Sampling and Connection Strategies for PRM Planners

1
Sampling and Connection Strategiesfor PRM
Planners

• Jean-Claude Latombe
• Computer Science Department
• Stanford University

2
Original Problem
3
The Solution Probabilistic Roadmap (PRM)
free space
4
The SolutionProbabilistic Roadmap (PRM)
free space
5
The New Issues

• Where to sample new milestones?? Sampling
  strategy
• Which milestones to connect?? Connection strategy

6
Examples

• Two-stage sampling
• Build initial roadmap with uniform sampling
• Perform additional sampling around poorly
  connected milestones
• Coarse Connection
• Maintain roadmaps connected components
• Attempt connection between 2 milestones only if
  they are in two distinct components

7
Multi-Query PRM
8
Single-Query PRM
9
Multi-Query PRM

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

10
Multi-Stage Strategies

• Rationale
• One can use intermediate sampling results to
  identify regions of the free space whose
  connectivity is more difficult to capture

11
Two-Stage Sampling
Kavraki, 94
12
Two-Stage Sampling
Kavraki, 94
13
Obstacle-Sensitive Strategies

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

14
Obstacle-Sensitive Strategies

• Ray casting from samples in obstacles
• Gaussian sampling

Amato, Overmars
Boor, Overmars, van der Stappen, 99
15
Multi-Query PRM

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

16
Narrow-Passage Strategies

• Rationale
• Finding the connectivity of the free space
  through narrow passage is the only hard problem.

17
Narrow-Passage Strategies

• Medial-Axis Bias
• Dilatation/contraction of the free space
• Bridge test

Amato, Kavraki
Baginski, 96 Hsu et al, 98
Hsu et al, 02
18
Bridge Test
19
Comparison with Gaussian Strategy
Bridge test
Gaussian
20
Other Examples
21
Running Times
22
Comments (JCL)

• 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 .

23
Single-Query PRM

• Diffusion
• Adaptive step
• Biased sampling
• Control-based sampling

24
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

25
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
• RRT strategy
• Pick a configuration q uniformly at random in
  c-space
• Select the milestone m the closest from q
• Expand from m

Hsu et al, 97
LaValle and Kuffner, 00
26
Adaptive-Step Strategies

• Rationale
• Makes big steps in wide-open area of the free
  space, and smaller steps in cluttered areas.

27
Adaptive-Step Strategies

• Shrinking-window strategy

mg
mb
Sanchez-Ante, 02
28
Single-Query PRM

• Diffusion
• Adaptive step
• Biased sampling
• Control-based sampling

29
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

30
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

31
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
32
The New Issues

• Where to sample new milestones?? Sampling
  strategy
• Which milestones to connect?? Connection strategy

33
Connection Strategies

• Multi-query PRMs ? Coarse connections
• Single-query PRMs ? Lazy collision checking

34
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.

35
Coarse Connnections

• Methods
• Connect only pairs of milestones that are not too
  far apart
• Connect each milestone to at most k other
  milestones
• Connect two milestones only if they are in two
  distinct components of the current roadmap (? the
  roadmap is a collection of acyclic graph)
• 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
36
Connection Strategies

• Multi-query PRMs ? Coarse connections
• Single-query PRMs ? Lazy collision checking

37
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

38
Lazy Collision Checking
X
Sanchez-Ante, 02
39
Lazy Collision Checking
Sanchez-Ante, 02
40
Possible New Strategy

• Rationale
• Single-query planners are often more suitable
  than multi-querys
• 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

41
Application to Probabilistic Conformational
Roadmap