probabilistic roadmaps
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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 PresentationTRANSCRIPT
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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 (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