deterministic sampling methods for spheres and so(3) anna yershova steven m. lavalle dept. of...
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Deterministic Sampling Methods for Spheres and SO(3)
Anna Yershova Steven M. LaValleDept. of Computer Science
University of Illinois Urbana, IL, USA
Motivation
One important special case and our main motivation:
Motion planning problems Optimization problems
Sampling over spheres arises in sampling-based algorithms for solving:
Problem of motion planning for a rigid body in .
Target applications are:
Robotics Computer graphics
Control theory Computational biology
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
Motion planning for 3D rigid body
Existing techniques: Sampling-based motion planning algorithms based on random
sequences[Amato, Wu, 96; Bohlin, Kavraki, 00;Kavraki, Svestka, Latombe,Overmars, 96;LaValle, Kuffner, 01;Simeon, Laumond, Nissoux, 00;Yu, Gupta, 98]
Drawbacks: Would we like to exchange probabilistic completeness for
resolution completeness? In some applications resolution completeness is crucial (e.g.
verification problems)
Motion planning for 3D rigid body
Deterministic sequences for have been shown to perform well in practice (sometimes even with the improvement in the performance over random sequences)
[Lindemann, LaValle, 2003], [Branicky, LaValle, Olsen, Yang 2001] [LaValle, Branicky, Lindemann] [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 95][Blumlinger 91], [Rote,Tichy 95] [Shoemake 85, 92] [Kuffner 04] [Mitchell 04]
The Goal
Parameterization of SO(3)
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
Close relationship between sampling on spheres and SO(3)
Uniformity Criteria on Spheres and SO(3)
Discrepancy of a point set: The largest empty volume
that can fit in between the points
Dispersion of a point set: The radius of the largest
empty ball
The Outline of the Rest of the Talk
Provide general approach for sampling over spheres
Develop 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
Properties and experimental evaluation of this sequence on the problems of motion planning
The Outline of the Rest of the Talk
Provide general approach for sampling over spheres
Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which:
achieves low dispersion and low discrepancy is deterministic is incremental has lattice structure can be efficiently generated
Properties and experimental evaluation of this sequence on the problems of motion planning
Regular polygons in R2:
Regular polyhedra in R3:
Regular polytopes in R4:
Regular polytopes in Rd , d > 4:
Properties of the vertices of Platonic solids in Rd: Form a distribution on S d
Provide uniform coverage of S d
Provide lattice structure, natural for building roadmaps for planning
…
Platonic Solids
simplex, cube, cross polytope,24-cell, 120-cell, 600-cell
simplex, cube, cross polytope
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
Platonic Solids
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
General Approach forDistributions on Spheres
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
The Outline of the Rest of the Talk
Provide general approach for sampling over spheres
Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which:
achieves low dispersion and low discrepancy is deterministic is incremental has lattice structure can be efficiently generated
Properties and experimental evaluation of this sequence on the problems of motion planning
Layered Sukharev Grid Sequencein d
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]
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
The Outline of the Rest of the Talk
Provide general approach for sampling over spheres
Develop a particular sequence (Layered Sukharev grid sequence) on spheres and SO(3) which:
achieves low dispersion and low discrepancy is deterministic is incremental has lattice structure can be efficiently generated
Properties and experimental evaluation of this sequence on the problems of motion planning
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.
Random Quaternions Random Euler Angles Layered Sukharev Grid Sequence
1088 nodes 3021 nodes 1067 nodes
ExperimentsPRM method
SO(3) configuration space
Averaged over 50 trials
ExperimentsPRM method
Random Quaternions Random Euler Angles Layered Sukharev Grid Sequence
909 nodes >80000 nodes 1013 nodes
SO(3) configuration space
Averaged over 50 trials
Conclusion We have proposed a general framework for uniform sampling over
spheres and SO(3)
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 polytope’s faces
Design deterministic sequences for more general topological spaces