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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

Randomized Motion Planning

Jean-Claude Latombe

Computer Science DepartmentStanford University

goal of motion planning
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
  • Bits of history
  • Approaches
  • Probabilistic Roadmaps
  • Applications
  • Conclusion
early work
Early Work

Shakey (Nilsson, 1969): Visibility graph

mathematical foundations

C = S1 x S1

Mathematical Foundations

Lozano-Perez, 1980: Configuration Space

computational analysis
Computational Analysis

Reif, 1979: Hardness (lower-bound results)

exact general purpose path planners
Exact General-Purpose Path Planners

- Schwarz and Sharir, 1983: Exact cell decomposition based on Collins technique

- Canny, 1987: Silhouette method

heuristic planners
Heuristic Planners

Khatib, 1986:

Potential Fields

other types of constraints
Other Types of Constraints

E.g., Visibility-Based Motion Planning

Guibas, Latombe, LaValle, Lin, and Motwani, 1997

  • Bits of history
  • Approaches
  • Probabilistic Roadmaps
  • Applications
  • Conclusion
criticality based motion planning
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 planning1
Criticality-Based Motion Planning
  • Advantages:
    • Completeness
    • Insight
  • Drawbacks:
    • Computational complexity
    • Difficult to implement
sampling based motion planning
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 planning1
Sampling-Based Motion Planning
  • Advantages:
    • Easy to implement
    • Fast, scalable to many degrees of freedom and complex constraints
  • Drawbacks:
    • Probabilistic completeness
    • Limited insight
  • Bits of history
  • Approaches
  • Probabilistic Roadmaps
  • Applications
  • Conclusion

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




Probabilistic Roadmap (PRM)

admissible space

[Kavraki, Svetska, Latombe,Overmars, 95]

sampling strategies
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

endgame region

m’ = f(m,u)



PRM With Dynamic Constraints in State x Time Space

[Hsu, Kindel, Latombe, and Rock, 2000]

relation to art gallery problems
Relation to Art-Gallery Problems

[Kavraki, Latombe, Motwani, Raghavan, 95]

desirable properties of a prm
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
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 space1

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

a few remarks
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
  • Bits of history
  • Approaches
  • Probabilistic Roadmaps
  • Applications
  • Conclusion
design for manufacturing and servicing
Design for Manufacturing and Servicing

General Motors

General Motors

General Electric

[Hsu, 2000]

graphic animation of digital actors
Graphic Animation of Digital Actors

The MotionFactory

[Koga, Kondo, Kuffner, and Latombe, 1994]

digital actors with visual sensing
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

humanoid robot
Humanoid Robot

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

space robotics
Space Robotics



air thrusters

gaz tank

air bearing

[Kindel, Hsu, Latombe, and Rock, 2000]

autonomous helicopter
Autonomous Helicopter

[Feron, 2000] (AA Dept., MIT)

interacting nonholonomic robots



(Grasp Lab - U. Penn)






Interacting Nonholonomic Robots
map building
Map Building

[Gonzalez, 2000]

map building1
Map Building

[Gonzalez, 2000]

map building2
Map Building

[Gonzalez, 2000]

radiosurgical planning
Radiosurgical Planning

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

radiosurgical planning1

•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


Radiosurgical Planning
sample case
Sample Case

50% Isodose Surface

80% Isodose Surface

Conventional system’s plan


reconfiguration planning for modular robots
Reconfiguration Planning for Modular Robots

Casal and Yim, 1999

Xerox, Parc

prediction of molecular motions
Prediction of Molecular Motions

Protein folding

Ligand-protein binding

[Apaydin, 2000]

[Singh, Latombe, and Brutlag, 1999]

  • Bits of history
  • Approaches
  • Probabilistic Roadmaps
  • Applications
  • 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
  • Relatively large standard deviation of planning time
  • No rigorous termination criterion when no solution is found
  • New challenging applications…
planning nice looking motions for digital actors
Planning Nice-Looking Motions for Digital Actors

Toy Story (Pixar/Disney)

Antz (Dreamworks)

A Bug’s Life (Pixar/Disney)

Tomb Raider 3 (Eidos Interactive)

The Legend of Zelda (Nintendo)

Final Fantasy VIII (SquareOne)

main common difficulty
Main Common Difficulty

Formulating motion constraints