Sampling and Searching Methods for Practical Motion Planning Algorithms

1. 29 August 2007 Sampling and Searching Methods for Practical Motion Planning Algorithms Anna Yershova PhD Preliminary Examination Dept. of Computer Science University of Illinois

2. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

3. Basic Motion Planning Problem”Moving Pianos” Given: • (geometric model of a robot) • (space of configurations, q, thatare applicable to ) • (the set of collision freeconfigurations) • Initial and goal configurations Task: • Compute a collision free path that connects initial and goal configurations

4. Extensions of Basic Motion Planning Problem Given: • , , • (kinematic closure constraints) • Initial and goal configurations Task: • Compute a collision free path that connects initial and goal configurations

5. Motion Planning Problemunder Differential Constraints Given: • , , • State space X • Input space U • state transition equation • Initial and goal states Task: • Compute a collision free path that connects initial and goal states

6. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

7. History of Motion Planning • Grid Sampling, AI Search (beginning of time-1977) • Experimental mobile robotics, etc. • Problem Formalization (1977-1983) • PSPACE-hardness (Reif, 1979) • Configuration space (Lozano-Perez, 1981) • 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)

8. Applications of Motion Planning • Manipulation Planning • Computational Chemistryand Biology • Medical applications • Computer Graphics(motions for digital actors) • Autonomous vehicles and spacecrafts

9. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

10. Sampling and Searching Framework Build a graph over the state (configuration) space that connects initial state to the goal: • INITIALIZATION • SELECTION METHOD • LOCAL PLANNING METHOD • INSERT AN EDGE IN THE GRAPH • CHECK FOR SOLUTION • RETURN TO STEP 2 xbest xnew xinit

11. Thesis Statement The performance of motion planning algorithms can be significantly improved by careful consideration of sampling issues. ADDRESSED ISSUES: STEP 2: nearest neighbor computation STEP 2: uniform sampling over configuration space STEPS 2,3:guided sampling for exploration STEP 3: motion primitives generation

12. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

13. State of Progress 100%Efficient Nearest Neighbor Searching 85% Uniform Deterministic Sampling Methods 75% Guided Sampling for Efficient Exploration 20% Motion Primitives Generation

14. MPNN: Nearest Neighbor Library For Motion Planning 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 • Efficient Nearest Neighbor Searching for Motion PlanningAnna Yershova and Steven M. LaValleIn Proc. IEEE International Conference on Robotics and Automation (ICRA 2002) Software:http://msl.cs.uiuc.edu/~yershova/mpnn/mpnn.tar.gz

15. Problem Formulation Given a d-dimensional manifold, T, and a set of data points in T. Preprocess these points so that, for any query point qT, the nearest data point to q can be found quickly. The manifolds of interest: • Euclidean one-space, represented by (0,1)  R . • Circle, represented by [0,1], in which 0  1 by identification. • P3, represented by S3 with antipodal points identified. Examples of topological spaces: cylinder torus projective plane

16. 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: a torus 4 6 7 q 8 5 9 10 3 2 1 11

17. 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 The kd-tree is a powerful data structure that is based on recursively subdividing a set of points with alternating axis-aligned hyperplanes. The classical kd-tree uses O(dnlgn) precomputation time, O(dn) space and answers queries in time logarithmic in n, but exponential in d. l1 l3 l2 l4 l5 l7 l6 l8 l10 l9

18. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

19. Library For Generating Deterministic Sequences Of Samples Over SO(3) Publications: • Deterministic sampling methods for spheres and SO(3) Anna Yershova and Steven M. LaValle,2004 IEEE International Conference on Robotics and Automation (ICRA 2004) • Incremental Grid Sampling Strategies in Robotics Stephen R. Lindemann, Anna Yershova, and Steven M. LaValle,Sixth International Workshop on the Algorithmic Foundations of Robotics (WAFR 2004) Software:http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz

20. Random Samples Halton sequence A Spectrum of Roadmaps Hammersley Points Lattice Grid

21. What uniformity criteria are best suited for Motion Planning Which of the roadmaps alone the spectrum is best suited for Motion Planning? Questions

22. Measuring the (Lack of) Quality • Let R (range space) denote a collection of subsets of a sphere • Discrepancy: “maximum volume estimation error over all boxes”

