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Randomized Motion Planning

Randomized Motion Planning. Jean-Claude Latombe Computer Science Department Stanford University. Goal of Motion Planning. Answer queries about connectivity of a space Classical example: find a collision-free path in robot configuration space among static obstacles

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Randomized Motion Planning

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  1. Randomized Motion Planning Jean-Claude Latombe Computer Science DepartmentStanford University

  2. Goal of Motion Planning • Answer queries about connectivity of a space • Classical example: find a collision-free path in robot configuration space among static obstacles • Examples of additional constraints: • Kinodynamic constraints • Visibility constraints

  3. Outline • Bits of history • Approaches • Probabilistic Roadmaps • Applications • Conclusion

  4. Early Work Shakey (Nilsson, 1969): Visibility graph

  5. C = S1 x S1 Mathematical Foundations Lozano-Perez, 1980: Configuration Space

  6. Computational Analysis Reif, 1979: Hardness (lower-bound results)

  7. Exact General-Purpose Path Planners - Schwarz and Sharir, 1983: Exact cell decomposition based on Collins technique - Canny, 1987: Silhouette method

  8. Heuristic Planners Khatib, 1986: Potential Fields

  9. Other Types of Constraints E.g., Visibility-Based Motion Planning Guibas, Latombe, LaValle, Lin, and Motwani, 1997

  10. Outline • Bits of history • Approaches • Probabilistic Roadmaps • Applications • Conclusion

  11. Criticality-Based Motion Planning • Principle: • Select a property P over the space of interest • Compute an arrangement of cells such that P stays constant over each cell • Build a search graph based on this arrangement • Example: Wilson’s Non-Directional Blocking Graphs for assembly planning • Other examples: • Schwartz-Sharir’s cell decomposition • Canny’s roadmap

  12. Criticality-Based Motion Planning • Advantages: • Completeness • Insight • Drawbacks: • Computational complexity • Difficult to implement

  13. Sampling-Based Motion Planning • Principle: • Sample the space of interest • Connect sampled points by simple paths • Search the resulting graph • Example:Probabilistic Roadmaps (PRM’s) • Other example:Grid-based methods (deterministic sampling)

  14. Sampling-Based Motion Planning • Advantages: • Easy to implement • Fast, scalable to many degrees of freedom and complex constraints • Drawbacks: • Probabilistic completeness • Limited insight

  15. Outline • Bits of history • Approaches • Probabilistic Roadmaps • Applications • Conclusion

  16. Motivation Computing an explicit representation of the admissible space is hard, but checking that a point lies in the admissible space is fast

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

  18. Sampling Strategies • Multi vs. single query strategies • Multi-stage strategies • Obstacle-sensitive strategies • Lazy collision checking • Probabilistic biases (e.g., potential fields)

  19. endgame region m’ = f(m,u) mg mb PRM With Dynamic Constraints in State x Time Space [Hsu, Kindel, Latombe, and Rock, 2000]

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

  21. Narrow Passage Issue

  22. Desirable Properties of a PRM • Coverage:The milestones should see most of the admissible space to guarantee that the initial and goal configurations can be easily connected to the roadmap • Connectivity:There should be a 1-to-1 map between the components of the admissible space and those of the roadmap

  23. Complexity Measures • e-goodness[Kavraki, Latombe, Motwani, and Raghavan, 1995] • Path clearance[Kavraki, Koulountzakis, and Latombe, 1996] • e-complexity[Overmars and Svetska, 1998] • Expansiveness[Hsu, Latombe, and Motwani, 1997]

  24. Expansiveness of Admissible Space

  25. Lookout of F1 Prob[failure] = K exp(-r) Expansiveness of Admissible Space The admissible space is expansive if each of its subsets has a large lookout

  26. Expansive Poorly expansive Two Very Different Cases

  27. A Few Remarks • Big computational saving is achieved at the cost of slightly reduced completeness • Computational complexity is a function of the shape of the admissible space, not the size needed to describe it • Randomization is not really needed; it is a convenient incremental scheme

  28. Outline • Bits of history • Approaches • Probabilistic Roadmaps • Applications • Conclusion

  29. Design for Manufacturing and Servicing General Motors General Motors General Electric [Hsu, 2000]

  30. Robot Programming and Placement [Hsu, 2000]

  31. Graphic Animation of Digital Actors The MotionFactory [Koga, Kondo, Kuffner, and Latombe, 1994]

  32. Digital Actors With Visual Sensing Simulated Vision Kuffner, 1999 • Segment environment • Render false-color scene offscreen • Scan pixels & record IDs Actor camera image Vision module image

  33. Humanoid Robot [Kuffner and Inoue, 2000] (U. Tokyo)

  34. Space Robotics robot obstacles air thrusters gaz tank air bearing [Kindel, Hsu, Latombe, and Rock, 2000]

  35. Total duration : 40 sec

  36. Autonomous Helicopter [Feron, 2000] (AA Dept., MIT)

  37. y2 q2 (Grasp Lab - U. Penn) d q1 y1 x2 x1 Interacting Nonholonomic Robots

  38. Map Building [Gonzalez, 2000]

  39. Next-Best View Computation

  40. Map Building [Gonzalez, 2000]

  41. Map Building [Gonzalez, 2000]

  42. Radiosurgical Planning Cyberknife System (Accuray, Inc.) CARABEAMER Planner [Tombropoulos, Adler, and Latombe, 1997]

  43. •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 Radiosurgical Planning

  44. Sample Case 50% Isodose Surface 80% Isodose Surface Conventional system’s plan CARABEAMER’s plan

  45. Reconfiguration Planning for Modular Robots Casal and Yim, 1999 Xerox, Parc

  46. Prediction of Molecular Motions Protein folding Ligand-protein binding [Apaydin, 2000] [Singh, Latombe, and Brutlag, 1999]

  47. Capturing Energy Landscape [Apaydin, 2000]

  48. Outline • Bits of history • Approaches • Probabilistic Roadmaps • Applications • Conclusion

  49. Conclusion • PRM planners have successfully solved many diverse complex motion problems with different constraints (obstacles, kinematics, dynamics, stability, visibility, energetic) • They are easy to implement • Fast convergence has been formally proven in expansive spaces. As computers get more powerful, PRM planners should allow us to solve considerably more difficult problems • Recent implementations solve difficult problems with many degrees of freedom at quasi-interactive rate

  50. Issues • Relatively large standard deviation of planning time • No rigorous termination criterion when no solution is found • New challenging applications…

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