# Sampling and Connection Strategies for PRM Planners by Kittibitti

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```									Sampling and Connection Strategies
for PRM Planners

Jean-Claude Latombe

Computer Science Department
Stanford University
Original Problem
q0        q1

q2
qn
t(s)
q3
q4
The “Solution”:
free space
The “Solution”:
local path free space

milestone

mg

mb
The New Issues

Where to sample new milestones?
 Sampling strategy

Which milestones to connect?
 Connection strategy
Examples

 Two-stage sampling:
1) Build initial roadmap with uniform sampling
2) Perform additional sampling around poorly
connected milestones

 Coarse Connection:
2) Attempt connection between 2 milestones only
if they are in two distinct components
Multi-Query PRM
Single-Query PRM

mg
mb
Multi-Query PRM

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

Rationale:
One can use intermediate sampling
results to identify regions of the free
space whose connectivity is more
difficult to capture
Two-Stage Sampling

[Kavraki, 94]
Two-Stage Sampling

[Kavraki, 94]
Obstacle-Sensitive Strategies

Rationale:
The connectivity of free space is more
difficult to capture near its boundary
than in wide-open area
Obstacle-Sensitive Strategies

Ray casting from samples in obstacles

[Amato, Overmars]

Gaussian sampling

[Boor, Overmars, van der Stappen, 99]
Multi-Query PRM

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

Rationale:
Finding the connectivity of the free
space through narrow passage is the
only hard problem.
Narrow-Passage Strategies

 Medial-Axis Bias
[Amato, Kavraki]

 Dilatation/contraction of the free space
[Baginski, 96; Hsu et al, 98]

 Bridge test
[Hsu et al, 02]
Bridge Test
Comparison with Gaussian Strategy

Gaussian      Bridge test
Other Examples
Running Times

The bridge test most likely yields a high
rejection rate of configurations
But, in general it results in a much
smaller number of milestones, hence
much fewer connections to be tested
Since testing connections is costly,
there can be significant computational
gain
More on this later ….
Single-Query PRM

mg
mb
•   Diffusion
•   Biased sampling
•   Control-based sampling
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
Diffusion Strategies

 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 milestone m the closest from q
 Expand from m
[LaValle and Kuffner, 00]

Rationale:
Makes big steps in wide-open area of
the free space, and smaller steps in
cluttered areas.
Shrinking-window strategy

mg
mb

[Sanchez-Ante, 02]
Single-Query PRM

mg
mb
•   Diffusion
•   Biased sampling
•   Control-based sampling
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
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
Control-Based Strategies

Rationale:
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]
The New Issues

Where to sample new milestones?
 Sampling strategy

Which milestones to connect?
 Connection strategy
Connection Strategies

 Multi-query PRMs
 Coarse connections

 Single-query PRMs
 Lazy collision checking
Coarse Connections

Rationale:
Since connections are expensive to test, pick
only those which have a good chance to test
collision-free and to contribute to the roadmap
connectivity.
Coarse Connnections

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
collection of acyclic graph)
4. Visibility-based roadmap: Keep a new milestone m if:
a)    m cannot be connected to any previous milestone and
b)    m can be connected to 2 previous milestones belonging to distinct
[Laumond and Simeon, 01]
Connection Strategies

 Multi-query PRMs
 Coarse connections

 Single-query PRMs
 Lazy collision checking
Lazy Collision Checking
Rationale:
 Connections between close milestones have high
probability of being collision-free
 Most of the time spent in collision checking is done 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
Lazy Collision Checking

mg

mb
X

[Sanchez-Ante, 02]
Lazy Collision Checking

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mb

[Sanchez-Ante, 02]
Possible New Strategy

 Rationale:
 Single-query planners are often more suitable than multi-query’s
 But there are some very good multi-query strategies
 Milestones are much less expensive to create than connections

 Pre-compute the milestones of the roadmap,                      with
uniform sampling, two-stage sampling, bridge test, and
dilatation/contraction of free space to place milestones well

 Process queries with single-query roadmaps
restricted to pre-computed milestones, with lazy
collision checking
Application to Probabilistic