Randomized Motion Planning JeanClaude Latombe Computer Science Department
Randomized Motion Planning Jean-Claude Latombe Computer Science Department Stanford University
Goal of Motion Planning u u u 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
Outline u Bits of history u Approaches u Probabilistic Roadmaps u Applications u Conclusion
Early Work Shakey (Nilsson, 1969): Visibility graph
Mathematical Foundations Lozano-Perez, 1980: Configuration Space C = S 1 x S 1
Computational Analysis Reif, 1979: Hardness (lower-bound results)
Exact General-Purpose Path Planners - Schwarz and Sharir, 1983: Exact cell decomposition based on Collins technique - Canny, 1987: Silhouette method
Heuristic Planners Khatib, 1986: Potential Fields
Other Types of Constraints E. g. , Visibility-Based Motion Planning Guibas, Latombe, La. Valle, Lin, and Motwani, 1997
Outline u Bits of history u Approaches u Probabilistic Roadmaps u Applications u Conclusion
Criticality-Based Motion Planning u u u 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
Criticality-Based Motion Planning u Advantages: § Completeness § Insight u Drawbacks: § Computational complexity § Difficult to implement
Sampling-Based Motion Planning u Principle: § Sample the space of interest § Connect sampled points by simple paths § Search the resulting graph u Example: Probabilistic Roadmaps (PRM’s) u Other example: Grid-based methods (deterministic sampling)
Sampling-Based Motion Planning u Advantages: – Easy to implement – Fast, scalable to many degrees of freedom and complex constraints u Drawbacks: – Probabilistic completeness – Limited insight
Outline u Bits of history u Approaches u Probabilistic Roadmaps u Applications u Conclusion
Motivation Computing an explicit representation of the admissible space is hard, but checking that a point lies in the admissible space is fast
Probabilistic Roadmap (PRM) admissible space milestone mg mb [Kavraki, Svetska, Latombe, Overmars, 95]
Sampling Strategies u Multi vs. single query strategies u Multi-stage strategies u Obstacle-sensitive strategies u Lazy collision checking u Probabilistic biases (e. g. , potential fields)
PRM With Dynamic Constraints in State x Time Space m’ = f(m, u) endgame region mg mb [Hsu, Kindel, Latombe, and Rock, 2000]
Relation to Art-Gallery Problems [Kavraki, Latombe, Motwani, Raghavan, 95]
Narrow Passage Issue
Desirable Properties of a PRM u u 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
Complexity Measures u e-goodness [Kavraki, Latombe, Motwani, and Raghavan, 1995] u Path clearance [Kavraki, Koulountzakis, and Latombe, 1996] u e-complexity [Overmars and Svetska, 1998] u Expansiveness [Hsu, Latombe, and Motwani, 1997]
Expansiveness of Admissible Space
Expansiveness of Admissible Space The admissible space is expansive if each of its subsets has a large lookout Lookout of F 1 Prob[failure] = K exp(-r)
Two Very Different Cases Poorly expansive Expansive
A Few Remarks u u u 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
Outline u Bits of history u Approaches u Probabilistic Roadmaps u Applications u Conclusion
Design for Manufacturing and Servicing General Motors General Electric [Hsu, 2000]
Robot Programming and Placement [Hsu, 2000]
Graphic Animation of Digital Actors The Motion Factory [Koga, Kondo, Kuffner, and Latombe, 1994]
Digital Actors With Visual Sensing Simulated Vision Kuffner, 1999 u Segment environment u Render false-color scene offscreen u Scan pixels & record IDs Actor camera image Vision module image
Humanoid Robot [Kuffner and Inoue, 2000] (U. Tokyo)
Space Robotics robot obstacles air thrusters gaz tank air bearing [Kindel, Hsu, Latombe, and Rock, 2000]
Total duration : 40 sec
Autonomous Helicopter [Feron, 2000] (AA Dept. , MIT)
Interacting Nonholonomic Robots q 2 y 2 d q 1 y 1 x 1 (Grasp Lab - U. Penn) x 2
Map Building [Gonzalez, 2000]
Next-Best View Computation
Map Building [Gonzalez, 2000]
Map Building [Gonzalez, 2000]
Radiosurgical Planning Cyberknife System (Accuray, Inc. ) CARABEAMER Planner [Tombropoulos, Adler, and Latombe, 1997]
Radiosurgical Planning • 2000 < Tumor < 2200 T B 1 B 2 2000 < B 2 + B 4 < 2200 2000 < B 3 < 2200 2000 < B 1 + B 3 + B 4 < 2200 2000 < B 1 + B 2 < 2200 C B 3 B 4 • 0 < Critical < 500 0 < B 2 < 500
Sample Case 50% Isodose Surface 80% Isodose Surface Conventional system’s plan CARABEAMER’s plan
Reconfiguration Planning for Modular Robots Casal and Yim, 1999 Xerox, Parc
Prediction of Molecular Motions Ligand-protein binding Protein folding [Singh, Latombe, and Brutlag, 1999] [Apaydin, 2000]
Capturing Energy Landscape [Apaydin, 2000]
Outline u Bits of history u Approaches u Probabilistic Roadmaps u Applications u Conclusion
Conclusion u u 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
Issues u u u Relatively large standard deviation of planning time No rigorous termination criterion when no solution is found New challenging applications …
Planning Minimally Invasive Surgery Procedures Amidst Soft-Tissue Structures
Planning Nice-Looking Motions for Digital Actors A Bug’s Life (Pixar/Disney) Tomb Raider 3 (Eidos Interactive) Toy Story (Pixar/Disney) The Legend of Zelda (Nintendo) Antz (Dreamworks) Final Fantasy VIII (Square. One)
Dealing with 1, 000 s of Degrees of Freedom Protein folding
Main Common Difficulty Formulating motion constraints
- Slides: 55