The Recent Impact of QMC Methods on Robot Motion Planning

1
The Recent Impact of QMC Methods on Robot Motion
Planning

• Steven M. LaValle Stephen R. Lindemann
• Anna Yershova
• Dept. of Computer Science
• University of Illinois
• Urbana, IL, USA

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The Goal of the Talk

• Introduce the MC2QMC community to the problem of
  robot motion planning
• Survey the state of the art of the sampling
  techniques in Motion Planning
• Discussion on the unique challenges and open
  problems that arise in Robotics

3
Talk Overview

• Motion Planning Problem
• QMC Philosophy in Motion Planning
• A Spectrum of Planners from Grids to Random
  Roadmaps
• Connecting Difficulty of Motion Planning with
  Sampling Quality
• QMC techniques and extensible lattices in the
  Motion Planning Planners
• Conclusions and Discussion

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Classical Motion Planning Problem Moving Pianos

• Given
• (geometric model of a robot)
• (space of configurations, q, thatare
  applicable to )
• (the set of collision
  freeconfigurations)
• Initial and goal configurations
• Task
• Compute a collision free path that connects
  initial and goal configurations

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Typical Configuration Spaces

• Translations in 2D, 3D 2, 3
• Rotations in 2D S1
• Rotations in 3D SO(3)
• Motions of chains of 2D objects (S 1)n
• Motion of 3D chains depending on the type of
  joints SE(3) x S 1 x
• Motions of closed chains algebraic variaties
• Motions of multiple robots (SE(3))n
• Humanoid type robot performing manipulation tasks
  up to 100 dimension configuration space
  containing a multiple copies of all the above
• Obstacles in these spaces represent collisions
  (with obstacles, self-collisions and collisions
  with other robots)

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Applications of Motion Planning

• (Coordinated) ManipulationPlanning
• Computational Chemistryand Biology
• Medical applications
• Computer Graphics(motions for digital actors)
• Autonomous vehicles and spacecrafts

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History of Motion Planning

• Grid Sampling, AI Search (beginning of time-1977)
• Experimental mobile robotics, etc.
• Problem Formalization (1977-1983)
• PSPACE-hardness (Reif, 1979)
• Configuration space (Lozano-Perez, 1981)
• Exact Solutions (1983-1988)
• Cylindrical algebraic decomposition (Schwartz,
  Sharir, 1983)
• Stratifications, roadmap (Canny, 1987)
• Sampling-based Planning (1988-present)
• Randomized potential fields (Barraquand, Latombe,
  1989)
• Ariadne's clew algorithm (Ahuactzin, Mazer, 1992)
• Probabilistic Roadmaps (PRMs) (Kavraki, Svestka,
  Latombe, Overmars, 1994)
• Rapidly-exploring Random Trees (RRTs) (LaValle,
  Kuffner, 1998)

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Probabilistic Roadmaps (PRMs)Kavraki, Latombe,
Overmars, Svestka, 1994

• Developed for high-dimensional spaces
• Avoid pitfalls of classical grid search
• Random sampling of Cfree
• Find neighbors of each sample(radius parameter)
• Local planner attempts connections
• Probabilistic completeness" achieved
• Other PRM variants Obstacle-Based PRM (Amato,
  Wu, 1996) Sensor-based PRM (Yu, Gupta, 1998)
  Gaussian PRM (Boor, Overmars, van der Stappen,
  1999) Medial axis PRMs (Wilmarth, Amato,
  Stiller, 1999 Pisula, Ho, Lin, Manocha, 2000
  Kavraki, Guibas, 2000) Contact space PRM (Ji,
  Xiao, 2000) Closed-chain PRMs (LaValle, Yakey,
  Kavraki, 1999 Han, Amato 2000) Lazy PRM
  (Bohlin, Kavraki, 2000) PRM for changing
  environments (Leven, Hutchinson, 2000)
  Visibility PRM (Simeon, Laumond, Nissoux, 2000).

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Rapidly-Exploring Random Trees (RRTs)LaValle,
Kuffner, 1998
movie
Other RRT variants Frazzoli, Dahleh, Feron,
2000 Toussaint, Basar, Bullo, 2000 Vallejo,
Jones, Amato, 2000 Strady, Laumond, 2000
Mayeux, Simeon, 2000 Karatas, Bullo, 2001 Li,
Chang, 2001 Kuner, Nishiwaki, Kagami, Inaba,
Inoue, 2000, 2001 Williams, Kim, Hofbaur, How,
Kennell, Loy, Ragno, Stedl, Walcott, 2001
Carpin, Pagello, 2002 Urmson, Simmons, 2003.
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Talk Overview

• Motion Planning Problem
• QMC Philosophy in Motion Planning
• A Spectrum of Planners from Grids to Random
  Roadmaps
• Connecting Difficulty of Motion Planning with
  Sampling Quality
• QMC techniques and extensible lattices in the
  Motion Planning Planners
• Conclusions and Discussion

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QMC Philosophy

• From 1989-2000 most of the community contributed
  planning success to randomization
• Questions
• Is randomization really the reason why
  challenging problems have been solved?
• Is random sampling in PRM advantageous?
• Approach
• Recognize that all machine implementations of
  random numbers produce deterministic sequences
• View sampling as an optimization problem
• Define criterion, and choose samples that
  optimize it for an intended application

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Talk Overview

• Motion Planning Problem
• QMC Philosophy in Motion Planning
• A Spectrum of Planners from Grids to Random
  Roadmaps
• Connecting Difficulty of Motion Planning with
  Sampling Quality
• QMC techniques and extensible lattices in the
  Motion Planning Planners
• Conclusions and Discussion

