anna yershova 1 , steven m. lavalle 2 , and julie c. mitchell 3

Anna Yershova1, Steven M. LaValle2,and Julie C. Mitchell3

1Dept. of Computer Science, Duke University2Dept. of Computer Science, University of Illinois at Urbana-Champaign

3Dept. of Mathematics, University of Wisconsin

December 8, 2008

Generating Uniform Incremental Grids

on SO(3) Using the Hopf Fibration

Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)

1

IntroductionIntroduction Motivation Problem Formulation

Properties and Representations of the space of rotations, SO(3)

Literature Overview Method Presentation Conclusions and Discussion

IntroductionIntroduction

Presentation OverviewPresentation OverviewPresentation OverviewPresentation Overview

2


Automotive Assembly

Computational Chemistryand Biology

Manipulation Planning

Medical applications

Computer Graphics(motions for digital actors)

Autonomous vehicles andspacecrafts

IntroductionIntroduction

3

Motivation

Sampling SO(3) Occurs in:Sampling SO(3) Occurs in:Sampling SO(3) Occurs in:Sampling SO(3) Occurs in:


Courtesy of Kineo CAMCourtesy of Kineo CAM

Our Main Motivation: Motion PlanningOur Main Motivation: Motion PlanningOur Main Motivation: Motion PlanningOur Main Motivation: Motion Planning

The graph over C-space should capture

the “path connectivity” of the space

4

IntroductionIntroduction Motivation


Our Main Motivation: Motion PlanningOur Main Motivation: Motion PlanningOur Main Motivation: Motion PlanningOur Main Motivation: Motion Planning

• The quality of the undelying samples affect the quality of the

graph

• SO(3) is often the C-space

5

IntroductionIntroduction Motivation


Problem FormulationProblem FormulationProblem FormulationProblem Formulation

Desirable properties of samples

over the SO(3):

• uniform

• deterministic

• incremental

• grid structure

• uniform

• deterministic

• incremental

• grid structure

6

IntroductionIntroduction Problem Formulation



over the SO(3):


• uniform

• deterministic

• incremental

• grid structure

• uniform

• deterministic

• incremental

• grid structure

Discrepancy: maximum volume estimation error

Dispersion: the radius of the largest empty ball

Uniform:

7




• uniform

• deterministic

• incremental

• grid structure

• uniform

• deterministic

• incremental

• grid structure

Deterministic:

The uniformity measures can be deterministically computed

Reason: resolution completeness

Deterministic:

The uniformity measures can be deterministically computed

Reason: resolution completeness

8


over the SO(3):




• uniform

• deterministic

• incremental

• grid structure

• uniform

• deterministic

• incremental

• grid structure

Incremental:

The uniformity measures are optimized with every new point

Reason: it is unknown how many points are needed to solve the problem in advance

Incremental:

The uniformity measures are optimized with every new point

Reason: it is unknown how many points are needed to solve the problem in advance

9


over the SO(3):




• uniform

• deterministic

• incremental

• grid structure

• uniform

• deterministic

• incremental

• grid structure

Grid:

Reason: Trivializes nearest neighbor computations

Grid:

Reason: Trivializes nearest neighbor computations

10


over the SO(3):



SO(3): Topology, Manifold StructureSO(3): Topology, Manifold StructureSO(3): Topology, Manifold StructureSO(3): Topology, Manifold Structure

• SO(3) is a Lie group

• SO(3) is diffeomorphic to S3 with antipodal points identified

• Haar measure on SO(3) corresponds to the surface measure on S3

• SO(3) has a fiber bundle structure

• Fibers represent SO(3) as a product of S1 and S2. Locally it is a Cartesian product

Remark: sampling on spheres and SO(3) are related

• SO(3) is a Lie group

• SO(3) is diffeomorphic to S3 with antipodal points identified

• Haar measure on SO(3) corresponds to the surface measure on S3

• SO(3) has a fiber bundle structure

• Fibers represent SO(3) as a product of S1 and S2. Locally it is a Cartesian product

Remark: sampling on spheres and SO(3) are related

11

SO(3) PropertiesSO(3) Properties


S3, SO(3)S1 S2

S3, SO(3)S1 S2

Fiber bundles

Mobius BandI S1

Mobius BandI S1

SO(3) Parameterizations and CoordinatesSO(3) Parameterizations and CoordinatesSO(3) Parameterizations and CoordinatesSO(3) Parameterizations and Coordinates

• Euler angles

• Axis angle representation (topology)

• Spherical coordinates (topology, Haar measure)

• Quaternions (topology, Haar measure, group operation)

• Hopf coordinates (topology, Haar measure, Hopf bundle)

• Euler angles

• Axis angle representation (topology)

• Spherical coordinates (topology, Haar measure)

• Quaternions (topology, Haar measure, group operation)

• Hopf coordinates (topology, Haar measure, Hopf bundle)

