CS 326 A: Probabilistic Roadmaps

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CS 326 A Probabilistic Roadmaps

• Provide an effective (fast, robust, easy to
  implement) computational framework to plan
  motions of robots with many degrees of
  freedom.dofs dimensions of configuration
  space
• Principle is very simple. Sample the
  configuration space at random. Keep the samples
  in the free space. Connect pairs of samples by
  simple paths
• Approach is not complete. Issues . What
  guarantee of performance does it offer?. How
  fast does it converge?

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Complexity of Complete Planning

• There is strong evidence that complete
  collision-free path planning among static
  obstacles takes time exponential in the N of
  dofs Reif, 79
• Two complete, general-purpose have been proposed
• One by Schwarz and Sharir (1983), which is twice
  exponential in N
• One by Canny (1987), which is simple exponential
• Neither has ever been implemented. It is very
  likely that none would work well for N greater
  than 3 or 4
• There are specific complete algorithms for robots
  with 2, 3, or 4 dofs

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What Can we Do?

• Massive use of heuristics, e.g., in the form of
  potential fields (functions that pull the robot
  toward the goal, and repulse it from obstacles),
  but huge local-minimum problem. Resulting
  planners usually offer no formal guarantees of
  performance
• Settle for a weaker notion of completeness, such
  as resolution completeness or probabilistic
  completeness. The former is based on a systematic
  discretization of the configuration space and
  does not work well when N is large

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Probabilistic Roadmaps

• Probabilistically complete with exponential rate
  of convergence is appropriate configuration
  spaces (expansive spaces)
• Rationale Computing the C-obstacles is too hard,
  while checking whether an arbitrary configuration
  is free, or not, can be efficiently done using
  one of the recent collision-checking or
  distance-computation techniques made available
• A probabilistic roadmap is a network of curve
  that approximate the connectivity of the free
  space

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So ...

• While I spent the previous class talking about
  the configuration space of a robot
• Probabilistic roadmaps are about avoid computing
  an explicit representation of the configuration
  space!
• But PRMs still require a good understanding of
  what a configuration space is

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Issues with Probabilistic Roadmaps

• Sampling strategy
• Precomputed roadmap or roadmap constructed on the
  fly?
• Rate of convergence
• Collision checking vs. distance computation

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Class of Today

• Basic framework Kavraki et als paper
• Collision-checking/distance-computation
  Quinlans paper

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Next Class

• Other sampling strategies
• Convergence of PRM planners

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Note on Terminology

• Except when mentioned otherwise, we will be
  dealing with holonomic robots.
• What is a holonomic robot? It is a robot that
  can move in any direction.Said otherwise, assume
  the robot is at some configuration q. There is an
  N-dimensional space of velocity vectors at q. All
  vector directions are feasible
• Counter-example Car-like robot