1. CS 326A: Motion Planning Probabilistic Roadmaps: Sampling and Connection Strategies

2. Two Types of Strategies  Where to sample new milestones?  Sampling strategy  Which milestones to connect?  Connection strategy  Goal: Minimize roadmap size to correctly answer motion-planning queries

3. Impact of the Sampling Strategy

4. Rationale for Non-Uniform Sampling Strategy Visibility is not uniformly favorable in free space Regions with poorer visibility should be more densely sampled good visibility poor visibility

5.  But how to identify poor visibility regions? ― What is the source of information?  Robot and workspace geometry ― How to exploit it?  Workspace-guided strategies  Filtering strategies  Adaptive strategies  Deformation strategies

6.  Workspace-guided strategies Identify narrow passages in the workspace and map them into the configuration space  Filtering strategies Sample many configurations, find interesting patterns, and retain only promising configurations  Adaptive strategies Adjust the sampling distribution (  ) on the fly  Deformation strategies Deform the free space, e.g., to widen narrow passages

7. Multi- vs. Single-Query Roadmaps  Multi-query roadmaps  Pre-compute roadmap  Re-use roadmap for answering queries  The roadmap must cover the free space well  Single-query roadmaps  Compute a roadmap from scratch for each new query  Often roadmap consists of 2 trees rooted at the query configurations

8. Workspace-Guided Strategies  Rationale: Most narrow passages in configuration space are caused by narrow passages in the workspace  Method: Detect narrow passages in the workspace (e.g., cell decomposition, medial-axis transform) Sample robot configurations that place selected robot points in workspace’s narrow passages - H. Kurniawati and D. Hsu. Workspace importance sampling for probabilistic roadmap planning. In Proc. IEEE/RSJ Int. Conf. on Intelligent Robots & Systems, pp. 1618–1623, 2004. - J.P. van den Berg and M. H. Overmars. Using Workspace Information as a Guide to Non- Uniform Sampling in Probabilistic Roadmap Planners. IJRR, 24(12):1055-1071, Dec. 2005.

9. Workspace-Guided Strategies Uniform sampling Workspace-guided sampling

10. ??

11. Non-Uniform Sampling Strategies  Workspace-guided strategies  Filtering strategies  Adaptive strategies  Deformation strategies

12. Filtering Strategies Main Idea:  Sample several configurations in the same region of configuration space  If a “pattern” is detected, then retain one of the configurations as a roadmap node  More sampling work, but better distribution of nodes  Less time is wasted in connecting “non-interesting” milestones Methods:  Gaussian sampling  Bridge Test  Hybrid - V. Boor, M. H. Overmars, and A. F. van der Stappen. The Gaussian sampling strategy for probabilistic roadmap planners. In Proc. 1999 IEEE Int. Conf. Robotics and Automation, 1999, pp. 1018–1023. - Z. Sun, D. Hsu, T. Jiang, H. Kurniawati, and J. Reif. Narrow passage sampling for probabilistic roadmap planners. IEEE Trans. on Robotics, 21(6):1105–1115, 2005.

13. Gaussian Sampling 1)Sample a configuration q uniformly at random from configuration space 2)Sample a real number x at random with Gaussian distribution N [0,  ] (x) 3)Sample a configuration q’ in the ball B(q,|x|) uniformly at random 4)If only one of q and q’ is in free space, retain the one in free space as a node; else retain none What is the effect? What is the intuition?

14. Example of Node Distribution

15. Uniform vs. Gaussian Sampling Milestones (13,000) created by uniform sampling before the narrow passage was adequately sampled Milestones (150) created by Gaussian sampling The gain is not in sampling fewer milestones, but in connecting fewer pairs of milestones

16. Bridge Test 1)Sample two conformations q and q’ using Gaussian sampling technique 2)If none is in free space, then if q m = (q+q’)/2 is in free space, then retain q m as a node 3)Else retain none What is the effect? What is the intuition?

17. Bridge Test

18. Example of Distribution

19. Gaussian Bridge test

20. 8-joint robot with mobile base Example of Distribution

21. 7-joint robot with fixed base Example of Distribution

22. Hybrid Sampling 1)Sample two configurations q and q’ using Gaussian sampling technique 2)If both are in free space, then retain one (any of the two) as a node with low probability (e.g., 0.1) 3)Else if only one is in free space, then retain it as a node with intermediate probability (e.g., 0.5) 4)Else if q m = (q+q’)/2 is in free space, then retain it as a node with probability 1

23. Uniform Bridge test Uniform + Bridge test

24. Non-Uniform Sampling Strategies  Workspace-guided strategies  Filtering strategies  Adaptive strategies  Deformation strategies

25. Adaptive Strategies Main idea: Use intermediate sampling results to identify regions of the free space whose connectivity is more difficult to capture  Time-varying sampling measure Methods: Connectivity expansion Diffusion

26. Connectivity Expansion Use work already done to detect poor- visibility regions [Kavraki, 94]

27. Connectivity Expansion Use work already done to detect low- visibility regions [Kavraki, 94]

28. Example of Distribution

29. s g Diffusion Strategy (Density-Based [Hsu et al, 97], RRT [LaValle and Kuffner, 00] )

30. s g [Sánchez-Ante, 2003] Adaptive-Step Sampling

31. Non-Uniform Sampling Strategies  Workspace-guided strategies  Filtering strategies  Adaptive strategies  Deformation strategies

32. Deformation Strategies Main idea: Deform the free space to make it more expansive Method: Free space dilatation

33. Motivating Experiment

34. Free Space Dilatation 1.Pre-computation: Slim the robot / obstacles 2.Planning:  Compute a path for slimmed robot  Deform this path for original robot -M. Saha, J.C. Latombe, Y.-C. Chang, F. Prinz. Finding Narrow Passages with Probabilistic Roadmaps: The Small-Step Retraction Method. Autonomous Robots, 19(3):301-319, Dec. 2005. -H.-L. Cheng, D. Hsu, J.-C. Latombe, and G. Sánchez-Ante. Multi-level free-space dilation for sampling narrow passages in PRM planning. Proc. IEEE Int. Conf. on Robotics & Automation, 2006.

35. Free Space Dilatation  ~ up to 2 orders of magnitude speedup Roadmap construction and repair fattened free space widened passage free space obstacle start goal

36. Time SBL* (secs) Time SBL (secs) (a)9.412295 (b)325955 (c)2.141 (d)492863 (e)65631 (f)13588>100000 (a) (b) (c) (d) (e) (f) Alpha 1.0 Up to 3 orders of magnitude speedup Some Results

37. Connection Strategies  Limit number of connections: Nearest-neighbor strategy Connected component strategy  Increase expansiveness: Library of local path shapes [Amato 98] Local search strategy [Isto 04]  Delay costly computation: Lazy collision checking [Sanchez-Ante, 02]

38. s g X [Sánchez-Ante, 2002] Lazy Collision Checking

39. s g [Sánchez-Ante, 2002] x10 speedup Lazy Collision Checking

40. Rationale of Lazy Collision Checking  Connections between close milestones have high probability of being free of collision  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