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