Slide 1

Probabilistic RoadmapsThe complexity of the robots free space is overwhelming

The cost of computing an exact representation of the configuration space of a free-flying 3D object, or a multi-joint articulated object is often prohibitive

But very fast algorithms exist that can check if a rigid or articulated object at a given configuration collides with obstacles (more next lecture)

Basic idea of Probabilistic Roadmaps (PRMs): Compute a very simplified representation of the free space by sampling configurations at random using some probability measureInitial idea: Potential Field + Random Walk Attract some points toward their goal Repulse other points by obstacles Use collision check to test collision Escape local minima by performing random walks4But many pathological cases 5Illustration of a Bad Potential LandscapeUqGlobal minimum6Probabilistic Roadmap (PRM)7Free/feasible spaceSpace nforbidden spaceProbabilistic Roadmap (PRM)8Configurations are sampled by picking coordinates at randomProbabilistic Roadmap (PRM)9Configurations are sampled by picking coordinates at randomProbabilistic Roadmap (PRM)10Sampled configurations are tested for collisionProbabilistic Roadmap (PRM)11The collision-free configurations are retained as milestonesProbabilistic Roadmap (PRM)12Each milestone is linked by straight paths to its nearest neighborsProbabilistic Roadmap (PRM)13Each milestone is linked by straight paths to its nearest neighborsProbabilistic Roadmap (PRM)14The collision-free links are retained as local paths to form the PRMProbabilistic Roadmap (PRM)15sgThe start and goal configurations are included as milestonesProbabilistic Roadmap (PRM)16The PRM is searched for a path from s to gsgMulti- vs. Single-Query PRMsMulti-query roadmaps Pre-compute roadmap Re-use roadmap for answering queries

Single-query roadmaps Compute a roadmap from scratch for each new query

This answer may occasionally be incorrectSampling strategyProcedure BasicPRM(s,g,N)Initialize the roadmap R with two nodes, s and gRepeat:Sample a configuration q from C with probability measure pIf q F then add q as a new node of RFor some nodes v in R such that v q doIf path(q,v) F then add (q,v) as a new edge of RUntil s and g are in the same connected component of R or R contains N+2 nodesIf s and g are in the same connected component of R thenReturn a path between themElseReturn NoPath18Requirements of PRM PlanningChecking sampled configurations and connections between samples for collision can be done efficiently. Hierarchical collision detection

A relatively small number of milestones and local paths are sufficient to capture the connectivity of the free space. Non-uniform sampling strategies

19PRM planners work well in practice. Why?

20PRM planners work well in practice. Why?Why are they probabilistic?

What does their success tell us?

How important is the probabilistic sampling measure p?2121Why 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 sensorsAt any moment, there exists an 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.]

22So ...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

Under which conditions is this the case?

23Relation to Monte Carlo Integrationxf(x)

abA = a bx1x2(xi,yi)

24Relation to Monte Carlo Integrationxf(x)

abA = a bx1x2(xi,yi)

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 failure25Visibility in F

Two configurations q and q see each other ifpath(q,q) FThe visibility set of q is V(q) = {q | path(q,q) F}26-Goodness of F

Let (X) stand for the volume of X F

Given (0,1], q F is -good if it sees at least an -fraction of F, i.e., if (V(q)) (F)

F is -good if every q in F is -good

Intuition: If F is -good, then with high probability a small set of configurations sampled at random will see most of FqFV(q)Here, 0.1827

F1F2Connectivity Issue28

F1F2Connectivity IssueLookout of F129

F1F2Connectivity IssueLookout of F1The -lookout of a subset F1 of F is the set of all configurations in F1 that see a -fraction of F2 = F\ F1

-lookout(F1) = {q F1 | (V(q)F2) (F2)} 30

F1F2Connectivity IssueLookout of F1The -lookout of a subset F1 of F is the set of all configurations in F1 that see a -fraction of F2 = F\ F1

-lookout(F1) = {q F1 | (V(q)F2) (F2)} F is (,,)-expansive if it is -good and each one of its subsets X has a -lookout whose volume is at least (X)Intuition: If F is favorably expansive, it should be relatively easy to capture its connectivity by a small network of sampled configurations31CommentsExpansiveness only depends on volumetric ratiosIt is not directly related to the dimensionality of the configuration spaceE.g., in 2-D the expansiveness of the free space can be made arbitrarily poor32Thanks to the wide passage at the bottom this space is favorably expansive Many narrow passages might be better than a single oneThis spaces expansiveness is worsethan if the passage was straightA convex set is maximally expansive,i.e., = = = 133Theoretical Convergence of PRM PlanningTheorem 1 Let F be (,,)-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

g = Pr(Failure)

Experimental convergence34Linking sequenceTheoretical Convergence of PRM PlanningTheorem 1 Let F be (,,)-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 2For any > 0, any N > 0, and any g in (0,1], there exists o and o such that if F is not (,,)-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 g.36What does the empirical success of PRM planning tell us?It tells us that F is often favorably expansive despite its overwhelming algebraic and geometric complexity37In retrospect, is this property surprising?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

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Most narrow passages in F are intentional but it is not easy to intentionally create complex narrow passages in FAlpha puzzle39PRM planners work well in practice. Why?Why are they probabilistic?

What does their success tell us?

How important is the probabilistic sampling measure ?

40How 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 setssmall lookout setsgood visibilitypoor visibility41Impact

sgGaussian[Boor, Overmars, van der Stappen, 1999]Connectivity expansion[Kavraki, 1994]42But 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 strategies43ConclusionThe 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 planning44How important is the randomness of the sampling source?Sampler = Uniform source S + Measure

Random

Pseudo-random

Deterministic [LaValle, Branicky, and Lindemann, 2004]

45Choice of the Source SAdversary argument in theoretical proofEfficiency

Practical convenience

sg46