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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


Outline
Outline

  • 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


Outline1
Outline

  • 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


Outline2
Outline

  • Bits of history

  • Approaches

  • Probabilistic Roadmaps

  • Applications

  • Conclusion


Motivation
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

milestone

mg

mb

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)

mg

mb

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


Two very different cases

Expansive

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


Outline3
Outline

  • 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

robot

obstacles

air thrusters

gaz tank

air bearing

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



Autonomous helicopter
Autonomous Helicopter

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


Interacting nonholonomic robots

y2

q2

(Grasp Lab - U. Penn)

d

q1

y1

x2

x1

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

T

T

B1

C

B2

B4

  • •0 < Critical < 500

    • 0 < B2 < 500

B3

Radiosurgical Planning


Sample case
Sample Case

50% Isodose Surface

80% Isodose Surface

Conventional system’s plan

CARABEAMER’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]



Outline4
Outline

  • Bits of history

  • Approaches

  • Probabilistic Roadmaps

  • Applications

  • Conclusion


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
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 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)



Main common difficulty
Main Common Difficulty

Formulating motion constraints


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