Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M...
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![Page 1: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/1.jpg)
Generating Uniform Incremental Grids on SO(3) Using the Hopf
Fibration
Anna Yershove, Steven M LaValle, and Julie C. Mitchell
Jory DennyCSCE 643
![Page 2: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/2.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 3: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/3.jpg)
SO(3)
A manifold representing the space of 3D rotations
Used in numerous fields Robotics
Aerospace Trajectory Design
Computational Biology
Generating uniform sampling would improve algorithms in these fields
![Page 4: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/4.jpg)
• Difficult to visualize
• Basically RP3 but with antipodal points identified
• Metric Distortion
• Like a world map distorts how Greenland looks
Why not set up a simple grid like in R2 or R3
![Page 5: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/5.jpg)
Deterministic Sampling Method Presented in this work
• Insures certain properties wanted by different fields currently using Uniform Random Sampling
– Incremental Generation
– Optimal Dispersion-reduction
– Explicit Neighborhood structure
– Low Metric Distortion
– Equivolumetric Partition of SO(3) into grid regions
![Page 6: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/6.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 7: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/7.jpg)
SO(3)
• Special Orthogonal Group representing rotations about the origin in R3
• Diffeomorphic to RP3
• RP3 = S3/(x~-x), or a three sphere with antipodal points identified
![Page 8: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/8.jpg)
Haar Measure
• Up to a scalar multiple there exists a unique measure on SO(3) that is invariant with respect to group actions
• Haar Measure of a set is equal to the haar measure of all rotations in the set
• Only way to obtain distortion free notions of distance and volume in SO(3)
![Page 9: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/9.jpg)
Quaternions
• Parameterization for rotations
• Let x=(x1, x
2, x
3, x
4) ϵ R4 be a unit quaternion, x1 +
x2i + x3j + x4k, ||X||=1
• Defines relationship between projective space and 3-sphere which allows metrics to respect Haar Measure
• example:shortest arc distance on the 3-sphere
– ρRP3(x, y) = cos-1|(x·y)|
• Easily represents points of 3-sphere but lacks convenience for surface/volume measures
![Page 10: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/10.jpg)
Spherical Coordinates for SO(3)
• (θ, φ, ψ) in which ψ has a range of π/2 (identifications), θ has a range of π, and φ has a range of 2π
• Defines a set of 2-spheres defined by θ and φ of radii sin(ψ)
• For quaternion:
– X1 = cos(ψ)
– X2 = sin(ψ)cos(θ)
– X3 = sin(ψ)sin(θ)cos(φ)
– X4 = sin(ψ)sin(θ)sin(φ)
![Page 11: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/11.jpg)
Spherical Coordinates for SO(3)
• Haar measure is volume
– dV = sin2(ψ)sin(θ)dθdφdψ
• But its not convenient for integration also difficult to use for computing composition of rotations
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Hopf Coordinates
• Unique for a 3-sphere
• Hopf Fibration – describes RP3 in terms of a circle and a 2-sphere, intuitively saying that RP3 is composed of non-intersecting fibers, one per 2-sphere
– Implies important relationship between 3-sphere and RP3
![Page 13: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/13.jpg)
Hopf Coordinates
• Written with (θ, φ, ψ) in which is the ψ parameterization of the circle and (θ, φ) describes the 2-sphere
• For Quaternion:
– X1 = cos(θ/2)cos(ψ/2)
– X2 = cos(θ/2)sin(ψ/2)
– X3 = sin(θ/2)cos(φ+ψ/2)
– X4 = sinθ(/2)sin(φ+ψ/2)
![Page 14: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/14.jpg)
Hopf Coordinates
• Haar Measure: surface volume
– dV = sinθdθdφdψ
• Good now for easy integration, but still inconvenient for expressing compositions of rotations
![Page 15: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/15.jpg)
Axis-Angle Representation
• Rotation, θ, about some unit axis, n = (n1, n
2,
n3), ||n||=1
• From Quaternions
– X = (cos(θ/2), sin(θ/2)n1, sin(θ/2)n
2,
sin(θ/2)n3)
![Page 16: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/16.jpg)
![Page 17: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/17.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 18: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/18.jpg)
Discrepancy
• Enforces two criteria
– No region of the space is left uncovered
– No region is too full
• Formally
– Choose a range space R as a collection of subsets of SO(3), Choose an R ϵ R, μ(R) is the Haar measure, P is a sample set
![Page 19: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/19.jpg)
Dispersion
• Eliminates the second criteria
• Its the measure of keeping samples apart