12

SO(3) PropertiesSO(3) Properties


Literature overviewLiterature overviewLiterature overviewLiterature overview

• Euclidean space, [0,1]d

• Spheres, Sd

• Special orthogonal group, SO(3)

13

Literature OverviewLiterature Overview


Euclidean Spaces, [0,1]Euclidean Spaces, [0,1]ddEuclidean Spaces, [0,1]Euclidean Spaces, [0,1]dd

+ uniform

+ deterministic

+ incremental

grid structure

+ uniform

+ deterministic

+ incremental

grid structure

+ uniform

+ deterministic

+ incremental

grid structure

+ uniform

+ deterministic

+ incremental

grid structure

+ uniform

deterministic

+ incremental

grid structure

+ uniform

deterministic

+ incremental

grid structure

+ uniform

+ deterministic

incremental

grid structure

+ uniform

+ deterministic

incremental

grid structure

+ uniform

+ deterministic

incremental

grid structure

+ uniform

+ deterministic

incremental

grid structure

Halton pointsHammersley

pointsRandom sequence

Sukharev grid A lattice14



Layered Sukharev Grid Sequence[Lindemann, LaValle 2003]

+ uniform

+ deterministic

+ incremental

grid structure

+ uniform

+ deterministic

+ incremental

grid structure

15

Euclidean Spaces, [0,1]Euclidean Spaces, [0,1]ddEuclidean Spaces, [0,1]Euclidean Spaces, [0,1]dd



Spheres, Spheres, SSdd, and SO(3), and SO(3)Spheres, Spheres, SSdd, and SO(3), 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 Sd, SO(3) uniform deterministic point sets for SO(3)

[Lubotzky, Phillips, Sarnak 86] [Mitchell 07]

No deterministic sequences to our knowledge

+ uniform

deterministic

+ incremental

grid structure

+ uniform

deterministic

+ incremental

grid structure

+ uniform

deterministic

incremental

grid structure

+ uniform

deterministic

incremental

grid structure

16



Our previous approach: SpheresOur previous approach: SpheresOur previous approach: SpheresOur previous approach: Spheres

~ uniform

deterministic

+ incremental

grid structure

~ uniform

deterministic

+ incremental

grid structure

Ordering on faces +Ordering inside faces

Make a Layered Sukharev Grid sequence inside each face Define the ordering across faces Combine these two into a sequence on the cube Project the faces of the cube outwards to form spherical tiling Use barycentric coordinates to define the sequence on the sphere

[Yershova, LaValle, ICRA 2004]

17



Our previous approach: Cartesian ProductsOur previous approach: Cartesian ProductsOur previous approach: Cartesian ProductsOur previous approach: Cartesian Products

X Y

Make grid cells inside X and Y Naturally extend the grid structure to X Y Define the cell ordering and the ordering inside each cell

XY

X Y

Ordering on cells,Ordering inside cells

1234

18



[Lindemann, Yershova, LaValle, WAFR 2004]

Our approach: Our approach: SO(3)SO(3)Our approach: Our approach: SO(3)SO(3)Method PresentationMethod Presentation

Hopf coordinates preserve the fiber bundle structure of R P3

Locally, R P3 is a product of S1 and S2

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The method for Cartesian products can then be extended to R P3

Need two grids, for S1 and S2

Healpix, [Gorski,05]Healpix, [Gorski,05]

Grid on S2

Grid on S2

Grid on S1

Grid on S1

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Grid on S2

Grid on S2

Grid on S1

Grid on S1

21





Ordering on cells, ordering on [0,1]3

Grid on S2

Grid on S2

Grid on S1

Grid on S1

+ uniform

deterministic

+ incremental

grid structure

+ uniform

deterministic

+ incremental

grid structure

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1. The dispersion of the sequence T on SO(3) at the resolution level l is:

in which is the dispersion of the sequence over S2.

Note: The best bound so far to our knowledge.

2. 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.

PropositionsPropositionsPropositionsPropositions

23

Method PresentationMethod Presentation


Illustration on Motion PlanningIllustration on Motion PlanningIllustration on Motion PlanningIllustration on Motion Planning Configuration space: SO(3)

(a)

(b)

24

Method PresentationMethod Presentation


ConclusionsConclusionsConclusionsConclusions

25

ConclusionsConclusions


1. We have designed a sequence of samples over the

SO(3) which are:

• uniform

• deterministic

• incremental

• grid structure

• uniform

• deterministic

• incremental

• grid structure

2. Main point: Hopf coordinates naturally preserve the

grid structure on SO(3). (Subgroup aglorithm by Shoemake

implicitly utilizes them)

ConclusionsConclusionsConclusionsConclusions

Thank you!

Thank you!

26

ConclusionsConclusions


anna yershova 1 , steven m. lavalle 2 , and julie c. mitchell 3

Documents

uniform incremental

so3 propertiesanna yershova

motivationsampling so3

undelying samples

surface measure

space of rotations

anna yershova1

motion planningthe quality