Sampling and Connection Strategies for PRM Planners

Sampling and Connection Strategiesfor 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 mg mb The “Solution”:Probabilistic Roadmap (PRM) free space

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: • Maintain roadmap’s connected components • Attempt connection between 2 milestones only if they are in two distinct components

Multi-Query PRM

mg mb Single-Query PRM

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]

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]

Bridge Test

Comparison with Gaussian Strategy Bridge test Gaussian

Other Examples

Running Times

Comments (JCL) • 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 • Adaptive step • 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]

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

Adaptive-Step Strategies • Shrinking-window strategy mg mb [Sanchez-Ante, 02]

mg mb Single-Query PRM • Diffusion • Adaptive step • 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 • Use task knowledge

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 mb Lazy Collision Checking X [Sanchez-Ante, 02]

mg mb 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

vi Pij vj Application to Probabilistic Conformational Roadmap

Sampling and Connection Strategies for PRM Planners