• Formally
– p is any metric on SO(3) that agrees with the Haar Measue
![Page 20: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/20.jpg)
Problem Formation
• Goal of the work is to define a sequence of elements from SO(3)
– Must be incremental
– Must be deterministic
– Minimizes the discrepancy and dispersion on SO(3)
– Has a grid structure
![Page 21: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/21.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 22: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/22.jpg)
Random Sequence of Rotations
• Depends on metric/representation being used
• Lacks deterministic uniformity
• Lacks explicit neighborhood structure
![Page 23: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/23.jpg)
Successive Orthogonal Images
• Generates lattice-like sets with a specified length step based on deterministic samples in both S1 and S2
• Lacks incremental quality
• Uses Hopf Coordinates
![Page 24: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/24.jpg)
Layered Sukharev Grid Sequence
• Minimizes discrepency by placing one resolution grid at a time
• Results in distortions
• Better for nonspherical coordinate systems
![Page 25: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/25.jpg)
HealPix
• Deterministic, uniform, multi-resolution, equal area partitioning for 2-sphere
• Focuses on measure preserving property from cylindrical coordinates
![Page 26: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/26.jpg)
![Page 27: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/27.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 28: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/28.jpg)
Overview of Approach
• Uses HealPix method to design grid on S2 and a ordinary grid for S1
• The work the combines the spaces by cross product
• The work allows for minimal discrepency, minimal dispersion, multiresolution, neighborhood structure, and deterministic method
• T1 and m
1 are the grid and base resolution for the
circle
• T2 and m
2 are the grid and base resolution for the
sphere
![Page 29: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/29.jpg)
Choosing the Base Resolution
• 2π/m1 = sqrt(4π/m
2); 2π is the
circumference of the circle, 4π is the surface area of the sphere
![Page 30: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/30.jpg)
Choosing the Base Ordering
• Ordering of the first set of points (number defined by base resolution) affects the quality of the sequence
• But because of a need to alternate at antipodal points the number of points needed to specify the initial ordering on is reduced
• For this work the order was manually set
– Fb a s e
:N->[1,...72] defines the optimal ordering
function
![Page 31: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/31.jpg)
The Sequence
• Start with the base ordering, for each successive m points (m = m
1*m
2) are placed in the same order
• Each grid region is subdivided into 8 grid regions at each pass and one point is assigned per grid region
• Those 8 grid regions are ismorphic to [0,1]3 or a cube
• Then a recursive descent into each region follows
• Order of the regions is defined by fc u b e
:N->[1,...8]
![Page 32: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/32.jpg)
Analysis
![Page 33: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/33.jpg)
Visualization of the Results
![Page 34: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/34.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 35: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/35.jpg)
Motion Planning Application
• Considered Robots which can only rotate
• Compares this method to basic PRM planner, and the layered Sukharev grid sequence
• Averaged over 50 trials, the new method performed only equivalent or a little better then PRM or Sukharev
![Page 36: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/36.jpg)
Outline
Introduction
Properties of SO(3)
Problem Formation
Previous Sampling Methods
Approach
Application: Motion Planning
Conclusion
![Page 37: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/37.jpg)
Conclusions and Future Work
• Implemented a deterministic incremental grid sequence on SO(3) that is highly uniform
• Creates equivolumetric partitions
• Need to complete a more extensive analysis of the method and benefits of the method
• Generalizing method for SO(n)
![Page 38: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/38.jpg)
Critique of the Paper
• Used a basic method to define there new approach as in they just combined two existing works
• Does not have any extensive analysis or results even if the two experiments they ran showed a slight improvement
• Very well written only had very minor punctuation/spelling errors
![Page 39: Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643.](https://reader035.fdocuments.net/reader035/viewer/2022081519/56649d7b5503460f94a5f39e/html5/thumbnails/39.jpg)
Thank you
Any questions?