1. 1 On the Probabilistic Foundations of Probabilistic Roadmaps Jean-Claude Latombe Stanford University joint work with David Hsu and Hanna Kurniawati National University of Singapore D. Hsu, J.C. Latombe, H. Kurniawati. On the Probabilistic Foundations of Probabilistic Roadmap Planning. IJRR, 25(7):627-643, 2006.

2. 2 Rationale of PRM Planners  The cost of computing an exact representation of a robot’s free space F is often prohibitive  Fast algorithms exist to check if a given configuration or path is collision-free  A PRM planner computes an extremely simplified representation of F in the form of a network of “local” paths connecting configurations sampled at random in F according to some probability measure

3. 3 This answer may occasionally be incorrect Connection strategy Sampling strategy Procedure BasicPRM (s,g,N) 1.Initialize the roadmap R with two nodes, s and g 2.Repeat: a.Sample a configuration q from C with probability measure  b.If q  F then add q as a new node of R c.For every node v in R such that v  q do If path (q,v)  F then add (q,v) as a new edge of R until s and g are in the same connected component of R or R contains N+2 nodes 3.If s and g are in the same connected component of R then Return a path between them 4.Else Return NoPath

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7. 7 PRM planners work well in practice. Why?  Why are they probabilistic?  What does their success tell us?  How important is the probabilistic sampling measure  ?  How important is the randomness of the sampling source?

8. 8 Why is PRM planning probabilistic?  A PRM planner ignores the exact shape of F. So, it acts like a robot building a map of an unknown environment with limited sensors  At any moment, there exists an implicit distribution (H,  ), where H is the set of all consistent hypotheses over the shapes of F For every x  H,  (x) is the probability that x is correct  The probabilistic sampling measure  used by the planner reflects this uncertainty. The goal is to minimize the expected number of iterations to connect s and g, whenever they lie in the same component of F

9. 9 Why is PRM planning probabilistic?  A PRM planner ignores the exact shape of F. So, it acts like a robot building a map of an unknown environment with limited sensors  At any moment, there exists an implicit distribution (H,  ), where H is the set of all consistent hypotheses over the shapes of F For every x  H,  (x) is the probability that x is correct  The probabilistic sampling measure  reflects this uncertainty. The goal is to minimize the expected number of remaining iterations to connect s and g, whenever they lie in the same component of F

10. 10 So...  PRM planning trades the cost of computing F exactly against the cost of dealing with uncertainty  This choice is beneficial only if a small roadmap has high probability to represent F well enough to answer planning queries correctly [Note the analogy with PAC learning]  Under which conditions is this the case?

11. 11 Relation to Monte Carlo Integration x f(x) a b A = a × b x1x1 x2x2 (x i,y i ) But a PRM planner must construct a path The connectivity of F may depend on small regions Insufficient sampling of such regions may lead the planner to failure

12. 12  Two configurations q and q’ see each other if path (q,q’)  F  The visibility set of q is V (q) = {q’ | path (q,q’)  F} Visibility in F [Kavraki et al., 1995]

13. 13 ε-Goodness of F [Kavraki et al., 1995]  Let μ(X) stand for the volume of X  F  Given ε  (0,1], q  F is ε-good if it sees at least an ε-fraction of F, i.e., if μ( V (q))  ε  μ(F)  F is  -good if every q in F is  -good  Intuition: If F is ε-good, then with high probability a small set of configurations sampled at random will see most of F

14. 14 F1F1 F2F2 Connectivity Issue

15. 15 F1F1 F2F2 Connectivity Issue Lookout of F 1 The β -lookout of a subset X of F is the set of all configurations in X that see a β -fraction of F\X β-lookout(X)  {q  X | μ ( V (q)\X)  β  μ (F\X)}

16. 16 F1F1 F2F2 (ε, ,β)-Expansiveness of F Lookout of F 1 F is (ε  )-expansive if it is ε -good and each one of its subsets X has a β -lookout whose volume is at least   μ (X) The β -lookout of a subset X of F is the set of all configurations in X that see a β -fraction of F\X β-lookout(X)  {q  X | μ ( V (q)\X)  β  μ (F\X)} [Hsu et al., 1997]

17. 17  Expansiveness only depends on volumetric ratios  It is not directly related to the dimensionality of the configuration space  In 2-D the expansiveness of the free space can be made arbitrarily poor  In n-D the passage can be made narrow in 1, 2,..., n-1 dimensions Comments

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19. 19 Theoretical Convergence of PRM Planning Theorem 1 [Hsu, Latombe, Motwani, 1997] Let F be (ε, ,β)-expansive, and s and g be two configurations in the same component of F. BasicPRM (s,g,N) with uniform sampling returns a path between s and g with probability converging to 1 at an exponential rate as N increases s g Linking sequence

20. 20 Theoretical Convergence of PRM Planning Theorem 1 [Hsu, Latombe, Motwani, 1997] Let F be (ε, ,β)-expansive, and s and g be two configurations in the same component of F. BasicPRM (s,g,N) with uniform sampling returns a path between s and g with probability converging to 1 at an exponential rate as N increases  = Pr(Failure)  Experimental convergence

21. 21 Theoretical Convergence of PRM Planning Theorem 1 [Hsu, Latombe, Motwani, 1997] Let F be (ε, ,  )-expansive, and s and g be two configurations in the same component of F. BasicPRM (s,g,N) with uniform sampling returns a path between s and g with probability converging to 1 at an exponential rate as N increases Theorem 2 [Hsu, Latombe, Kurniawati, 2006] For any ε  0, any N  0, and any  in (0,1], there exists  o and  o such that if F is not (ε, ,  )-expansive for  0 and  0, then there exists s and g in the same component of F such that BasicPRM (s,g,N) fails to return a path with probability greater than .

