Randomized Motion Planning

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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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Presentation Transcript

Randomized Motion Planning

Jean-Claude Latombe

Computer Science DepartmentStanford University

Goal of Motion Planning
• Classical example: find a collision-free path in robot configuration space among static obstacles
• Kinodynamic constraints
• Visibility constraints
Outline
• Bits of history
• Approaches
• Applications
• Conclusion
Early Work

Shakey (Nilsson, 1969): Visibility graph

C = S1 x S1

Mathematical Foundations

Lozano-Perez, 1980: Configuration Space

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
• 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
Criticality-Based Motion Planning
• 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
• Other example:Grid-based methods (deterministic sampling)
Sampling-Based Motion Planning
• Easy to implement
• Fast, scalable to many degrees of freedom and complex constraints
• Drawbacks:
• Probabilistic completeness
• Limited insight
Outline
• Bits of history
• Approaches
• Applications
• Conclusion
Motivation

Computing an explicit representation of the admissible

space is hard, but checking that a point lies in the

milestone

mg

mb

[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)

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

[Kavraki, Latombe, Motwani, Raghavan, 95]

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]

Lookout of F1

Prob[failure] = K exp(-r)

expansive if each of its subsets has a large lookout

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
• Applications
• Conclusion
Design for Manufacturing and Servicing

General Motors

General Motors

General Electric

[Hsu, 2000]

Graphic Animation of Digital Actors

The MotionFactory

[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

Humanoid Robot

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

Space Robotics

robot

obstacles

air thrusters

gaz tank

air bearing

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

Autonomous Helicopter

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

y2

q2

(Grasp Lab - U. Penn)

d

q1

y1

x2

x1

Interacting Nonholonomic Robots
Map Building

[Gonzalez, 2000]

Map Building

[Gonzalez, 2000]

Map Building

[Gonzalez, 2000]

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

•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

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

Protein folding

Ligand-protein binding

[Apaydin, 2000]

[Singh, Latombe, and Brutlag, 1999]

Outline
• Bits of history
• Approaches
• 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 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

Formulating motion constraints