On the Probabilistic Foundations of Probabilistic Roadmaps D' Hsu, J'C' Latombe, H' Kurniawati' On t

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On the Probabilistic Foundations of Probabilistic
RoadmapsD. 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 robots 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

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Procedure BasicPRM(s,g,N)

• Initialize the roadmap R with two nodes, s and g
• Repeat
• Sample a configuration q from C with probability
  measure p
• If q ? F then add q as a new node of R
• 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
  R
• until s and g are in the same connected
  component of R or R contains N2 nodes
• If s and g are in the same connected component of
  R then
• Return a path between them
• Else
• Return NoPath

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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 p?
• 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,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
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
x1
x2
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Visibility in FKavraki et al., 1995

• 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

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e-Goodness of FKavraki et al., 1995

• Let µ(X) stand for the volume of X ? F
• Given e ? (0,1, q ? F is e-good if it sees at
  least an e-fraction of F, i.e., if µ(V(q)) ?
  e?µ(F)
• F is e-good if every q in F is e-good
• Intuition If F is e-good, then with high
  probability a small set of configurations
  sampled at random will see most of F

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Connectivity Issue
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Connectivity Issue
The ß-lookout of a subset F1 of F is the set of
all configurations in F1 that see a ß-fraction of
F2 F\ F1 ß-lookout(F1) q ? F1 µ(V(q)?F2)
? ß?µ(F2)
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Connectivity Issue
The ß-lookout of a subset F1 of F is the set of
all configurations in F1 that see a ß-fraction of
F2 F\ F1 ß-lookout(F1) q ? F1 µ(V(q)?F2)
? ß?µ(F2)
F is (e,a,b)-expansive if it is e-good and each
one of its subsets X has a ß-lookout whose volume
is at least a?µ(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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Comments

• 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
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Thanks to the wide passage at the bottom this
space favorably expansive
Many narrow passages might be better than a
single one
This spaces expansiveness is worsethan if the
passage was straight
A convex set is maximally expansive,i.e., e a
b 1
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Theoretical Convergence of PRM Planning

• Theorem 1 Hsu, Latombe, Motwani, 1997
• Let F be (e,a,ß)-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

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Theoretical Convergence of PRM Planning

• Theorem 1 Hsu, Latombe, Motwani, 1997
• Let F be (e,a,b)-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 e gt 0, any N gt 0, and any g in (0,1,
  there exists ao and bo such that if F is not
  (e,a,b)-expansive for a gt a0 and b gt b0, 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 p?
• How important is the randomness of the sampling
  source?

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How important is the probabilistic sampling
measure p?

• 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 lookout sets
small visibility sets
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Impact
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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How important is the randomness of the sampling
source?

• Sampler Uniform source S Measure p
• Random
• Pseudo-random
• Deterministic LaValle, Branicky, and Lindemann,
  2004

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Choice of the Source S

• Adversary argument in theoretical proof
• Efficiency
• Practical convenience

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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