CS 326 A: Probabilistic Roadmaps

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CS 326 A Probabilistic Roadmaps

• Convergence of PRM plannersHow quickly does the
  probability of failure converge toward 0 (failure
  not finding a path when one exists?
• Smarter sampling strategies99 of the time of
  a PRM planner is spent checking collision. Can a
  smarter strategy reduce the size of a roadmap,
  hence the time spent in collision checking?

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Probabilistic Roadmap
milestones
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Good Precomputed Roadmap

• CoverageIt should be easy to connect new start
  and goal configurations to milestones of the
  precomputed roadmapConcept of e-goodness of the
  free space
• ConnectivityThere should be a single connected
  component of the roadmap in every connected
  component of the free spaceConcept of
  expansiveness of the free space

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e-Goodness of the Free Space F

• Let 0
• A free configuration q is e-good if it sees an e
  fraction of the volume of F
• F is e-good if every free configuration is e-good
  
• If F is e-good , the volume of the subset of F
  not seen by any of s milestones picked uniformly
  at random has a probability proportional to
  exp(-s) of being greater than eVol(F)

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Narrow Passages
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Narrow Passage in 6-D Configuration Space
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Lookout of a Subset

• Let S be an arbitrary subset of the free space F
• Let 0
• The lookout of S is the subset of points in S
  that see a b fraction of the volume of F\S
• In the above example, F1 has a small lookout

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Expansive Free Space

• F is expansive if every subset S of F has a large
  lookout. More formally
• Let 0
• F is expansive if the volume of the lookout of
  any subset S of F is an a fraction of the volume
  of S
• In an expansive free space F, the probability
  that a connected component contains several
  components of a roadmap of s milestones is
  proportional to exp(-s)

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Linking Sequence

• Any milestone of a roadmap of size s is likely to
  have a linking sequence of length t
• A linking sequence of length t spans a large
  volume
• If two linking sequences lie in the same
  connected component of the free space, then their
  intersection is likely to have a large and a
  milestone is likely to be in this volume

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Sampling Strategies

• Uniform sampling
• Two-stage strategies uniform sampling
  refinement in difficult areas
• Push non-free samples into free space (generate
  more milestones near the free spaces boundary)
• Wave propagation from given free configurations

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Sampling Strategy for Narrow Passages