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 free-flying 3D object, or a multijoint articulated object is often prohibitive
 But very fast algorithms exist that can check if a rigid or
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
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
But many pathological cases …
Illustration of a Bad Potential
“Landscape”
U
q
Global minimum
Probabilistic Roadmap (PRM)
Space
n
forbidden space
Free/feasible space
7
Probabilistic Roadmap (PRM)
Configurations are sampled by picking coordinates at random
8
Probabilistic Roadmap (PRM)
Configurations are sampled by picking coordinates at random
9
Probabilistic Roadmap (PRM)
Sampled configurations are tested for collision
10
Probabilistic Roadmap (PRM)
The collision-free configurations are retained as milestones
11
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
13
Probabilistic Roadmap (PRM)
The collision-free links are retained as local paths to form the PRM
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Probabilistic Roadmap (PRM)
The start and goal configurations are included as milestones
g
s
15
Probabilistic Roadmap (PRM)
The PRM is searched for a path from s to g
g
s
16
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
Procedure BasicPRM(s,g,N)
1.
2.
3.
4.
Initialize the roadmap R with two nodes, s and g
Repeat:
Sampling strategy
a. Sample a configuration q from C with probability measure p
b. If q  F then add q as a new node of R
c. 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
N+2 nodes
If s and g are in the same connected component of R then
Return a path between them
Else
Return NoPath
This answer may occasionally be incorrect
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
PRM planners work well in
practice. Why?
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?
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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.]
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
 Under which conditions is this the case?
Relation to Monte Carlo
Integration
x2
f(x)
I=  f(x)dx
x1
A=a×b
b
# red
I
A
# red # black
(xi,yi)
x1
a
x2 x
Relation to Monte Carlo
Integration
x2
f(x)
I=
f(x)dx

But a PRM planner must construct a path
x1
The connectivity of F mayAdepend
= a ×onb small
b
regions
# red
Insufficient sampling of such
regions
may lead  A
I
the planner to failure
# red # black
(xi,yi)
x1
a
x2 x
Visibility in F
• 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}
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))  εμ(F)
F
V(q)
• F is ε-good if every q in F
is ε-good
q
• Intuition:
If F is ε-good, then with high
probability a small set of configurations sampled
at random will see most of F
Here, ε ≈ 0.18
Connectivity Issue
F1
F2
Connectivity Issue
F1
Lookout of F1
F2
Connectivity Issue
F1
The β-lookout of a subset F1 of F is the set of all
configurationsFin
2 F1 that see a β-fraction of F2 =
F\ F1
β-lookout(F1) = {q  F1 | μ(V(q)F2)  βμ(F2)}
Lookout of F1
Connectivity Issue
The β-lookout of a subset F1 of F is the set of all
configurationsFin
2 F1 that see a β-fraction of F2 =
F\ F1
F1
β-lookout(F1) = {q  F1 | μ(V(q)F2)  βμ(F2)}
Lookout of F1
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 αμ(X)
Intuition: If F is favorably expansive, it should be relatively easy to
capture its connectivity by a small network of sampled configurations
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
Thanks to the wide passage at the
bottom this space is favorably
expansive
This space’s expansiveness is worse
than if the passage was straight
Many narrow passages might be
better than a single one
A convex set is maximally expansive,
i.e., ε = α = β = 1
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
Linking sequence
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.
What does the empirical success
of PRM planning tell us?
It tells us that F is often favorably expansive
despite its overwhelming algebraic and
geometric complexity
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
Most narrow passages in F are
intentional …
… but it is not easy to
intentionally create
complex narrow
passages in F
Alpha puzzle
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 probabilistic
sampling measure π?
 Visibility is usually not uniformly favorable across F
good visibility
small lookout sets
small visibility sets
poor visibility
 Regions with poorer visibility should be sampled more densely
(more connectivity information can be gained there)
Impact
s
Gaussian
[Boor, Overmars,
van der Stappen, 1999]
g
Connectivity expansion
[Kavraki, 1994]
 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
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
How important is the randomness of
the sampling source?
Sampler = Uniform source S + Measure π
 Random
 Pseudo-random
 Deterministic [LaValle, Branicky, and Lindemann, 2004]
Choice of the Source S
 Adversary argument in theoretical proof
 Efficiency
 Practical
gconvenience
s