Sampling and Connection Strategies for PRM Planners Jean-Claude Latombe Computer Science Department Stanford University

Original Problem q 1 q 3 q 0 q n q 4 q 2  (s)

The “Solution”: Probabilistic Roadmap (PRM) free space

The “Solution”: Probabilistic Roadmap (PRM) free space mbmbmbmb mgmgmgmg milestone local path

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 mbmbmbmb mgmgmgmg

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  Gaussian sampling [Boor, Overmars, van der Stappen, 99] [Amato, Overmars]

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 [Hsu et al, 02] [Amato, Kavraki] [Baginski, 96; Hsu et al, 98]

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 mbmbmbmb mgmgmgmg 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 [LaValle and Kuffner, 00] [Hsu et al, 97]

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

Adaptive-Step Strategies mbmbmbmb mgmgmgmg [Sanchez-Ante, 02]  Shrinking-window strategy

Single-Query PRM mbmbmbmb mgmgmgmg 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 mbmbmbmb mgmgmgmg [Sanchez-Ante, 02] X

Lazy Collision Checking mbmbmbmb mgmgmgmg [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 vivi vjvj P ij