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