Probabilistic Roadmaps

CS 326A: Motion Planning

The complexity of the robot’s free space is overwhelming

The cost of computing an exact representation of the configuration space of a multi-joint articulated object is often prohibitive

But very fast algorithms exist that can check if an 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 measure

Initial 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 walks

But many pathological cases …

Illustration of a Bad Potential “Landscape”

U

q

Global minimum

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Probabilistic Roadmap (PRM)Free/feasible space

Space nforbidden space

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Probabilistic Roadmap (PRM)Configurations are sampled by picking coordinates at random

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Probabilistic Roadmap (PRM)Configurations are sampled by picking coordinates at random

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Probabilistic Roadmap (PRM)Sampled configurations are tested for collision

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Probabilistic Roadmap (PRM)The collision-free configurations are retained as milestones

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Probabilistic Roadmap (PRM)Each milestone is linked by straight paths to its nearest neighbors

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Probabilistic Roadmap (PRM)Each milestone is linked by straight paths to its nearest neighbors

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Probabilistic Roadmap (PRM)The collision-free links are retained as local paths to form the PRM

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Probabilistic Roadmap (PRM)

s

g

The start and goal configurations are included as milestones

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Probabilistic Roadmap (PRM)The PRM is searched for a path from s to g

s

g

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Multi- vs. Single-Query PRMs

Multi-query roadmaps Pre-compute roadmap Re-use roadmap for answering queries

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

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This answer may occasionally be incorrect

Sampling strategy

Procedure BasicPRM(s,g,N)

1. Initialize the roadmap R with two nodes, s and g2. Repeat:

a. Sample a configuration q from C with probability measure b. If q F then add q as a new node of Rc. For some nodes v in R such that v q do

If 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 nodes

3. If s and g are in the same connected component of R then

Return a path between them4. Else

Return NoPath

Requirements of PRM Planning

1. Checking sampled configurations and connections between samples for collision can be done efficiently. Hierarchical collision detection

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

PRM planners work well in practice Why?

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PRM planners work well in practice. Why?

Why are they probabilistic?

What does their success tell us?

How important is the probabilistic sampling measure ?

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,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.]

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?

Relation to Monte Carlo Integration

x

f(x)

2

1

x

x

I = f (x)dx

a

bA = a × b

x1 x2

(xi,yi)

Ablack#red#

red#I

Relation to Monte Carlo Integration

x

f(x)

2

1

x

x

I = f (x)dx

a

bA = a × b

x1 x2

(xi,yi)

Ablack#red#

red#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

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

ε-Goodness of F

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)) εμ(F)

F is ε-good if every q in F is ε-good

Intuition: If F is ε-good, then with high probability a small set of configurations sampled at random will see most of F

q

F

V(q)

Here, ε ≈ 0.18

F1 F2

Connectivity Issue

F1 F2

Connectivity Issue

Lookout of F1

F1 F2

Connectivity Issue

Lookout of F1

The β-lookout of 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) βμ(F2)}

F1 F2

Connectivity Issue

Lookout of F1

The β-lookout of 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) βμ(F2)}

F is (ε,α,β)-expansive if it is ε-good and each one of its subsets X has a β-lookout whose volume is at least αμ(X)

Intuition: If F is favorably expansive, it should be relatively easy to capture its connectivity by a small network of sampled configurations

Expansiveness only depends on volumetric ratios

It is 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

Comments

Thanks to the wide passage at the bottom this space is 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 = α = β = 1

Theoretical Convergence 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 exponential rate as N increases

g = Pr(Failure)

Experimental convergence

Linking sequence

Theoretical Convergence 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 exponential rate as N increases

Theorem 2

For any ε > 0, any N > 0, and any g in (0,1], there exists αo and βo such that if F is not (ε,α,β)-expansive for α > α0 and β > β0, 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.

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

In 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

Most narrow passages in F are intentional …

… but it is not easy to intentionally create complex narrow passages in F

Alpha puzzle

PRM planners work well in practice. Why?

Why are they probabilistic?

What does their success tell us?

How important is the probabilistic sampling measure π?

How important is the probabilistic sampling measure π?

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 sets

good visibility

poor visibility

Impact

s g

Gaussian[Boor, Overmars,

van der Stappen, 1999]

Connectivity expansion[Kavraki, 1994]

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

Conclusion 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

How important is the randomness of the sampling source?

Sampler = Uniform source S + Measure π

Random

Pseudo-random

Deterministic [LaValle, Branicky, and Lindemann, 2004]

Adversary argument in theoretical proof Efficiency

Practical convenience

Choice of the Source S

s g