1. An Efficient Motion PlannerBased on Random Sampling Jean-Claude Latombe Computer Science DepartmentStanford University

2. Main Collaborators • Lydia Kavraki (Rice U.) • David Hsu (U. of North Carolina, Chapel Hill) • Gildardo Sanchez (ITESM, Mexico) • James Kuffner (U. of Tokyo) • Rajeev Motwani (Stanford U.)

3. Goal of Motion Planning Answer queries about the connectivity of a space

4. Collision-free Kino-dynamic Stability Visibility Possible Constraints

5. The Beginning … Shakey (Nilsson, 1969): Visibility graph

6. Configuration Space Represent the robot as a point in a parameter space

7. Why Sampling-Based Planning? • Computing an explicit representation of the collision-free space is extremely time consuming and impractical • There exist fast collision-checking algorithms to test whether any given configuration or short path is collision-free, or not (0.001 sec or less)

8. Outline • General Approach • Specific Planner • Experimental Results • Other Applications

9. milestone mg mb Probabilistic Roadmap (PRM) admissible space [Kavraki, Svetska, Latombe,Overmars, 95]

10. Relation to Art-Gallery Problems [Kavraki, Latombe, Motwani, Raghavan, 95]

11. Easy Narrow Passage Issue Difficult

12. Probabilistic completeness  Fast and reliable Probabilistic Completeness Under generally satisfied assumptions, if a solution path exists, the probability that a PRM planner fails to find one goes to 0 exponentially in the number of milestones. Full completeness  Too costly Heuristic  Too unreliable

13. Key Techniques • Collision checking / Distance computation • Sampling strategies

14. Key Techniques • Collision checking / Distance computation • Hierarchical approach • Feature-based approach • Sampling strategies

15. Hierarchical Collision Checking

16. Three-Dimensional Case

17. Collision Checking

18. Collision Checking

19. Performance • Collision checking takes between 0.0001 and .002 seconds for 2 objects of 500,000 triangles each on a 1-GHz Pentium III • Collision checking is faster when objects collide or are far apart, and gets slower when they get closer without colliding • Overall collision checking time grows roughly as the log of the number of triangles

20. Key Techniques • Collision checking / Distance computation • Sampling strategies • Multi-stage strategies • Obstacle-sensitive strategies • Multiple vs. single query strategies • Configuration vs. control sampling • Single vs. bi-directional sampling • Lazy collision checking • Probabilistic biases (e.g., medial axis transform)

21. Outline • General Approach • Specific Planner • Experimental Results • Other Applications

22. SBL Planner • Single-query Does not pre-compute a roadmap [Hsu, Latombe, Motwani, 1997] • Bi-directional sampling Constructs a roadmap by growing two trees of milestones rooted at the input query configuration [Hsu, 2000] • Lazy collision checking Postpone collision-checking operations until absolutely needed [Bohlin and Kavraki, 2000]

23. SBL Planner

24. SBL Planner m m is picked at random among the milestones with a probabilistic distribution inverse to the local density of sampling

25. SBL Planner

26. SBL Planner

27. SBL Planner

28. SBL Planner X

29. SBL Planner The collision-checking work is memorized

30. Why Postponing Collision Checking? • The a priori probability that a short edge be collision-free is rather large

31. Why Postponing Collision Checking? • The a priori probability that a short edge be collision-free is rather large • The test of an edge is most expensive when it is actually collision-free • Most edges of a roadmap do not end up in a solution path

32. Remedy • remove as many vertices as possible • add vertices as needed • Problems • too few vertices: get stuck • too many vertices: slow Path Optimization

33. Outline • General Approach • Specific Planner • Experimental Results • Other Applications

34. Single-Robot Examples nrob = 3,000 and nobs = 50,000 nrob = 5,000 and nobs = 21,000 nrob = 5,000; nobs = 83,000 nrob = 3,000; nobs = 50 nrob = 3,000 and nobs = 100

35. Videos nrobot =5,000; nobst = 21,000 Tav = 0.6 s

36. Videos nrobot =3,000; nobst = 50,000 Tav = 0.17 s nrobot =5,000; nobst = 83,000 Tav = 4.42 s

37. Videos nrobot =3,000; nobst = 100 Tav = 6.99 s nrobot =3,000; nobst = 50,000 Tav = 4.45 s

38. Experimental Data on One Example nrob = 5,000 nobs = 21,000 (1 GHz Pentium III processor)

39. 1e 1d 1c 1b 1a Average Performance Averages over 100 runs (1GHz Pentium III processor)

40. Convergence of SBL

41. Impact of Lazy Collision Checking Average performance with lazy collision checking Average performance without lazy collision checking

42. Multi-Robot Spot Welding

43. Typical Problem

44. Video

45. Average Running Times (1 GHz processor)

46. Centralized vs. Decoupled Planning Averages over 20 runs

47. Outline • General Approach • Specific Planner • Experimental Results • Other Applications

48. Design for Manufacturing/Servicing General Motors General Motors General Electric [Hsu, 2000]

49. Radio-Surgical Planning Cyberknife System (Accuray, Inc.) CARABEAMER Planner [Tombropoulos, Adler, and Latombe, 1997] Visibility constraints

50. •2000 < Tumor < 2200 • 2000 < B2 + B4 < 2200 • 2000 < B4 < 2200 • 2000 < B3 + B4 < 2200 • 2000 < B3 < 2200 • 2000 < B1 + B3 + B4 < 2200 • 2000 < B1 + B4 < 2200 • 2000 < B1 + B2 + B4 < 2200 • 2000 < B1 < 2200 • 2000 < B1 + B2 < 2200 T T B1 C B2 B4 • •0 < Critical < 500 • 0 < B2 < 500 B3 Radio-Surgical Planning