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Part II
Probabilistic Planners
PMSM Gen.-Feb. 2007
M. Vendittelli 1
Complete Planners
find a collision-free path whenever one exists and return failure otherwise
method C time complexity space extensions
comp.
CB i
generalized polygons
Visibility
Graph C = R2 O(n2) O(n2) C = R3,
polyhedral CB i
Generalized
CB i generalized polygons
Voronoi C = R2 O(n log n) O(n)
Diagram 3D GVD
optimal:
C = R3
Polygonal C= R2 NP-hard O(n)
polyhedral CB i
Cell Dec. trapezoidal
(sweep-line): O(n
log n)
PMSM Gen.-Feb. 2007
n: #vertices of CB
M. Vendittelli 2
Resolution Complete Planners
discretize the space and return a path whenever one exists in
this representation
example: approximate cell decomposition
2dim(C)-tree decomposition
2dim(C)h leaves (h: height of the tree)
labeling when C = R2 takes linear time in the number of
constraints defining C-obstacles
produce too many cells for high dimensional C
PMSM Gen.-Feb. 2007
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Planners Based on Potential Fields
very efficient but not complete: they may fail to find a free
path, even if one exists
construction of a navigation function is difficult; known
solution only for CBi of simple shapes and/or when
dim(C) = 2 or 3
local minima remain an important cause of inefficiency
some engineering allows constructing planners which are both
quite efficient and reasonably reliable
PMSM Gen.-Feb. 2007
M. Vendittelli 4
Deterministic Planning Methods for Articulated Robots
Silhouette → simple exponential time (O(2dim(C) ))
Collins decomposition → double exp. time (O(2^(2dim(C))))
extension of Freeway method for a manipulator with only
revolute joints
Approximate Cell Decomposition: obtain an alternative labeling
method by discretizing the motion of each joint into small
intervals
Potential Fields: define workspace potential for a set of
“control points”
PMSM Gen.-Feb. 2007
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Sampling-Based Motion Planning
idea
sample the space of interest
connect sampled points by simple paths (local paths)
search the resulting graph
examples
grid-based methods
(deterministic sampling)
probabilistically complete planners
randomized potential fields
Probabilisitc Roadmaps (PRMs)
PMSM Gen.-Feb. 2007
M. Vendittelli 6
Probabilistically Complete Planners
the probability that the planner will find a solution
whenever one exists is a function that goes to 1 as
running time increases
initial idea randomized potential fields
initialization (i=1)
stuck
i++ best-first and i=k
random walk (random/det. sampli ng) backtrack
stuck and i < k reset i to 1
was able to solve problems up to 31 dof, but too may heuristic
prameters to be adjusted for each problem
PMSM Gen.-Feb. 2007
M. Vendittelli 7
Probabilistic Roadmap (PRM)
local path Cfree
milestone
qgoal
qinit
[Kavraki, Svetska, Latombe,Overmars, 95]
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advantages
easy to implement
fast, scalable to many degrees of freedom and complex
constraints
drawbacks
probabilistic completeness
limited insight
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M. Vendittelli 9
motivation
computing an explicit representation of Cfree is hard but
checking sampled configurations and connections between
samples for collision can be done efficiently
hierarchical collision checking
a relatively small number of milestones and local paths are
sufficient to capture the connectivity of the free space
exponential convergence if free space has appropriate
properties (probabilistic completeness)
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M. Vendittelli 10
Desirable Properties of a PRM
coverage
the milestones should see most of the admissible space to guarantee
that the initial and goal configurations can be easily connected to
the roadmap
bad good
connectivity
there should be a 1-to-1 map between the components of Cfree and those of
the roadmap
bad good
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Issues
where to sample new milestones?
sampling strategy
which milestones to connect?
connection strategy
goals
minimize roadmap size
achieve good coverage and connectivity
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Main Distinction
multi-query roadmaps
pre-compute roadmap
re-use roadmap for answering queries
qi
qi
qi qg
qg
qg
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single-query roadmaps
compute a roadmap from scratch for each new query
qg
qi
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M. Vendittelli 14
Sampling in Multi-Query Strategies
• Multi-stage sampling
• Obstacle-sensitive sampling
• Narrow-passage sampling
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Multi-Stage Strategies
idea use intermediate sampling results to identify regions of the
free space whose connectivity is more difficult to capture
example two-stage sampling
[Kavraki, 94]
PMSM Gen.-Feb. 2007
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Obstacle-Sensitive Strategies
rationale
the connectivity of free space is more difficult to capture near its
boundary than in wide-open area
ray casting from samples in obstacles
[Amato, Overmars]
Gaussian sampling
[Boor, Overmars, van der Stappen, 99]
PMSM Gen.-Feb. 2007
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Narrow-Passage Strategies
rationale
finding the connectivity of the free space through narrow passages
is the only hard problem
Medial-Axis Bias
[Amato, Kavraki]
Bridge test
[Hsu et al, 02]
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Comparison with Gaussian Strategy
Gaussian Bridge test
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other examples
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Sampling in Single-Query Strategies
qg
qi
• Diffusion
• Adaptive step
• Biased sampling
• Control-based sampling
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M. Vendittelli 21
Diffusion Strategies
rationale
the trees of milestones should diffuse throughout
the free space to guarantee that the planner will
find a path with high probability, if one exists
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Density-based strategy
associate a sampling density to each milestone in the trees
pick a milestone m at random with probability inverse to density
expand from m
[Hsu et al, 97]
RRT strategy
pick a configuration q uniformly at random in C-space
select the closest milestone m to q
expand from m
[LaValle and Kuffner, 00]
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M. Vendittelli 23
Adaptive-Step Strategies
idea
make big steps in wide-open area of the free space,
and smaller steps in cluttered areas
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shrinking-window strategy
qg
qi
[Sanchez-Ante, 02]
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Biased Strategies
rationale
use heuristic knowledge extracted from the workspace
example
define a potential field U and bias tree growth along the
steepest descent of U
use task knowledge
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Control-Based Strategies
idea
directly satisfy differential kinodynamic constraints
method
represent motion in state (configuration x velocity) space
pick control input at random
integrate motion over short interval of time
[Kindel, Hsu, et al, 00] [LaValle and Kuffner, 00]
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Connection Strategies
multi-query PRMs
coarse connections
single-query PRMs
lazy collision checking
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Coarse Connections
rationale
since connections are expensive to test, pick only those which
have a good chance to be collision-free and to contribute to
the roadmap connectivity
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methods
1. connect only pairs of milestones that are not too far apart
2. connect each milestone to at most k other milestones
3. connect two milestones only if they are in two distinct components of
the current roadmap ( the roadmap is a collection of acyclic graph)
4. Visibility-based roadmap keep a new milestone m if:
m cannot be connected to any previous milestone and
m can be connected to 2 previous milestones belonging to distinct
components of the roadmap
[Laumond and Simeon, 01]
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Lazy Collision Checking
rationale
connections between close milestones have high
probability of being collision-free
most of the time spent in collision checking is used
to test connections
most collision-free connections will not be part of
the final path
testing connections is more expensive for collision-
free connections
hence, postpone the tests of connections until they
are absolutely needed
PMSM Gen.-Feb. 2007
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qg
X
qi
[Sanchez-Ante, 02]
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qg
qi
[Sanchez-Ante, 02]
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M. Vendittelli 33
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