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 Examples of additional constraints: Kinodynamic constraints Visibility constraints

Outline Bits of history Approaches Probabilistic Roadmaps Applications Conclusion

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

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

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, LaValle, Lin, and Motwani, 1997

Outline Bits of history Approaches Probabilistic Roadmaps Applications Conclusion

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

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

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)

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

Outline Bits of history Approaches Probabilistic Roadmaps Applications 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 mb mg [Kavraki, Svetska, Latombe,Overmars, 95]

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

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

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

Narrow Passage Issue

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

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]

Expansiveness of Admissible Space

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

Two Very Different Cases Expansive Poorly expansive

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

Outline Bits of history Approaches Probabilistic Roadmaps Applications Conclusion

Design for Manufacturing and Servicing General Motors 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 Segment environment Render false-color scene offscreen Scan pixels & record IDs Actor camera image Vision module image

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

[Kindel, Hsu, Latombe, and Rock, 2000] 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 y1 x2 d x1 y2 q1 q2 (Grasp Lab - U. Penn)

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 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 • 0 < Critical < 500 0 < B2 < 500 T C B1 B2 B3 B4

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 Bits of history Approaches Probabilistic Roadmaps Applications Conclusion

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

Issues 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) Toy Story (Pixar/Disney) Antz (Dreamworks) The Artist’s perspective on animated characters Animation Variables (AVARS) FEATURES: Kinematic hierarchy of geometric primitives Moving joints change shape and/or position Situated in some scenario or storyline Graphical representation of a “personality” Top 3 movies and video games Contrasts the different approaches to motion: MOVIES: off-line ray-tracing hand-animated GAMES: real-time polygon rend. Captured or hand-animated canned motions + real-time blending, warping, IK If you have a good algorithm for generating motion for off-line animation, it will apply to real-time games in 10-15 years (if you believe Moore’s law) Tomb Raider 3 (Eidos Interactive) The Legend of Zelda (Nintendo) Final Fantasy VIII (SquareOne)

Dealing with 1,000s of Degrees of Freedom Protein folding

Main Common Difficulty Formulating motion constraints