Sampling and Searching Methods for Practical Motion Planning Algorithms

1. Sampling and Searching Methods for Practical Motion Planning Algorithms Anna Yershova 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 • Research Statement • Technical Approach • Efficient Nearest Neighbor Searching • Guided Sampling for Efficient Exploration • Uniform Deterministic Sampling Methods • 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. 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

10. Research 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

11. Nearest Neighbor Searching for Motion Planning Software:http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz

12. 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

13. 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

14. 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

15. Dynamic-Domain RRTs

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

17. Voronoi Bias for the Original RRT

18. KD-Tree Bias for the RRT

19. KD-Tree Bias for the RRT

20. KD-Tree Bias for the RRT

21. Library For Generating Deterministic Sequences Of Samples Over SO(3) Software:http://msl.cs.uiuc.edu/~yershova/sampling/sampling.tar.gz

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

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

24. 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”

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

26. 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

27. 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]

28. 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

29. Motion Primitives Generation Reachability graph

30. Dubin’s Car Reachability Graph

31. 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

32. 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.

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

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

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

36. 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]

37. 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

38. 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?

39. Thank you!