Rationale of PRM Planners

1. On the Probabilistic Foundations of Probabilistic RoadmapsD. Hsu, J.C. Latombe, H. Kurniawati. On the Probabilistic Foundations of Probabilistic Roadmap Planning. IJRR, 25(7):627-643, 2006.

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. Sampling strategy This answer may occasionally be incorrect Procedure BasicPRM(s,g,N) • Initialize the roadmap R with two nodes, s and g • Repeat: • Sample a configuration q from C with probability measure p • If q  F then add q as a new node of R • For some nodes v in R such that v q do If path(q,v)  Fthen 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 • If s and g are in the same connected component of R then Return a path between them • Else Return NoPath

4. PRM planners work well in practice. Why?

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

6. 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 existsan implicit distribution (H,s), where H is the set of all consistent hypotheses over the shapes of F For every x  H, s(x) is the probability that x is correct The probabilistic sampling measure p reflects this uncertainty. [Its goal is to minimize the expected number of remaining iterations to connect s and g, whenever they lie in the same component of F.] Why is PRM planning probabilistic?

7. 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? So ...

8. Relation to Monte Carlo Integration f(x) A = a × b b (xi,yi) a x 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 x1 x2

9. Visibility in F[Kavraki et al., 1995] • 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}

10. F V(q) q Here, ε ≈ 0.18 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 e-good if every q in F is e-good • Intuition: If F is ε-good, then with high probability a small set of configurations sampled at random will see most of F

11. F1 F2 Connectivity Issue

12. F1 F2 Lookout of F1 Connectivity Issue The β-lookoutof a subset F1 of F is the set of all configurations in F1 that see a β-fraction of F2 = F\ F1 β-lookout(F1) = {q  F1 | μ(V(q)F2)  βμ(F2)}

13. F1 F2 Lookout of F1 Connectivity Issue The β-lookoutof a subset F1 of F is the set of all configurations in F1 that see a β-fraction of F2 = F\ F1 β-lookout(F1) = {q  F1 | μ(V(q)F2)  βμ(F2)} F is(ε,a,b)-expansiveif it is ε-good and each one of its subsets X has a β-lookout whose volume is at least aμ(X) Intuition: If F is favorably expansive, it should be relatively easy to capture its connectivity by a small network of sampled configurations

14. Comments • Expansiveness only depends on volumetric ratios • Itis not directly related to the dimensionality of the configuration space E.g., in 2-D the expansiveness of the free space can be made arbitrarily poor

15. Thanks to the wide passage at the bottom this space favorably expansive Many narrow passages might be better than a single one This space’s expansiveness is worsethan if the passage was straight A convex set is maximally expansive,i.e., ε = a = b = 1

16. Experimental convergence g= Pr(Failure) Theoretical Convergence of PRM Planning Theorem 1 [Hsu, Latombe, Motwani, 1997] Let F be (ε,a,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

17. Theoretical Convergence of PRM Planning Theorem 1 [Hsu, Latombe, Motwani, 1997] Let F be (ε,a,b)-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 g in (0,1], there exists ao and bo such that if F is not (ε,a,b)-expansive for a > a0 and b > b0, 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 g.

18. 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

19. Not really! Narrow passages are unstable features under small random perturbations of the robot/workspace geometry  Poorly expansive space are unlikely to occur by accident In retrospect, is this property surprising?

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

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

22. good visibility poor visibility How important is the probabilistic sampling measure p? • 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 lookout sets small visibility sets

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

24. 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

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

26. g s Choice of the Source S • Adversary argument in theoretical proof • Efficiency • Practical convenience

27. 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