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On the Probabilistic Foundations of Probabilistic

Roadmaps

Jean-Claude LatombeStanford University

joint work with David Hsu and Hanna Kurniawati

National University of Singapore

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

643, 2006.

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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

3This answer may occasionally be incorrect

Connection strategy

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 every node v in R such that v q do

If path(q,v) F then 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

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

Return a path between them

4. ElseReturn NoPath

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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 ?

How important is the randomness of the sampling source?

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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,), where • H is the set of all consistent hypotheses over the shapes of F • For every x H, (x) is the probability that x is correct

The probabilistic sampling measure used by the planner reflects this uncertainty. The goal is to minimize the expected number of iterations to connect s and g, whenever they lie in the same component of F

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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,), where • H is the set of all consistent hypotheses over the shapes of F

• For every x H, (x) is the probability that x is correct

The probabilistic sampling measure reflects this uncertainty. The 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

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[Kavraki et al., 1995]

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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)) εμ(F)

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

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

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F1 F2

Connectivity Issue

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F1 F2

Connectivity Issue

Lookout of F1

The β-lookout of a subset X of F is the set of all configurations in X that see a β-fraction of F\X

β-lookout(X) {q X | μ(V(q)\X) βμ(F\X)}

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F1 F2

(ε,,β)-Expansiveness of F

Lookout of F1

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

The β-lookout of a subset X of F is the set of all configurations in X that see a β-fraction of F\X

β-lookout(X) {q X | μ(V(q)\X) βμ(F\X)}

[Hsu et al., 1997]

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Expansiveness only depends on volumetric ratios

It is not directly related to the dimensionality of the configuration space

In 2-D the expansiveness of the free space can be made arbitrarily poor

In n-D the passage can be made narrow in 1, 2, ..., n-1 dimensions

Comments

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Theoretical Convergence of PRM Planning

Theorem 1 [Hsu, Latombe, Motwani, 1997]

Let F be (ε,,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

s g

Linking sequence

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Theoretical Convergence of PRM Planning

Theorem 1 [Hsu, Latombe, Motwani, 1997]

Let F be (ε,,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

= Pr(Failure)

Experimental convergence

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Theoretical Convergence of PRM Planning

Theorem 1 [Hsu, Latombe, Motwani, 1997]

Let F be (ε,,)-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 in (0,1], there exists o and o such that if F is not (ε,,)-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 .

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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

Revealing this property might well be the most important contribution of PRM planning

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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

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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

[Chaudhuri and Koltun, 2006] PRM planning with uniform sampling has polynomial smoothed running time in spaces of fixed dimensions (Recall that the worst-case running time of PRM planning is unbounded as a function of the input’s combinatorial complexity)

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 ?

How important is the randomness of the sampling source?

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How 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 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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Workspace-Guided Strategies Main idea:

• Most narrow passages in configuration space derive from narrow passages in the workspace

Methods: • Detect narrow passages in the workspace • Sample more densely configurations that place

selected robot points in workspace’s narrow passages

Uniform sampling

Workspace-guided sampling[Kurniawati and Hsu, 2004]

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??

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Filtering Strategies Main Idea:

• Sample several configurations in the same region • If a “pattern” is detected, then retain one of the

configurations as a roadmap node(~ a more sophisticated probe) More sampling work, but better distribution of nodes Less time is wasted in connecting nodes

Methods:• Gaussian sampling• Bridge Test• Visibility PRM

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Adaptive Strategies

Main idea:• Use previous sampling results to identify which

regions to sample next Time-varying sampling measure

Methods:• Connectivity expansion• Diffusion (EST, RRT, SRT)• Active learning

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Deformation Strategies

Main idea:• Deform the free space

to make it more expansive

Method:• Free space

dilatation

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Deformation Strategies

Main idea:• Deform the free space

to make it more expansive

Method:• Free space

dilatation

dilated free space

[Saha et al, 2004]

Relation with Gaussian sampling

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How important is the randomness of the sampling

source?

Sampler = Uniform source S + Measure

Random Pseudo-random Deterministic [LaValle, Branicky, and Lindemann,

2004]

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Choice of the Source S

Adversary argument in theoretical proof

Efficiency (or lack of) Robustness (or lack of) Practical convenience (or lack of)

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Efficiency

s g

corridor width

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Robustness

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Practical Convenience

Think about implementing the Gaussian strategy with deterministic sampling

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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

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How to improve PRM planning?

By improving the sampling strategy!

How? By answering :What is the right granularity of a PRM collision-checking probe?

Build roadmaps where nodes and edges are not fully tested and labeled by probabilities(e.g., lazy collision checking strategies)

Adapt the granularity of the probe (e.g., to get more information on the local shape of F)

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Open Problems for PRM Planners

Free space made of multiple subspaces of different dimensionalities, like in legged locomotion on rough terrain or arm manipulation

[Bretl and Hauser]

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Open Problems for PRM Planners

Motion spaces of linkages with huge number of degrees of freedom

[Redon, 2004]

Millipede-like robot with 13,000 joints

[Amato et al., Apaydin et al., Singhal et al., Cortes et al.]

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Connection Strategies

Which nodes to connect? Which shapes of local paths to use?

Limit the number of connections:• Nearest-neighbor strategy • Connected component strategy Large impact

Increase expansiveness:• Multiple fixed shapes of local path [Amato 98]

small impact• Local search strategy [Geraerts and Overmars, 2005, Isto 04]

uneven impact

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Poor expansiveness is caused by narrow passages

But narrow passages do not necessarily imply poor expansiveness

Comments

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Many narrow passages can lead to a more expansive F than a single one

Comments

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Windy passages are more difficult than straight ones

Comments

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A convex free space is (1,1,1)-expansive

Comments

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SBL SBL*

(a) 12295 9.4

(b) 5955 32

(c) 41 2.1

(d) 863 492

(e) 631 65

(f) >100000

13588

(a) (b) (c)

(d) (e)(f)

Alpha 1.0

Experimental Results with Free Space Dilatation

Time (s) [not including robot thinning]