Sampling and Connection Strategies for PRM Planners by Kittibitti

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									Sampling and Connection Strategies
         for PRM Planners


         Jean-Claude Latombe

      Computer Science Department
          Stanford University
Original Problem
           q0        q1

                               q2
      qn
                      t(s)
                          q3
                q4
      The “Solution”:
Probabilistic Roadmap (PRM)
              free space
               The “Solution”:
         Probabilistic Roadmap (PRM)
                local path free space

milestone

                                        mg

    mb
          The New Issues

Where to sample new milestones?
  Sampling strategy

Which milestones to connect?
  Connection strategy
                  Examples

 Two-stage sampling:
    1) Build initial roadmap with uniform sampling
    2) Perform additional sampling around poorly
       connected milestones


 Coarse Connection:
    1) Maintain roadmap’s connected components
    2) Attempt connection between 2 milestones only
       if they are in two distinct components
Multi-Query PRM
     Single-Query PRM



                        mg
mb
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



  [Amato, Overmars]


Gaussian sampling



 [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
 [Amato, Kavraki]




 Dilatation/contraction of the free space
  [Baginski, 96; Hsu et al, 98]




 Bridge test
  [Hsu et al, 02]
Bridge Test
Comparison with Gaussian Strategy




      Gaussian      Bridge test
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 ….
     Single-Query PRM



                               mg
mb
          •   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
[Hsu et al, 97]


 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]
     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]
     Single-Query PRM



                               mg
mb
          •   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
               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:
  1. Connect only pairs of milestones that are not too far apart
  2. Connect each milestone to at most k other milestones
  3. 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 and
     b)    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
               Lazy Collision Checking




                                     mg

      mb
                             X



[Sanchez-Ante, 02]
               Lazy Collision Checking




                                     mg

      mb




[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 milestones of 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
Application to Probabilistic
 Conformational Roadmap


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