# Probabilistic Planners

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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
M. Vendittelli                                                        3
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
M. Vendittelli                                                          5
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

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

local path        Cfree

milestone

qgoal
qinit

[Kavraki, Svetska, Latombe,Overmars, 95]

PMSM                                                  Gen.-Feb. 2007
M. Vendittelli                                                     8
 easy to implement
 fast, scalable to many degrees of freedom and complex
constraints
drawbacks
 probabilistic completeness
 limited insight

PMSM                                                   Gen.-Feb. 2007
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)

PMSM                                                    Gen.-Feb. 2007
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

connectivity
there should be a 1-to-1 map between the components of Cfree and those of

PMSM                                                          Gen.-Feb. 2007
M. Vendittelli                                                             11
Issues

 where to sample new milestones?
 sampling strategy

 which milestones to connect?
 connection strategy

goals
 achieve good coverage and connectivity

PMSM                                            Gen.-Feb. 2007
M. Vendittelli                                               12
Main Distinction

qi
qi

qi                     qg
qg
qg

PMSM                                             Gen.-Feb. 2007
M. Vendittelli                                                13
 compute a roadmap from scratch for each new query

qg
qi

PMSM                                               Gen.-Feb. 2007
M. Vendittelli                                                  14
Sampling in Multi-Query Strategies

•   Multi-stage sampling
• Obstacle-sensitive sampling
• Narrow-passage sampling

PMSM                                                     Gen.-Feb. 2007
M. Vendittelli                                                        15
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
M. Vendittelli                                                       16
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
M. Vendittelli                                                             17
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]

PMSM                                                         Gen.-Feb. 2007
M. Vendittelli                                                            18
Comparison with Gaussian Strategy

Gaussian              Bridge test

PMSM                                                  Gen.-Feb. 2007
M. Vendittelli                                                     19
other examples

PMSM                              Gen.-Feb. 2007
M. Vendittelli                                 20
Sampling in Single-Query Strategies

qg

qi

•   Diffusion
•   Biased sampling
•   Control-based sampling

PMSM                                                     Gen.-Feb. 2007
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

PMSM                                          Gen.-Feb. 2007
M. Vendittelli                                             22
   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]

PMSM                                                                   Gen.-Feb. 2007
M. Vendittelli                                                                      23

idea
make big steps in wide-open area of the free space,
and smaller steps in cluttered areas

PMSM                                           Gen.-Feb. 2007
M. Vendittelli                                              24
 shrinking-window strategy

qg

qi

[Sanchez-Ante, 02]

PMSM                               Gen.-Feb. 2007
M. Vendittelli                                  25
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

PMSM                                                     Gen.-Feb. 2007
M. Vendittelli                                                        26
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]

PMSM                                                       Gen.-Feb. 2007
M. Vendittelli                                                          27
Connection Strategies

 multi-query PRMs
 coarse connections

 single-query PRMs
 lazy collision checking

PMSM                                       Gen.-Feb. 2007
M. Vendittelli                                          28
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

PMSM                                                    Gen.-Feb. 2007
M. Vendittelli                                                       29
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

[Laumond and Simeon, 01]

PMSM                                                             Gen.-Feb. 2007
M. Vendittelli                                                                30
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
M. Vendittelli                                              31
qg
X
qi

[Sanchez-Ante, 02]

PMSM                     Gen.-Feb. 2007
M. Vendittelli                        32
qg

qi

[Sanchez-Ante, 02]

PMSM                 Gen.-Feb. 2007
M. Vendittelli                    33

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