Deterministic Sampling Methods for Spheres and SO(3)

1
Deterministic Sampling Methods for Spheres and
SO(3)

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

2
Motivation
Sampling over spheres arises in sampling-based
algorithms for solving

• Motion planning problems
• Optimization problems

Target applications are

• Robotics
• Computer graphics

• Control theory
• Computational biology

One important special case and our main
motivation

• Problem of motion planning for a rigid body in
  .

3
Motion planning for 3D rigid body

• Given
• Geometric models of a robot and obstacles in 3D
  world
• Configuration space
• Initial and goal configurations
• Task
• Compute a collision free path that connects
  initial and goal configurations

4
Motion planning for 3D rigid body

• Existing techniques
• Sampling-based motion planning algorithms based
  on random sequences
• Amato, Wu, 96 Bohlin, Kavraki, 00Kavraki,
  Svestka, Latombe,Overmars, 96LaValle, Kuffner,
  01Simeon, Laumond, Nissoux, 00Yu, Gupta, 98
• Drawbacks
• Resolution completeness is crucial
  (manufacturing, verification problems)
• If the planner does not return an answer, what
  are the guarantees on the existence of a solution?

5
The Goal

• Deterministic sequences for have been shown
  to perform well in practice (sometimes even with
  the improvement in the performance over random
  sequences)
• LaValle, Branicky, Lindemann 03 Matousek 99
  Niederreiter 92
• Problem
• Uniformity measure is induced by the metric, and
  therefore, partially by the topology of the space
• Cannot be applied to configuration spaces with
  different topology
• The Goal
• Extend deterministic sequences to spheres and
  SO(3)
• Arvo 95Blumlinger 91, Rote,Tichy 95
  Shoemake 85, 92 Kuffner 04 Mitchell 04

6
Parameterization of SO(3)

• Uniformity depends on the parameterization.
• Haar measure defines the volumes of the subsets
  of locally compact topological groups, so that
  they are invariant up to a rotation
• The parameterization of SO(3) with quaternions
  respects the bi-invariant Haar measure on SO(3)
• Quaternions can be viewed as all the points lying
  on S 3 with the antipodal points identified
• Close relationship between sampling on spheres
  and SO(3)

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Uniformity Criteria for Spheres and SO(3)

• Let R (range space) denote a collection of
  subsets of a sphere
• Discrepancy maximum volume estimation error
  over all boxes

R
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Uniformity Criteria for Spheres and SO(3)

• Let ? denote metric on a sphere
• Dispersion radius of the largest empty ball

9
The Outline of the Rest of the Talk

• Literature on sampling over spheres
• General approach for sampling over spheres
• A particular sequence (Layered Sukharev grid
  sequence) on spheres and SO(3) which
• is deterministic
• achieves low dispersion and low discrepancy
• is incremental
• has lattice structure
• can be efficiently generated
• Extension of this sequence to cross product
  spaces and SE(3)
• Properties and experimental evaluation of this
  sequence on the problems of motion planning

10
The Outline of the Rest of the Talk

• Literature on sampling over spheres
• General approach for sampling over spheres
• A particular sequence (Layered Sukharev grid
  sequence) on spheres and SO(3) which
• is deterministic
• achieves low dispersion and low discrepancy
• is incremental
• has lattice structure
• can be efficiently generated
• Extension of this sequence to cross product
  spaces and SE(3)
• Properties and experimental evaluation of this
  sequence on the problems of motion planning

11
Literature on sampling spheres and SO(3)

• Random sequences
• subgroup method for random sequences SO(3)
• almost optimal discrepancy random sequences for
  spheres
• Beck, 84 Diaconis, Shahshahani 87 Wagner,
  93 Bourgain, Linderstrauss 93
• Deterministic point sets
• optimal discrepancy point sets for SO(3)
• uniform deterministic point sets for SO(3)
• Lubotzky, Phillips, Sarnak 86 Mitchell 04
• No deterministic sequences to our knowledge

12
The Outline of the Rest of the Talk

• Literature on sampling over spheres
• General approach for sampling over spheres
• A particular sequence (Layered Sukharev grid
  sequence) which
• is deterministic
• achieves low dispersion and low discrepancy
• is incremental
• has lattice structure
• can be efficiently generated
• Extension of this sequence to cross product
  spaces and SE(3)
• Properties and experimental evaluation of these
  sequences on the problems of motion planning

13
Platonic Solids

• Regular polygons in R2
• Regular polyhedra in R3
• Regular polytopes in R4
• Regular polytopes in Rd , d gt 4
• Properties of the vertices of Platonic solids in
  R(d 1)
• Form a distribution on S d
• Provide uniform coverage of S d
• Provide lattice structure, natural for building
  roadmaps for planning

simplex, cube, cross polytope,24-cell, 120-cell,
600-cell
simplex, cube, cross polytope
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Platonic Solids