23. Measuring the (Lack of) Quality • Let  denote metric on a sphere • Dispersion: “radius of the largest empty ball”

24. The Goal for Motion Planning • We want to develop sampling schemes with the following properties: • uniform (low dispersion or discrepancy) • lattice structure • incremental quality (it should be a sequence) • on the configuration spaces with different topologies

25. Layered Sukharev Grid Sequencein [0, 1]d • Places Sukharev grids one resolution at a time • Achieves low dispersion at each resolution • Achieves low discrepancy • Has explicit neighborhoodstructure [Lindemann, LaValle 2003]

26. 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

27. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

28. Dynamic-Domain RRTs Publications: • Planning for closed chains without inverse kinematicsAnna Yershova and Steven M. LaValle, To be submitted to ICRA 2008 • Adaptive Tuning of the Sampling Domain for Dynamic-Domain RRTsL. Jaillet, A. Yershova, S. M. LaValle and T. Simeon, In Proc. IEEE International Conference on Intelligent Robots and Systems (IROS 2005) • Dynamic-Domain RRTs: Efficient Exploration by Controlling the Sampling DomainA. Yershova, L. Jaillet, T. Simeon, and S. M. LaValle, In Proc. IEEE International Conference on Robotics and Automation (ICRA 2005)

29. Bug Trap Which one will perform better? Small Bounding Box Large Bounding Box

30. Voronoi Bias for the Original RRT

31. KD-Tree Bias for the RRT

32. KD-Tree Bias for the RRT

33. KD-Tree Bias for the RRT

34. Presentation Overview • Motion Planning Problem • Basic Motion Planning Problem • Extensions of Basic Motion Planning • Motion Planning under Differential Constraints • State of the Art • Thesis Statement • Technical Approach • Efficient Nearest Neighbor Searching • Uniform Deterministic Sampling Methods • Guided Sampling for Efficient Exploration • Motion Primitives Generation • Conclusions and Discussion

35. Motion Primitives Generation Reachability graph

36. Dubin’s Car Reachability Graph

37. Motion Primitives Generation Numerical integration can be costly for complex control models. In several works it has been demonstrated that the performance of motion planning algorithms can be improved by orders of magnitude by having good motion primitives

38. Motion Primitives Generation Motivating example 1: Autonomous Behaviors for Interactive Vehicle Animations Jared Go, Thuc D. Vu, James J. Kuffner Generated spacecraft trajectories in a field of moving asteroid obstacles.

39. Motion Primitives Generation Criteria: • Hand-picked “pleasing to the eye” trajectories • Efficient performance of the online planner

40. Motion Primitives Generation Motivating example 2: Optimal, Smooth, Nonholonomic Mobile Robot Motion Planning in State Lattices M. Pivtoraiko, R.A. Knepper, and A. Kelly

41. Motion Primitives Generation The controls are chosen to reach the points on the state lattice Criteria: • Well separated trajectories • Efficiency in performance

42. Motivational Literature • Robotics literature: [Kehoe, Watkins, Lind 2006] [Anderson, Srinivasa 2006] [Pivtoraiko, Knepper, Kelly 2006] [Green, Kelly 2006] [Go, Vu, Kuffner 2004] [Frazzoli, Dahleh, Feron 2001] • Motion Capture literature [Laumond, Hicheur, Berthoz 2005] [Gleicher]

43. Proposed problem • Formulate the criteria of “goodness” for motion primitives in the context of Motion Planning • Automatically generate the motion primitives • Propose Efficient Motion Planning algorithms using the motion primitives

44. Things to investigate: • Dispersion, discrepancy in state space? • In trajectory space? • Robustness with respect to the obstacles? • Complexity of the set of trajectories? • Is it extendable to second order systems?

45. Thank you!

46. Appendix

47. 4 6 l1 7 8 5 9 10 3 2 1 11 2 5 4 11 8 1 3 9 10 6 7 Kd-trees. Construction l9 l1 l5 l6 l3 l2 l3 l2 l10 l4 l5 l7 l6 l8 l7 l4 l8 l10 l9

48. l1 q 2 5 4 11 8 1 3 9 10 6 7 Kd-trees. Query 4 6 l9 l1 7 l5 l6 l3 8 l2 l3 l2 5 9 10 3 l10 l4 l5 l7 l6 l8 l7 2 1 l4 11 l8 l10 l9

49. l1 l3 l2 l4 l5 l7 l6 l8 l10 l9 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 q 2 5 4 11 8 1 3 9 10 6 7 Algorithm Presentation

50. 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 further from the query point then 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.