Anna Yershova 1 , Steven M. LaValle 2 , and Julie C. Mitchell 3
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Transcript of 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)
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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
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Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Automotive Assembly
Computational Chemistryand Biology
Manipulation Planning
Medical applications
Computer Graphics(motions for digital actors)
Autonomous vehicles andspacecrafts
IntroductionIntroduction
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Motivation
Sampling SO(3) Occurs in:Sampling SO(3) Occurs in:Sampling SO(3) Occurs in:Sampling SO(3) Occurs in:
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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
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IntroductionIntroduction Motivation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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
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IntroductionIntroduction Motivation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Problem FormulationProblem FormulationProblem FormulationProblem Formulation
Desirable properties of samples
over the SO(3):
• uniform
• deterministic
• incremental
• grid structure
• uniform
• deterministic
• incremental
• grid structure
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IntroductionIntroduction Problem Formulation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Desirable properties of samples
over the SO(3):
Problem FormulationProblem FormulationProblem FormulationProblem Formulation
• uniform
• deterministic
• incremental
• grid structure
• uniform
• deterministic
• incremental
• grid structure
Discrepancy: maximum volume estimation error
Dispersion: the radius of the largest empty ball
Uniform:
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IntroductionIntroduction Problem Formulation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Problem FormulationProblem FormulationProblem FormulationProblem Formulation
• 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
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Desirable properties of samples
over the SO(3):
IntroductionIntroduction Problem Formulation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Problem FormulationProblem FormulationProblem FormulationProblem Formulation
• 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
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Desirable properties of samples
over the SO(3):
IntroductionIntroduction Problem Formulation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Problem FormulationProblem FormulationProblem FormulationProblem Formulation
• uniform
• deterministic
• incremental
• grid structure
• uniform
• deterministic
• incremental
• grid structure
Grid:
Reason: Trivializes nearest neighbor computations
Grid:
Reason: Trivializes nearest neighbor computations
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Desirable properties of samples
over the SO(3):
IntroductionIntroduction Problem Formulation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on 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
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SO(3) PropertiesSO(3) Properties
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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)
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SO(3) PropertiesSO(3) Properties
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Literature overviewLiterature overviewLiterature overviewLiterature overview
• Euclidean space, [0,1]d
• Spheres, Sd
• Special orthogonal group, SO(3)
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Literature OverviewLiterature Overview
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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
Literature OverviewLiterature Overview
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Layered Sukharev Grid Sequence[Lindemann, LaValle 2003]
+ uniform
+ deterministic
+ incremental
grid structure
+ uniform
+ deterministic
+ incremental
grid structure
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Euclidean Spaces, [0,1]Euclidean Spaces, [0,1]ddEuclidean Spaces, [0,1]Euclidean Spaces, [0,1]dd
Literature OverviewLiterature Overview
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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
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Literature OverviewLiterature Overview
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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]
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Literature OverviewLiterature Overview
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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
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Literature OverviewLiterature Overview
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
[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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Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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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Our approach: Our approach: SO(3)SO(3)Our approach: Our approach: SO(3)SO(3)Method PresentationMethod Presentation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
The method for Cartesian products can then be extended to R P3
Need two grids, for S1 and S2
Grid on S2
Grid on S2
Grid on S1
Grid on S1
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Our approach: Our approach: SO(3)SO(3)Our approach: Our approach: SO(3)SO(3)Method PresentationMethod Presentation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
The method for Cartesian products can then be extended to R P3
Need two grids, for S1 and S2
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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Our approach: Our approach: SO(3)SO(3)Our approach: Our approach: SO(3)SO(3)Method PresentationMethod Presentation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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
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Method PresentationMethod Presentation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
Illustration on Motion PlanningIllustration on Motion PlanningIllustration on Motion PlanningIllustration on Motion Planning Configuration space: SO(3)
(a)
(b)
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Method PresentationMethod Presentation
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
ConclusionsConclusionsConclusionsConclusions
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ConclusionsConclusions
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)
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!
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ConclusionsConclusions
Anna Yershova, et. al.Anna Yershova, et. al. Uniform Incremental Grids on SO(3)