Sampling and Connection Strategies for PRM Planners

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Sampling and Connection Strategies for PRM Planners. Jean-Claude Latombe Computer Science Department Stanford University. q. 2. q. q. q. q. q. t (s). 0. 1. n. 3. 4. Original Problem. The “Solution”: Probabilistic Roadmap (PRM). free space. local path. milestone. m g. m b.

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Sampling and Connection Strategiesfor PRM Planners

Jean-Claude Latombe

Computer Science Department

Stanford University

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Original Problem
The New Issues
• Where to sample new milestones? Sampling strategy
• Which milestones to connect? Connection strategy
Examples
• Two-stage sampling:
• Build initial roadmap with uniform sampling
• Perform additional sampling around poorly connected milestones
• Coarse Connection:
• Attempt connection between 2 milestones only if they are in two distinct components
Multi-Query PRM
• Multi-stage sampling
• Obstacle-sensitive sampling
• Narrow-passage sampling
Multi-Stage Strategies

Rationale:

One can use intermediate sampling results to identify regions of the free space whose connectivity is more difficult to capture

Two-Stage Sampling

[Kavraki, 94]

Two-Stage Sampling

[Kavraki, 94]

Obstacle-Sensitive Strategies

Rationale:

The connectivity of free space is more difficult to capture near its boundary than in wide-open area

Obstacle-Sensitive Strategies
• Ray casting from samples in obstacles
• Gaussian sampling

[Amato, Overmars]

[Boor, Overmars, van der Stappen, 99]

Multi-Query PRM
• Multi-stage sampling
• Obstacle-sensitive sampling
• Narrow-passage sampling
Narrow-Passage Strategies

Rationale:

Finding the connectivity of the free space through narrow passage is the only hard problem.

Narrow-Passage Strategies
• Medial-Axis Bias
• Dilatation/contraction of the free space
• Bridge test

[Amato, Kavraki]

[Baginski, 96; Hsu et al, 98]

[Hsu et al, 02]

• The bridge test most likely yields a high rejection rate of configurations
• But, in general it results in a much smaller number of milestones, hence much fewer connections to be tested
• Since testing connections is costly, there can be significant computational gain
• More on this later ….

mg

mb

Single-Query PRM
• Diffusion
• Biased sampling
• Control-based sampling
Diffusion Strategies

Rationale:

The trees of milestones should diffuse throughout the free space to guarantee that the planner will find a path with high probability, if one exists

Diffusion Strategies
• Density-based strategy
• Associate a sampling density to each milestone in the trees
• Pick a milestone m at random with probability inverse to density
• Expand from m
• RRT strategy
• Pick a configuration q uniformly at random in c-space
• Select the milestone m the closest from q
• Expand from m

[Hsu et al, 97]

[LaValle and Kuffner, 00]

Rationale:

Makes big steps in wide-open area of the free space, and smaller steps in cluttered areas.

• Shrinking-window strategy

mg

mb

[Sanchez-Ante, 02]

mg

mb

Single-Query PRM
• Diffusion
• Biased sampling
• Control-based sampling
Biased Strategies

Rationale:

Use heuristic knowledge extracted from the workspace

Example:

• Define a potential field U and bias tree growth along the steepest descent of U
Biased Strategies

Rationale:

Use heuristic knowledge extracted from the workspace

Example:

• Define a potential field U and bias tree growth along the steepest descent of U
Control-Based Strategies

Rationale:

Directly satisfy differential kinodynamic constraints

Method:

• Represent motion in state (configuration x velocity) space
• Pick control input at random
• Integrate motion over short interval of time

[Kindel, Hsu, et al, 00] [LaValle and Kuffner, 00]

The New Issues
• Where to sample new milestones? Sampling strategy
• Which milestones to connect? Connection strategy
Connection Strategies
• Multi-query PRMs Coarse connections
• Single-query PRMs Lazy collision checking
Coarse Connections

Rationale:

Since connections are expensive to test, pick only those which have a good chance to test collision-free and to contribute to the roadmap connectivity.

Coarse Connnections

Methods:

• Connect only pairs of milestones that are not too far apart
• Connect each milestone to at most k other milestones
• Connect two milestones only if they are in two distinct components of the current roadmap ( the roadmap is a collection of acyclic graph)
• Visibility-based roadmap: Keep a new milestone m if:
• m cannot be connected to any previous milestone and
• m can be connected to 2 previous milestones belonging to distinct components of the roadmap

[Laumond and Simeon, 01]

Connection Strategies
• Multi-query PRMs  Coarse connections
• Single-query PRMs Lazy collision checking
Lazy Collision Checking

Rationale:

• Connections between close milestones have high probability of being collision-free
• Most of the time spent in collision checking is done to test connections
• Most collision-free connections will not be part of the final path
• Testing connections is more expensive for collision-free connections
• Hence: Postpone the tests of connections until they are absolutely needed

mg

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Lazy Collision Checking

X

[Sanchez-Ante, 02]

mg

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Lazy Collision Checking

[Sanchez-Ante, 02]

Possible New Strategy
• Rationale:
• Single-query planners are often more suitable than multi-query’s
• But there are some very good multi-query strategies
• Milestones are much less expensive to create than connections
• Pre-compute the milestonesof the roadmap, with uniform sampling, two-stage sampling, bridge test, and dilatation/contraction of free space to place milestones well
• Process queries with single-query roadmaps restricted to pre-computed milestones, with lazy collision checking