1/31/2012 Idea: Reduce the Robot to a Point Configuration Space of an � Configuration Space Articulated Robot Free space Two-Revolute-Joint Robot � A configuration of a robot q 2 is a list of non-redundant parameters that fully q 2 specify the position and orientation of each of its i t ti f h f it bodies � In this robot, one possible choice is: (q 1 , q 2 ) The configuration space (C-space) has 2 dimensions ? 1
1/31/2012 Every robot maps to a point in Every robot maps to a point in its configuration space ... its configuration space ... ~40 D ~40 D 15 D 15 D 6 D 6 D 12 D 12 D q 0 q 0 q 1 q 1 q 0 q 0 q 1 q 1 q n q n q 3 q 3 ~65-120 D ~65-120 D q 4 q 4 ... and every robot path is a Issues!! curve in configuration space � Dimensionality of configuration space q 0 q 0 q 1 q 1 � Geometric complexity of free region q n � Plan in configuration space, but compute in workspace q 3 q 4 10 Rationale of Probabilistic Roadmap (PRM) Planners � The cost of computing an exact representation of the configuration space of a multi-joint articulated object is Probabilistic Roadmaps often prohibitive. (Sampling-Based Planning) (S li B d Pl i ) � But very fast algorithms exist that can � But very fast algorithms exist that can check if an articulated object at a given configuration collides with obstacles. � 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 11 12 2
1/31/2012 Probabilistic Roadmap (PRM) Probabilistic Roadmap (PRM) n -D space forbidden space feasible space Configurations are sampled by picking coordinates at random g g g g s s 13 14 Probabilistic Roadmap (PRM) Probabilistic Roadmap (PRM) Configurations are sampled by picking coordinates at random Sampled configurations are tested for feasibility g g g g s s 15 16 Probabilistic Roadmap (PRM) Probabilistic Roadmap (PRM) Feasible configurations are retained as “milestones” Each milestone is linked by straight paths to its nearest neighbors g g g g s s 17 18 3
1/31/2012 Probabilistic Roadmap (PRM) Probabilistic Roadmap (PRM) The feasible links are retained to form the PRM The feasible links are retained to form the PRM g g g g s s 19 20 Probabilistic Roadmap (PRM) Procedure BasicPRM (s,g,N) The PRM is built until s and g are connected 1. Initialize the roadmap R with two nodes, s and g 2. Repeat: Sampling strategy a. Sample a configuration q from C with probability measure π b. If q ∈ F then add q as a new node of R c. For some nodes v in R such that v ≠ q do g g Connection If path (q,v) ∈ F then add (q,v) as a new edge of R If th ( ) F th dd ( ) d f R strategy until s and g are in the same connected component of R or R contains N+2 nodes s 3. If s and g are in the same connected component of R then Return a path between them 4. Else Return NoPath This answer may occasionally be incorrect 21 22 PRM planners work well in Why is PRM planning practice. Why? probabilistic? � A PRM planner ignores the exact shape of F. � Why are they probabilistic? So, it acts like a robot building a map of an � What does their success tell us? unknown environment with limited sensors � The probabilistic sampling measure π reflects Th b bili ti li fl t � How important is the probabilistic this uncertainty. The goal is to minimize the sampling measure π ? expected number of remaining iterations to connect s and g, whenever they lie in the same � How important is the randomness component of F of the sampling source? 23 4
1/31/2012 Monte Carlo Integration So ... � PRM planning trades the cost of computing F x exactly against the cost of dealing with f(x) 2 ∫ I= f(x)dx uncertainty, by incrementally sampling milestones and connecting them in order to x 1 “learn” the connectivity of F y A = a × b A = a × b b � This choice is beneficial only if a small roadmap #brown has high probability to represent F well enough I ×A ≈ #brown+#gray to answer planning queries correctly and such a small roadmap has high probability to be generated x 1 x 2 a x � Under which conditions is this the case? 26 Connectivity Issue Experiment 27 Visibility in F Connectivity Issue Two configurations q and q’ see each other if path (q,q’) ∈ F S 1 S 2 30 5
1/31/2012 Connectivity Issue � Expansiveness only depends on volumetric ratios S 1 S 2 S 1 S 2 � It is not directly related to the dimensionality of the configuration space Lookout of S 1 In 2-D the expansiveness of the free space can be made arbitrarily poor F is expansive if each one of its subsets X has a “large” lookout 31 32 Probabilistic Completeness of Which Ones are Most Difficult? PRM Planning Theorem 1 Let F be ( ε,α,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 exponential rate as N increases γ = Pr(Failure) ≤ Experimental convergence 33 34 Probabilistic Completeness of Intuition PRM Planning Theorem 1 If F is favorably expansive, then it is easy to capture its Let F be ( ε,α,β )-expansive, and s and g be two configurations connectivity by a small network of sampled configurations in the same component of F. BasicPRM (s,g,N) with uniform sampling returns a path g s between s and g with probability converging to 1 at an exponential rate as N increases exponential rate as N increases Theorem 2 For any ε > 0, any N > 0, and any g in (0,1], there exists α 0 and β 0 such that if F is not ( ε,α,β )-expansive for α > α 0 and β > β 0 , then there exists s and γ in the same component of F Linking sequence such that BasicPRM (s,g,N) fails to return a path with probability greater than γ . 6
1/31/2012 Probabilistic Completeness of PRM Planning Theorem 1 Let F be ( ε,α,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 In general, a PRM planner is unable to g p exponential rate as N increases exponential rate as N increases detect that no path exists Theorem 2 For any ε > 0, any N > 0, and any g in (0,1], there exists α 0 and β 0 such that if F is not ( ε,α,β )-expansive for α > α 0 and β > β 0 , then there exists s and γ in the same component of F such that BasicPRM (s,g,N) fails to return a path with probability greater than γ . 38 What does the empirical In retrospect, success of PRM planning tell us? is this property surprising? � Not really! Narrow passages are unstable It tells us that F has often good visibility features under small random perturbations of the properties despite its overwhelming p p p g robot/workspace geometry robot/workspace geometry geometric complexity 39 Most narrow passages in F are Impact of Sampling Strategy intentional … … but it is not easy to intentionally y create complex g s narrow passages in F Alpha puzzle Gaussian Connectivity expansion [Boor, Overmars, 41 42 [Kavraki, 1994] van der Stappen, 1999] 7
1/31/2012 Key Topics for Future Lectures � Sampling/connection strategies � Fast collision checking 43 8
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