Probabilistic Roadmaps

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Probabilistic Roadmaps CS 326A: Motion Planning

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CS 326A: Motion Planning. Probabilistic Roadmaps. 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 - PowerPoint PPT Presentation

Transcript of Probabilistic Roadmaps

Page 1: Probabilistic Roadmaps

Probabilistic Roadmaps

CS 326A: Motion Planning

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The complexity of the robot’s free space is overwhelming

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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

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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

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But many pathological cases …

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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

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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

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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 ?

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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.]

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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?

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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

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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

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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

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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

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F1 F2

Connectivity Issue

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F1 F2

Connectivity Issue

Lookout of F1

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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)}

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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

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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

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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

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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

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Linking sequence

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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.

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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

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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

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Most narrow passages in F are intentional …

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

Alpha puzzle

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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 π?

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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

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Impact

s g

Gaussian[Boor, Overmars,

van der Stappen, 1999]

Connectivity expansion[Kavraki, 1994]

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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

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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

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How important is the randomness of the sampling source?

Sampler = Uniform source S + Measure π

Random

Pseudo-random

Deterministic [LaValle, Branicky, and Lindemann, 2004]

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Adversary argument in theoretical proof Efficiency

Practical convenience

Choice of the Source S

s g