Sampling Methods in Robot Motion Planning

1
Sampling Methods in Robot Motion Planning

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

2
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

3
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

4
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)
• Complete 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)

5
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).

6
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.
7
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

8
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

9
QMC Applications

• Optimization problem (finding a maximum of a
  function)
• given continuous real function, f, defined on
  0, 1d
• solution take a point sequence (xn) ? 0, 1d,
  define m1 f(x1), and recursively set
• Integration problem in higher dimensions (finding
  average)
• given continuous real function, f, defined on
  0, 1d
• solution
• QMC methods proved to be very successful in
  Computer Graphics
• mental images was awarded a Technical
  Achievement Academy Award (Oscar) for developing
  a rendering software in such movies as The
  Matrix, Spider Man, Harry Potter.

10
Basic Definitions

• Sample types over
• Literature landmarks 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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Measuring the (Lack of) Quality

• Global quality measure, used for integration

12
Measuring the (Lack of) Quality

• Local quality measure, used for optimization

13
Optimal Sequences and Point Sets

• Low discrepancy sequence
• Low discrepancy point set
• Low dispersion sequence/point set
• Implied constants may be big, for example for
  dispersion
• Low discrepancy implies good dispersion, but not
  necessarily optimal

14
Sukharev Sampling Criterion
15
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

16
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).

17
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?

20
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

21
Connecting Sample Quality to Problem Difficulty
Problem Quality Measure Difficulty Measure Theoretical Bound
integration discrepancy bounded Hardy-Krause variation Koksma-Hlawka inequality
optimization dispersion modulus of continuity N92
motion planning dispersion corridor thickness our analysis
22
Decidability of Configuration Spaces
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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

26
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

27
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)

29
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

30
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 for PRMs
32
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

33
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.
34
What is the Role of Sampling in RRTs?Lindemann,
LaValle 2004

• Random samples induce Voronoi bias exploration in
  RRTs
• Is this the best way to approximate the Voronoi
  regions?
• Attempts to design other sampling techniques
• use k samples at each iteration to estimate the
  vertex with the biggest Voronoi region
• reuse these k samples for some number of
  iterations
• deterministic samples can be used

35
What is the Role of Sampling in RRTs?

• Produces less nodes, less collision checks
• Not numerically robust
• Computations are still expensive

36
Discussions

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

37
Discussions

• How to develop stratified and adaptive sampling
  for motion planning?