Randomized Motion Planning

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Randomized Motion Planning

• Jean-Claude Latombe
• Computer Science DepartmentStanford University

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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

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Outline

• Bits of history
• Approaches
• Probabilistic Roadmaps
• Applications
• Conclusion

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Early Work
Shakey (Nilsson, 1969) Visibility graph
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Mathematical Foundations
Lozano-Perez, 1980 Configuration Space
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Computational Analysis
Reif, 1979 Hardness (lower-bound results)
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Exact General-Purpose Path Planners
- Schwarz and Sharir, 1983 Exact cell
decomposition based on Collins technique
- Canny, 1987 Silhouette method
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Heuristic Planners
Khatib, 1986 Potential Fields
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Other Types of Constraints
E.g., Visibility-Based Motion Planning Guibas,
Latombe, LaValle, Lin, and Motwani, 1997
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Outline

• Bits of history
• Approaches
• Probabilistic Roadmaps
• Applications
• Conclusion

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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 Wilsons Non-Directional Blocking
  Graphs for assembly planning
• Other examples
• Schwartz-Sharirs cell decomposition
• Cannys roadmap

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Criticality-Based Motion Planning

• Advantages
• Completeness
• Insight
• Drawbacks
• Computational complexity
• Difficult to implement

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Sampling-Based Motion Planning

• Principle
• Sample the space of interest
• Connect sampled points by simple paths
• Search the resulting graph
• ExampleProbabilistic Roadmaps (PRMs)
• Other exampleGrid-based methods (deterministic
  sampling)

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Sampling-Based Motion Planning

• Advantages
• Easy to implement
• Fast, scalable to many degrees of freedom and
  complex constraints
• Drawbacks
• Probabilistic completeness
• Limited insight

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Outline

• Bits of history
• Approaches
• Probabilistic Roadmaps
• Applications
• Conclusion

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

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Probabilistic Roadmap (PRM)
admissible space
Kavraki, Svetska, Latombe,Overmars, 95
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Sampling Strategies

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

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PRM With Dynamic Constraints in State x Time Space
Hsu, Kindel, Latombe, and Rock, 2000
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Relation to Art-Gallery Problems
Kavraki, Latombe, Motwani, Raghavan, 95
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Narrow Passage Issue
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Desirable Properties of a PRM

• CoverageThe milestones should see most of the
  admissible space to guarantee that the initial
  and goal configurations can be easily connected
  to the roadmap
• ConnectivityThere should be a 1-to-1 map
  between the components of the admissible space
  and those of the roadmap

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Complexity Measures

• e-goodnessKavraki, Latombe, Motwani, and
  Raghavan, 1995
• Path clearanceKavraki, Koulountzakis, and
  Latombe, 1996
• e-complexityOvermars and Svetska, 1998
• ExpansivenessHsu, Latombe, and Motwani, 1997

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Expansiveness of Admissible Space
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Expansiveness of Admissible Space
The admissible space is expansive if each
of its subsets has a large lookout
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Two Very Different Cases
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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

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Outline

• Bits of history
• Approaches
• Probabilistic Roadmaps
• Applications
• Conclusion

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Design for Manufacturing and Servicing
General Motors
General Motors
General Electric
Hsu, 2000
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Robot Programming and Placement
Hsu, 2000
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Graphic Animation of Digital Actors
The MotionFactory
Koga, Kondo, Kuffner, and Latombe, 1994
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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
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Humanoid Robot
Kuffner and Inoue, 2000 (U. Tokyo)
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Space Robotics
robot
obstacles
air thrusters
gaz tank
air bearing
Kindel, Hsu, Latombe, and Rock, 2000
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Total duration 40 sec
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Autonomous Helicopter
Feron, 2000 (AA Dept., MIT)
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Interacting Nonholonomic Robots
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Map Building
Gonzalez, 2000
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Next-Best View Computation
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Map Building
Gonzalez, 2000
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Map Building
Gonzalez, 2000
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Radiosurgical Planning
Cyberknife System (Accuray, Inc.)
CARABEAMER Planner
Tombropoulos, Adler, and Latombe, 1997

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Radiosurgical Planning
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Sample Case
50 Isodose Surface
80 Isodose Surface
Conventional systems plan
CARABEAMERs plan
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Reconfiguration Planning for Modular Robots
Casal and Yim, 1999
Xerox, Parc
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Prediction of Molecular Motions
Protein folding
Ligand-protein binding
Apaydin, 2000
Singh, Latombe, and Brutlag, 1999
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Capturing Energy Landscape
Apaydin, 2000
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Outline

• Bits of history
• Approaches
• Probabilistic Roadmaps
• Applications
• Conclusion

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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

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Issues

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

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Planning Minimally Invasive SurgeryProcedures
Amidst Soft-Tissue Structures
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Planning Nice-Looking Motions for Digital Actors
A Bugs Life (Pixar/Disney)
Toy Story (Pixar/Disney)
Antz (Dreamworks)
Tomb Raider 3 (Eidos Interactive)
Final Fantasy VIII (SquareOne)
The Legend of Zelda (Nintendo)
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Dealing with 1,000s of Degrees of Freedom
Protein folding
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Main Common Difficulty

• Formulating motion constraints

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