4 11 1 3 9 10 6 7 Kd-trees for [0,1]d The kd-tree is a data structure based on recursively subdividing a set of points with alternating axis-aligned hyperplanes. l1 l3 l2 l4 l5 l7 l6 l8 l10 l9 Anna Yershova Thesis Defense

  23. Technical Approach Nearest Neighbor Search l1 2 5 4 11 8 q 1 3 9 10 6 7 Query phase for [0,1]2 4 6 l9 7 l5 l6 l1 8 l3 l2 5 l3 l2 9 10 3 l10 l8 l7 l4 l5 l7 l6 2 1 l4 11 l8 l10 l9 Anna Yershova Thesis Defense

  24. Technical Approach Nearest Neighbor Search l1 4 6 l9 7 l5 l6 8 l3 l2 5 9 10 3 l10 l8 l7 2 1 l4 11 2 5 4 11 8 1 3 9 10 6 7 Kd-trees with modified metric Main idea: construction: unchanged procedure query: modify metric between the query point and enclosing rectangles in the kd-tree l1 l3 l2 [0,1]xS1 l4 l5 l7 l6 l8 l10 l9 Anna Yershova Thesis Defense

  25. Technical Approach Nearest Neighbor Search l1 l2 4 6 l9 7 l5 3 l6 l8 8 l3 l2 5 1 l4 9 10 3 l10 l8 l7 2 1 l4 11 2 5 4 11 8 q 1 3 9 10 6 7 Query phase with modified metric l1 l3 l2 [0,1]xS1 l4 l5 l7 l6 l8 l10 l9 Anna Yershova Thesis Defense

  26. Technical Approach Nearest Neighbor Search Analysis of the algorithm Proposition 1. The algorithm correctly returns the nearest neighbor. Proof idea: The points of kd-tree not visited by an algorithm will always be farther from the query point than some point already visited. Proposition 2. For n points in dimension d, the construction time is O(dnlgn), the space is O(dn), and the query time is logarithmic in n, but exponential in d. Proof idea: This follows directly from the well-known complexity of the basic kd-tree. Anna Yershova Thesis Defense

  27. Technical Approach Nearest Neighbor Search Experiments For 50,000 data points, 100 queries were made: Anna Yershova Thesis Defense

  28. Technical Approach Nearest Neighbor Search Experiments Anna Yershova Thesis Defense

  29. Technical Approach Nearest Neighbor Search Outcomes Publications: • Improving Motion Planning Algorithms by Efficient Nearest Neighbor Searching Anna Yershova and Steven M. LaValleIEEE Transactions on Robotics 23(1):151-157, February 2007 Publicly available library:http://msl.cs.uiuc.edu/~yershova/mpnn/mpnn.tar.gz • Also implemented in Move3D at LAAS, and KineoWorksTM Anna Yershova Thesis Defense

  30. Technical Approach Uniform Deterministic Sampling Presentation Overview • Introduction • Motion Planning • ISS Framework • Thesis Overview • Technical Contributions • Nearest Neighbor Searching • Uniform Deterministic Sampling (partly in collaboration with Julie C. Mitchell) • Guided Sampling • Conclusions and Discussion Anna Yershova Thesis Defense

  31. Technical Approach Uniform Deterministic Sampling Motivation The graph over C-space should capture the “path connectivity” of the space Anna Yershova Thesis Defense

  32. Technical Approach Uniform Deterministic Sampling Problem Formulation Desirable properties of samples over the C-space: • uniform • deterministic • incremental • grid structure Anna Yershova Thesis Defense

  33. Technical Approach Uniform Deterministic Sampling Problem Formulation Desirable properties of samples over the C-space: Dispersion: the radius of the largest empty balls Discrepancy:maximum volume estimation error • uniform • deterministic • incremental • grid structure Anna Yershova Thesis Defense

  34. Technical Approach Uniform Deterministic Sampling Problem Formulation Desirable properties of samples over the C-space: • uniform • deterministic • incremental • grid structure The uniformity measures can be deterministically computed Reason: resolution completeness Anna Yershova Thesis Defense

  35. Technical Approach Uniform Deterministic Sampling Problem Formulation Desirable properties of samples over the C-space: • uniform • deterministic • incremental • grid structure 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 Thesis Defense

  36. Technical Approach Uniform Deterministic Sampling Problem Formulation Desirable properties of samples over the C-space: • uniform • deterministic • incremental • grid structure Reason: Trivializes nearest neighbor computations Anna Yershova Thesis Defense

  37. Technical Approach Uniform Deterministic Sampling Problem Formulation Desirable properties of samples over the C-space: • Euclidean space, [0,1]d • Spheres, Sd • Projective space, R P3 • Cartesian products of the above • uniform • deterministic • incremental • grid structure Anna Yershova Thesis Defense

  38. Technical Approach Uniform Deterministic Sampling Literature overview • Euclidean space, [0,1]d • Spheres, Sd • Special orthogonal group, SO(3) Anna Yershova Thesis Defense

  39. Technical Approach Uniform Deterministic Sampling 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 Thesis Defense

  40. Technical Approach Uniform Deterministic Sampling Literature Overview: Euclidean Spaces, [0,1]d Layered Sukharev Grid Sequence [Lindemann, LaValle 2003] + uniform + deterministic + incremental + grid structure Anna Yershova Thesis Defense

  41. Technical Approach Uniform Deterministic Sampling 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 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 Thesis Defense

  42. Technical Approach Uniform Deterministic Sampling Our 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 Ordering on faces + Ordering inside faces Anna Yershova Thesis Defense

  43. Technical Approach Uniform Deterministic Sampling 3 2 XY 1 4 XY Ordering on cells, Ordering inside cells Our 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 Anna Yershova Thesis Defense

  44. Technical Approach Uniform Deterministic Sampling Our approach: SO(3) • Hopf coordinates preserve the fiber bundle structure of RP3 • Locally, RP3 is a product of S1 and S2 Joint work with J.C.Mitchell Anna Yershova Thesis Defense

  45. Technical Approach Uniform Deterministic Sampling Our approach:SO(3) • The method for Cartesian products can then be applied to R P3 • Need two grids, for S1 and S2 Grid on S2 Grid on S1 Healpix, [Gorski,05] Anna Yershova Thesis Defense

  46. Technical Approach Uniform Deterministic Sampling Our approach:SO(3) • The method for Cartesian products can then be applied to R P3 • Need two grids, for S1 and S2 Grid on S1 Grid on S2 Anna Yershova Thesis Defense

  47. Technical Approach Uniform Deterministic Sampling Our approach:SO(3) • The method for Cartesian products can then be applied to R P3 • Need two grids, for S1 and S2 • Ordering on faces, ordering on [0,1]3 + uniform + deterministic + incremental + grid structure Grid on S1 Grid on S2 Anna Yershova Thesis Defense

  48. Technical Approach Uniform Deterministic Sampling Propositions • 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. Anna Yershova Thesis Defense

  49. Technical Approach Uniform Deterministic Sampling Propositions • The dispersion of the sequence Tat the resolution level lis: in which is the dispersion of the sequence over S2. Anna Yershova Thesis Defense

  50. Technical Approach Uniform Deterministic Sampling Experiments • Configuration spaces: SO(3) and SE(3) = R3 x SO(3) (a) (b) Anna Yershova Thesis Defense

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