Sampling and Connection Sampling and Connection StrategiesStrategies

for PRM Plannersfor PRM Planners

Jean-Claude Latombe

Computer Science DepartmentStanford University

Abridged and Modified Version (D.H.)

see JCL’s website for the full version

The (New) IssuesThe (New) Issues

Where to sample new milestones? Sampling strategy

Which milestones to connect? Connection strategy

ExamplesExamples

Two-stage sampling:1) Build initial roadmap with uniform sampling2) Perform additional sampling around poorly

connected milestones

Coarse Connection:1) Maintain roadmap’s connected components2) Attempt connection between 2 milestones

only if they are in two distinct components

Multi-Query PRMMulti-Query PRM

Single-Query PRMSingle-Query PRM

mmbb

mmgg

Multi-Query PRMMulti-Query PRM

• Multi-stage sampling• Obstacle-sensitive sampling• Narrow-passage sampling

Multi-Stage StrategiesMulti-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 SamplingTwo-Stage Sampling

[Kavraki, 94]

Two-Stage SamplingTwo-Stage Sampling

[Kavraki, 94]

Obstacle-Sensitive StrategiesObstacle-Sensitive Strategies

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

Obstacle-Sensitive StrategiesObstacle-Sensitive Strategies

Ray casting from samples in obstacles

Gaussian sampling

[Boor, Overmars, van der Stappen, 99]

[Amato, Overmars]

Multi-Query PRMMulti-Query PRM

• Multi-stage sampling• Obstacle-sensitive sampling• Narrow-passage sampling

Narrow-Passage StrategiesNarrow-Passage Strategies

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

Narrow-Passage StrategiesNarrow-Passage Strategies

Medial-Axis Bias

Dilatation/contraction of the free space

Bridge test[Hsu et al, 02]

[Amato, Kavraki]

[Baginski, 96; Hsu et al, 98]

Bridge TestBridge Test

Comparison with Gaussian Comparison with Gaussian StrategyStrategy

Gaussian Bridge test

Single-Query PRMSingle-Query PRM

mmbb

mmgg

Diffusion StrategiesDiffusion 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 StrategiesDiffusion 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[LaValle and Kuffner, 00]

[Hsu et al, 97]

Adaptive-Step StrategiesAdaptive-Step Strategies

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

Adaptive-Step StrategiesAdaptive-Step Strategies

mmbb

mmgg

[Sanchez-Ante, 02]

Shrinking-window strategy

Single-Query PRMSingle-Query PRM

mmbb

mmgg

Coarse ConnectionsCoarse 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 ConnnectionsCoarse Connnections

Methods:1. Connect only pairs of milestones that are not too far apart2. Connect each milestone to at most k other milestones3. Connect two milestones only if they are in two distinct

components of the current roadmap ( the roadmap is a collection of acyclic graph)

4. Visibility-based roadmap: Keep a new milestone m if:a) m cannot be connected to any previous milestone andb) m can be connected to 2 previous milestones belonging to

distinct components of the roadmap[Laumond and Simeon, 01]