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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

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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]

Narrow passage issue

Narrow Passage Issue

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 space

Expansiveness of Admissible Space

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

Two very different cases


Poorly expansive

Two Very Different Cases

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]

Robot programming and placement

Robot Programming and Placement

[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]

Randomized motion planning 1356862

Total duration : 40 sec

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]

Next best view computation

Next-Best View Computation

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]

Capturing energy landscape

Capturing Energy Landscape

[Apaydin, 2000]



  • 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 minimally invasive surgery procedures amidst soft tissue structures

Planning Minimally Invasive SurgeryProcedures Amidst Soft-Tissue Structures

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)

Dealing with 1 000s of degrees of freedom

Dealing with 1,000s of Degrees of Freedom

Protein folding

Main common difficulty

Main Common Difficulty

Formulating motion constraints

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