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Probabilistic RoadmapsKavraki, Latombe,
Overmars, Svestka, 1994

• Developed for high-dimensional spaces
• Avoid pitfalls of classical grid search
• Random sampling of Cfree
• Find neighbors of each sample(radius parameter)
• Local planner attempts connections
• Probabilistic completeness" achieved
• Other PRM variants Obstacle-Based PRM (Amato,
  Wu, 1996) Sensor-based PRM (Yu, Gupta, 1998)
  Gaussian PRM (Boor, Overmars, van der Stappen,
  1999) Medial axis PRMs (Wilmarth, Amato,
  Stiller, 1999 Pisula, Ho, Lin, Manocha, 2000
  Kavraki, Guibas, 2000) Contact space PRM (Ji,
  Xiao, 2000) Closed-chain PRMs (LaValle, Yakey,
  Kavraki, 1999 Han, Amato 2000) Lazy PRM
  (Bohlin, Kavraki, 2000) PRM for changing
  environments (Leven, Hutchinson, 2000)
  Visibility PRM (Simeon, Laumond, Nissoux, 2000).

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A Spectrum of Roadmaps

• Random Samples Halton
  sequence

Hammersley Points Lattice
Grid
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A Spectrum of Planners

• Grid-Based Roadmaps (grids, Sukharev grids)
• optimal dispersion poor discrepancy explicit
  neighborhood structure
• Lattice-Based Roadmaps (lattices, extensible
  lattices)
• optimal dispersion near-optimal discrepancy
  explicit neighborhood structure
• Low-Discrepancy/Low-Dispersion (Quasi-Random)
  Roadmaps (Halton sequence, Hammersley point set)
• optimal dispersion and discrepancy irregular
  neighborhood structure
• Probabilistic (Pseudo-Random) Roadmaps
• non-optimal dispersion and discrepancy irregular
  neighborhood structure
• Literature 1916 Weyl 1930 van der Corput 1951
  Metropolis 1959 Korobov 1960 Halton,
  Hammersley 1967 Sobol' 1971 Sukharev 1982
  Faure 1987 Niederreiter 1992 Niederreiter 1998
  Niederreiter, Xing 1998 Owen, Matousek2000
  Wang, Hickernell

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Questions

• What uniformity criteria are best suited for
  Motion Planning
• Which of the roadmaps alone the spectrum is best
  suited for Motion Planning?

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Talk Overview

• Motion Planning Problem
• QMC Philosophy in Motion Planning
• A Spectrum of Planners from Grids to Random
  Roadmaps
• Connecting Difficulty of Motion Planning with
  Sampling Quality
• QMC techniques and extensible lattices in the
  Motion Planning Planners
• Conclusions and Discussion

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Connecting Sample Quality to Problem Difficulty
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Decidability of Configuration Spaces
x
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Undecidability Results
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Comparing to Random Sequences
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The Goal for Motion Planning

• We want to develop sampling schemes with the
  following properties
• uniform (low dispersion or discrepancy)
• lattice structure
• incremental quality (it should be a sequence)
• on the configuration spaces with different
  topologies

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Talk Overview

• Motion Planning Problem
• QMC Philosophy in Motion Planning
• A Spectrum of Planners from Grids to Random
  Roadmaps
• Connecting Difficulty of Motion Planning with
  Sampling Quality
• QMC techniques and extensible lattices in the
  Motion Planning Planners
• Conclusions and Discussion

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Layered Sukharev Grid Sequencein 0, 1d

• Places Sukharev grids one resolution at a time
• Achieves low dispersion at each resolution
• Achieves low discrepancy
• Has explicit neighborhoodstructure
• Lindemann, LaValle 2003

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Sequences for SO(3)

• Important points
• Uniformity depends on the parameterization.
• Haar measure defines the volumes of the sets in
  the space, so that they are invariant up to a
  rotation
• The parameterization of SO(3) with quaternions
  respects the unique (up to scalar multiple) Haar
  measure for SO(3)
• Quaternions can be viewed as all the points lying
  on S 3 with the antipodal points identified
• Notions of dispersion and discrepancy can be
  extended to the surface of the sphere
• Close relationship between sampling on spheres
  and SO(3)

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Sukharev Grid on S d

• Take a cube in Rd1
• Place Sukharev grid on each face
• Project the faces of the cube outwards to form
  spherical tiling
• Place a Sukharev grid on each spherical face

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Layered Sukharev Grid Sequence for Spheres

• Take a Layered Sukharev Grid sequence inside each
  face
• Define the ordering on faces
• Combine these two into a sequence on the sphere

Ordering on faces Ordering inside faces
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Experimental Results

• Random sequence produces slightly more node in
  the roadmap than QMC sequences and the Layered
  Sukharev Grid sequence
• The amount of the computation is saved in Layered
  Sukharev Grid sequence due to efficient
  generation and fast nearest neighbor search
• All the improvements observed so far are not very
  significant

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Conclusions

• Random sampling in the PRMs seems to offer no
  advantages over the deterministic sequences
• Deterministic sequences can offer advantages in
  terms of dispersion, discrepancy and neighborhood
  structure for motion planning

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Discussion

• Are there sequences that will give a significant
  superior performance for motion planning?
• How to develop importance sampling sequences?
• How to develop deterministic techniques for
  sampling over general topological spaces that
  arise in motion planning?
• What to do in higher dimensions?
• How to derandomize other motion planning
  algorithms?