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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 strategies for prm planners

Sampling and Connection Strategiesfor PRM Planners

Jean-Claude Latombe

Computer Science Department

Stanford University

original problem

q

2

q

q

q

q

q

t(s)

0

1

n

3

4

Original Problem
the new issues
The New Issues
  • Where to sample new milestones? Sampling strategy
  • Which milestones to connect? Connection strategy
examples
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 prm1
Multi-Query PRM
  • Multi-stage sampling
  • Obstacle-sensitive sampling
  • Narrow-passage sampling
multi stage strategies
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
Two-Stage Sampling

[Kavraki, 94]

two stage sampling1
Two-Stage Sampling

[Kavraki, 94]

obstacle sensitive strategies
Obstacle-Sensitive Strategies

Rationale:

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

obstacle sensitive strategies1
Obstacle-Sensitive Strategies
  • Ray casting from samples in obstacles
  • Gaussian sampling

[Amato, Overmars]

[Boor, Overmars, van der Stappen, 99]

multi query prm2
Multi-Query PRM
  • Multi-stage sampling
  • Obstacle-sensitive sampling
  • Narrow-passage sampling
narrow passage strategies
Narrow-Passage Strategies

Rationale:

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

narrow passage strategies1
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]

comments jcl
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 ….
single query prm1

mg

mb

Single-Query PRM
  • Diffusion
  • Adaptive step
  • Biased sampling
  • Control-based sampling
diffusion strategies
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 strategies1
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
Adaptive-Step Strategies

Rationale:

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

adaptive step strategies1
Adaptive-Step Strategies
  • Shrinking-window strategy

mg

mb

[Sanchez-Ante, 02]

single query prm2

mg

mb

Single-Query PRM
  • Diffusion
  • Adaptive step
  • Biased sampling
  • Control-based sampling
biased strategies
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
biased strategies1
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
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 issues1
The New Issues
  • Where to sample new milestones? Sampling strategy
  • Which milestones to connect? Connection strategy
connection strategies
Connection Strategies
  • Multi-query PRMs Coarse connections
  • Single-query PRMs Lazy collision checking
coarse connections
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
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 strategies1
Connection Strategies
  • Multi-query PRMs  Coarse connections
  • Single-query PRMs Lazy collision checking
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
lazy collision checking1

mg

mb

Lazy Collision Checking

X

[Sanchez-Ante, 02]

lazy collision checking2

mg

mb

Lazy Collision Checking

[Sanchez-Ante, 02]

possible new strategy
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