• Problem
• In higher dimensions there are only few regular
  polytopes
• How to obtain evenly distributed points for n
  points in Rd
• Is it possible to avoid distortions?
• General idea
• Borrow the structure of the regular polytopes and
  transform generated points on the surface of the
  sphere

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General Approach forDistributions on Spheres

• Take a good distribution of points on the surface
  of a polytope
• Project the faces of the polytope outward to form
  spherical tiling
• Use the same baricentric coordinates on spherical
  faces as they are on polytope faces

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Example. Sukharev Grid on S 1

• Take a square in R2
• Place Sukharev grid on each edge
• Project the edges of the square outwards to form
  circle tiling
• Place a Sukharev grid on each circular edge
• Important note similar procedure applies for any
  S d

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Example. Sukharev Grid on S 2

• Take a cube in R3
• 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
• Important note similar procedure applies for any
  S d

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Properties of Spherical Sukharev Grids

• Advantages
• distortions are easy to calculate
• lattice structure is beneficial for motion
  planning
• calculations are efficient
• easily extendable to sequences
• Disadvantages
• distortions grow with dimension

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The Outline of the Rest of the Talk

• Literature on sampling over spheres
• General approach for sampling over spheres
• A particular sequence (Layered Sukharev grid
  sequence) which
• is deterministic
• achieves low dispersion and low discrepancy
• is incremental
• has lattice structure
• can be efficiently generated
• Extension of this sequence to cross product
  spaces and SE(3)
• Properties and experimental evaluation of these
  sequences on the problems of motion planning

20
Layered Sukharev Grid Sequencein 0, 1d

• Places Sukharev grids one resolution at a time
• Achieves low dispersion and low discrepancy at
  each resolution
• Performs well in practice
• Can be easily adapted forspheres and SO(3)
• Lindemann, LaValle 2003

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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
22
The Outline of the Rest of the Talk

• Literature on sampling over spheres
• General approach for sampling over spheres
• A particular sequence (Layered Sukharev grid
  sequence) which
• is deterministic
• achieves low dispersion and low discrepancy
• is incremental
• has lattice structure
• can be efficiently generated
• Extension of this sequence to cross product
  spaces and SE(3)
• Properties and experimental evaluation of these
  sequences on the problems of motion planning

23
Layered Sukharev Grid Sequence for X ? Y

• Take cell structure in X and Y
• Define a cell structure in X ? Y
• Determine the cell ordering and the ordering
  inside each cell

Y
X
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Layered Sukharev Grid Sequence for SE(3)

• SE(3) SO(3) ? R3
• The measure can be defined as ?SO(3) ? ? R3
• This measure corresponds to the left-invariant
  Haar measure on SE(3)
• That is, defined construction will respect this
  Haar measure on SE(3)

25
The Outline of the Rest of the Talk

• Literature on sampling over spheres
• General approach for sampling over spheres
• A particular sequence (Layered Sukharev grid
  sequence) which
• is deterministic
• achieves low dispersion and low discrepancy
• is incremental
• has lattice structure
• can be efficiently generated
• Extension of this sequence to cross product
  spaces and SE(3)
• Properties and experimental evaluation of these
  sequences on the problems of motion planning

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Properties

• The dispersion of the sequence Ts at the
  resolution level l containing
  points is
• The relationship between the discrepancy of the
  sequence T at the resolution level l taken over
  d-dimensional spherical canonical rectangles and
  the discrepancy of the optimal sequence, To, is
• The sequence T has the following properties
• The position of the i-th sample in the sequence T
  can be generated in O(log i) time.
• For any i-th sample any of the 2d nearest grid
  neighbors from the same layer can be found in
  O((log i)/d) time.

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ExperimentsPRM method

• SO(3) configuration space
• Averaged over 50 trials

Random Quaternions Random Euler Angles Layered Sukharev Grid Sequence
1088 nodes 3021 nodes 1067 nodes
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ExperimentsPRM method

• SO(3) configuration space
• Averaged over 50 trials

Random Quaternions Random Euler Angles Layered Sukharev Grid Sequence
909 nodes gt80000 nodes 1013 nodes
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ExperimentsPRM method

• SE(3) configuration space
• Averaged over 50 trials

Random Quaternions Layered Sukharev Grid Sequence
3460 nodes 3285 nodes
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ExperimentsPRM method

• SE(3) configuration space
• Averaged over 50 trials

Random Quaternions Layered Sukharev Grid Sequence
3481 nodes 3202 nodes
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Conclusion

• We have proposed a general framework for uniform
  sampling over spheres, SO(3), and cross product
  spaces
• We have developed and implemented a particular
  sequence which extends the layered Sukharev grid
  sequence designed for a unit cube
• We have tested the performance of this sequence
  in a PRM-like motion planning algorithm
• We have demonstrated that the sequence is a
  useful alternative to random sampling, in
  addition to the advantages that it has

Future Work

• Reduce the amount of distortion introduced with
  more dimensions and with the size of polytopes
  faces
• Design deterministic sequences for other
  configuration spaces