1. Sampling and Connection Strategies for PRM Planners Jean-Claude Latombe Computer Science Department Stanford University Abridged and Modified Version (D.H.) see JCL’s website for the full version

2. The (New) Issues  Where to sample new milestones?  Sampling strategy  Which milestones to connect?  Connection strategy

3. 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

4. Multi-Query PRM

5. Single-Query PRM mbmbmbmb mgmgmgmg

6. Multi-Query PRM Multi-stage sampling Obstacle-sensitive sampling Narrow-passage sampling

7. Multi-Stage Strategies Rationale: One can use intermediate sampling results to identify regions of the free space whose connectivity is more difficult to capture

8. Two-Stage Sampling [Kavraki, 94]

9. Two-Stage Sampling [Kavraki, 94]

10. Obstacle-Sensitive Strategies Rationale: The connectivity of free space is more difficult to capture near its boundary than in wide-open area

11. Obstacle-Sensitive Strategies  Ray casting from samples in obstacles  Gaussian sampling [Boor, Overmars, van der Stappen, 99] [Amato, Overmars]

12. Multi-Query PRM Multi-stage sampling Obstacle-sensitive sampling Narrow-passage sampling

13. Narrow-Passage Strategies Rationale: Finding the connectivity of the free space through narrow passage is the only hard problem.

14. 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]

15. Bridge Test

16. Comparison with Gaussian Strategy Gaussian Bridge test

17. Single-Query PRM mbmbmbmb mgmgmgmg

18. 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

19. 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]

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

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

22. Single-Query PRM mbmbmbmb mgmgmgmg

23. 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.

24. 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]