1 probabilistic roadmaps cs 326a: motion planning
Post on 19-Dec-2015
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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 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 ?
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 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 aμ(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 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,a,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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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 2For 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?
How important is the randomness of the sampling source?
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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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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
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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