CS 326A: Motion Planning

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CS 326A Motion Planning

• Probabilistic Roadmaps
• Sampling and Connection Strategies

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Two Types of Strategies

• Where to sample new milestones?? Sampling
  strategy
• Which milestones to connect?? Connection
  strategy
• Goal
• Minimize roadmap size to correctly
  answermotion-planning queries

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Impact of the Sampling Strategy
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Rationale for Non-Uniform Sampling Strategy

• Visibility is not uniformly favorable in free
  space
• Regions with poorer visibility should be more
  densely sampled

good visibility
poor visibility
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• 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

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• Workspace-guided strategiesIdentify narrow
  passages in the workspace and map them into the
  configuration space
• Filtering strategiesSample many configurations,
  find interesting patterns, and retain only
  promising configurations
• Adaptive strategiesAdjust the sampling
  distribution (p) on the fly
• Deformation strategiesDeform the free space,
  e.g., to widen narrow passages

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Multi- vs. Single-Query Roadmaps

• Multi-query roadmaps? Pre-compute roadmap?
  Re-use roadmap for answering queries? The
  roadmap must cover the free space well
• Single-query roadmaps? Compute a roadmap from
  scratch for each new query? Often roadmap
  consists of 2 trees rooted at the query
  configurations

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Workspace-Guided Strategies

• Rationale Most narrow passages in configuration
  space are caused by narrow passages in the
  workspace
• Method
• Detect narrow passages in the workspace (e.g.,
  cell decomposition, medial-axis transform)
• Sample robot configurations that place selected
  robot points in workspaces narrow passages
• - H. Kurniawati and D. Hsu. Workspace importance
  sampling for probabilistic roadmap planning. In
  Proc. IEEE/RSJ Int. Conf. on Intelligent Robots
  Systems, pp. 16181623, 2004.
• - J.P. van den Berg and M. H. Overmars. Using
  Workspace Information as a Guide to Non-Uniform
  Sampling in Probabilistic Roadmap Planners. IJRR,
  24(12)1055-1071, Dec. 2005.

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Workspace-Guided Strategies
Workspace-guided sampling
Uniform sampling
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??
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Non-Uniform Sampling Strategies

• Workspace-guided strategies
• Filtering strategies
• Adaptive strategies
• Deformation strategies

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Filtering Strategies

• Main Idea
• Sample several configurations in the same region
  of configuration space
• If a pattern is detected, then retain one of
  the configurations as a roadmap node
• More sampling work, but better distribution of
  nodes
• Less time is wasted in connecting
  non-interesting milestones
• Methods
• Gaussian sampling
• Bridge Test
• Hybrid

- V. Boor, M. H. Overmars, and A. F. van der
Stappen.The Gaussian sampling strategy for
probabilistic roadmap planners. In Proc. 1999
IEEE Int. Conf. Robotics and Automation, 1999,
pp. 10181023. - Z. Sun, D. Hsu, T. Jiang, H.
Kurniawati, and J. Reif . Narrow passage
sampling for probabilistic roadmap planners.
IEEE Trans. on Robotics, 21(6)11051115, 2005.
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Gaussian Sampling

• Sample a configuration q uniformly at random from
  configuration space
• Sample a real number x at random with Gaussian
  distribution N0,s(x)
• Sample a configuration q in the ball B(q,x)
  uniformly at random
• If only one of q and q is in free space, retain
  the one in free space as a node else retain none

What is the effect? What is the intuition?
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Example of Node Distribution
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Uniform vs. Gaussian Sampling
Milestones (13,000) created by uniform sampling
before the narrow passage was adequately sampled
Milestones (150) created by Gaussian sampling
The gain is not in sampling fewer milestones,
but in connecting fewer pairs of milestones
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Bridge Test

• Sample two conformations q and q using Gaussian
  sampling technique
• If none is in free space, then
• if qm (qq)/2 is in free space, then retain
  qm as a node
• Else retain none

What is the effect? What is the intuition?
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Bridge Test
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Example of Distribution
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Example of Distribution
Bridge test
Gaussian
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Example of Distribution
8-joint robot with mobile base
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Example of Distribution
7-joint robot with fixed base
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Hybrid Sampling

• Sample two configurations q and q using Gaussian
  sampling technique
• If both are in free space, then retain one (any
  of the two) as a node with low probability (e.g.,
  0.1)
• Else if only one is in free space, then retain it
  as a node with intermediate probability (e.g.,
  0.5)
• Else if qm (qq)/2 is in free space, then
  retain it as a node with probability 1

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Uniform
Uniform Bridge test
Bridge test
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Non-Uniform Sampling Strategies

• Workspace-guided strategies
• Filtering strategies
• Adaptive strategies
• Deformation strategies

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Adaptive Strategies

• Main idea
• Use intermediate sampling results to identify
  regions of the free space whose connectivity is
  more difficult to capture ? Time-varying
  sampling measure
• Methods
• Connectivity expansion
• Diffusion

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Connectivity Expansion

• Use work already done to detect poor-visibility
  regions

Kavraki, 94
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Connectivity Expansion

• Use work already done to detect low-visibility
  regions

Kavraki, 94
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Example of Distribution
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Diffusion Strategy(Density-Based Hsu et al,
97, RRT LaValle and Kuffner, 00)
g
s
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Adaptive-Step Sampling
g
s
Sánchez-Ante, 2003
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Non-Uniform Sampling Strategies

• Workspace-guided strategies
• Filtering strategies
• Adaptive strategies
• Deformation strategies

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Deformation Strategies

• Main idea
• Deform the free space to make it more expansive
• Method
• Free space dilatation

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Motivating Experiment
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Free Space Dilatation

• Pre-computationSlim the robot / obstacles
• Planning
• Compute a path for slimmed robot
• Deform this path for original robot

• M. Saha, J.C. Latombe, Y.-C. Chang, F. Prinz.
  Finding Narrow Passages
• with Probabilistic Roadmaps The Small-Step
  Retraction Method.
• Autonomous Robots, 19(3)301-319, Dec. 2005.
• H.-L. Cheng, D. Hsu, J.-C. Latombe, and G.
  Sánchez-Ante . Multi-level
• free-space dilation for sampling narrow passages
  in PRM planning.
• Proc. IEEE Int. Conf. on Robotics Automation,
  2006.

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Free Space Dilatation
Roadmap construction and repair
widened passage
fattened free space
? up to 2 orders of magnitude speedup
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Some Results
Up to 3 orders of magnitude speedup
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Connection Strategies

• Limit number of connections
• Nearest-neighbor strategy
• Connected component strategy
• Increase expansiveness
• Library of local path shapes Amato 98
• Local search strategy Isto 04
• Delay costly computation
• Lazy collision checking Sanchez-Ante, 02

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Lazy Collision Checking
X
Sánchez-Ante, 2002
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Lazy Collision Checking
x10 speedup
Sánchez-Ante, 2002
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Rationale of Lazy Collision Checking

• Connections between close milestones have high
  probability of being free of collision
• 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