22. 22 What does the empirical success of PRM planning tell us?  It tells us that F is often favorably expansive despite its overwhelming algebraic/geometric complexity  Revealing this property might well be the most important contribution of PRM planning

23. 23 In retrospect, is this property surprising?  Not really! Narrow passages are unstable features under small random perturbations of the robot/workspace geometry

24. 24 In retrospect, is this property surprising?  Not really! Narrow passages are unstable features under small random perturbations of the robot/workspace geometry  [Chaudhuri and Koltun, 2006]  PRM planning with uniform sampling has polynomial smoothed running time in spaces of fixed dimensions (Recall that the worst-case running time of PRM planning is unbounded as a function of the input’s combinatorial complexity)  Poorly expansive space are unlikely to occur by accident

25. 25 Most narrow passages in F are intentional … … but it is not easy to intentionally create complex narrow passages in F Alpha puzzle

26. 26 PRM planners work well in practice. Why?  Why are they probabilistic?  What does their success tell us?  How important is the probabilistic sampling measure  ?  How important is the randomness of the sampling source?

27. 27 How important is the probabilistic sampling measure  ?  Visibility is usually not uniformly favorable across F  Regions with poorer visibility should be sampled more densely (more connectivity information can be gained there) small visibility sets small lookout sets good visibility poor visibility

28. 28 Impact s g Gaussian [Boor, Overmars, van der Stappen, 1999] Connectivity expansion [Kavraki, 1994]

29. 29  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

30. 30 Workspace-Guided Strategies  Main idea: Most narrow passages in configuration space derive from narrow passages in the workspace  Methods: Detect narrow passages in the workspace Sample more densely configurations that place selected robot points in workspace’s narrow passages Uniform sampling Workspace-guided sampling [Kurniawati and Hsu, 2004]

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32. 32 Filtering Strategies  Main Idea: Sample several configurations in the same region If a “pattern” is detected, then retain one of the configurations as a roadmap node (~ a more sophisticated probe)  More sampling work, but better distribution of nodes  Less time is wasted in connecting nodes  Methods: Gaussian sampling Bridge Test Visibility PRM

33. 33 Adaptive Strategies  Main idea: Use previous sampling results to identify which regions to sample next  Time-varying sampling measure   Methods: Connectivity expansion Diffusion (EST, RRT, SRT) Active learning

34. 34 Deformation Strategies  Main idea: Deform the free space to make it more expansive  Method: Free space dilatation

35. 35 Deformation Strategies  Main idea: Deform the free space to make it more expansive  Method: Free space dilatation dilated free space [Saha et al, 2004] Relation with Gaussian sampling

36. 36 How important is the randomness of the sampling source? Sampler = Uniform source S + Measure   Random  Pseudo-random  Deterministic [LaValle, Branicky, and Lindemann, 2004]

37. 37 Choice of the Source S  Adversary argument in theoretical proof  Efficiency (or lack of)  Robustness (or lack of)  Practical convenience (or lack of)

38. 38 Efficiency s g corridor width

39. 39 Robustness

40. 40 Practical Convenience  Think about implementing the Gaussian strategy with deterministic sampling

41. 41 Conclusion  In PRM, the word probabilistic matters.  The success of PRM planning depends mainly and critically on favorable visibility in F  The probability measure used for sampling F derives from the uncertainty on the shape of F  By exploiting the fact that visibility is not uniformly favorable across F, sampling measures have major impact on the efficiency of PRM planning  In contrast, the impact of the sampling source – random or deterministic – is small

42. 42 How to improve PRM planning?  By improving the sampling strategy!  How? By answering : What is the right granularity of a PRM collision-checking probe?  Build roadmaps where nodes and edges are not fully tested and labeled by probabilities (e.g., lazy collision checking strategies)  Adapt the granularity of the probe (e.g., to get more information on the local shape of F)

43. 43 Open Problems for PRM Planners  Free space made of multiple subspaces of different dimensionalities, like in legged locomotion on rough terrain or arm manipulation [Bretl and Hauser]

44. 44 Open Problems for PRM Planners  Motion spaces of linkages with huge number of degrees of freedom [Redon, 2004] Millipede-like robot with 13,000 joints [Amato et al., Apaydin et al., Singhal et al., Cortes et al.]

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47. 47 Connection Strategies  Which nodes to connect?  Which shapes of local paths to use?  Limit the number of connections: Nearest-neighbor strategy Connected component strategy  Large impact  Increase expansiveness: Multiple fixed shapes of local path [Amato 98]  small impact Local search strategy [Geraerts and Overmars, 2005, Isto 04]  uneven impact

48. 48  Poor expansiveness is caused by narrow passages  But narrow passages do not necessarily imply poor expansiveness Comments

49. 49  Many narrow passages can lead to a more expansive F than a single one Comments

50. 50  Windy passages are more difficult than straight ones Comments

51. 51  A convex free space is (1,1,1)-expansive Comments

52. 52 SBLSBL* (a)122959.4 (b)595532 (c)412.1 (d)863492 (e)63165 (f)>10000013588 (a) (b) (c) (d) (e) (f) Alpha 1.0 Experimental Results with Free Space Dilatation Time (s) [not including robot thinning]