Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration Anna Yershove, Steven M LaValle, and Julie C. Mitchell Jory Denny CSCE 643

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

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

Difficult to visualize Basically RP 3 but with antipodal points identified Metric Distortion Like a world map distorts how Greenland looks Why not set up a simple grid like in R 2 or R 3

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

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

SO(3) Special Orthogonal Group representing rotations about the origin in R 3 Diffeomorphic to RP 3 RP 3 = S 3 /(x~-x), or a three sphere with antipodal points identified

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)

Quaternions Parameterization for rotations Let x=(x 1, x 2, x 3, x 4 ) R 4 be a unit quaternion, x 1 + x 2 i + x 3 j + x 4 k, ||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 – ρRP 3 (x, y) = cos -1 |(x·y)| Easily represents points of 3-sphere but lacks convenience for surface/volume measures

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: – X 1 = cos(ψ) – X 2 = sin(ψ)cos(θ) – X 3 = sin(ψ)sin(θ)cos(φ) – X 4 = sin(ψ)sin(θ)sin(φ)

Spherical Coordinates for SO(3) Haar measure is volume – dV = sin 2 (ψ)sin(θ)dθdφdψ But its not convenient for integration also difficult to use for computing composition of rotations

Hopf Coordinates Unique for a 3-sphere Hopf Fibration – describes RP 3 in terms of a circle and a 2-sphere, intuitively saying that RP 3 is composed of non-intersecting fibers, one per 2-sphere – Implies important relationship between 3- sphere and RP 3

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: – X 1 = cos(θ/2)cos(ψ/2) – X 2 = cos(θ/2)sin(ψ/2) – X 3 = sin(θ/2)cos(φ+ψ/2) – X 4 = sinθ(/2)sin(φ+ψ/2)

Hopf Coordinates Haar Measure: surface volume – dV = sinθdθdφdψ Good now for easy integration, but still inconvenient for expressing compositions of rotations

Axis-Angle Representation Rotation, θ, about some unit axis, n = (n 1, n 2, n 3 ), ||n||=1 From Quaternions – X = (cos(θ/2), sin(θ/2)n 1, sin(θ/2)n 2, sin(θ/2)n 3 )

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

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 R, μ(R) is the Haar measure, P is a sample set

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

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

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

Random Sequence of Rotations Depends on metric/representation being used Lacks deterministic uniformity Lacks explicit neighborhood structure

Successive Orthogonal Images Generates lattice-like sets with a specified length step based on deterministic samples in both S 1 and S 2 Lacks incremental quality Uses Hopf Coordinates

Layered Sukharev Grid Sequence Minimizes discrepency by placing one resolution grid at a time Results in distortions Better for nonspherical coordinate systems

HealPix Deterministic, uniform, multi-resolution, equal area partitioning for 2-sphere Focuses on measure preserving property from cylindrical coordinates

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

Overview of Approach Uses HealPix method to design grid on S 2 and a ordinary grid for S 1 The work the combines the spaces by cross product The work allows for minimal discrepency, minimal dispersion, multiresolution, neighborhood structure, and deterministic method T 1 and m 1 are the grid and base resolution for the circle T 2 and m 2 are the grid and base resolution for the sphere

Choosing the Base Resolution 2π/m 1 = sqrt(4π/m 2 ); 2π is the circumference of the circle, 4π is the surface area of the sphere

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 – F b a s e :N->[1,...72] defines the optimal ordering function

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 f c u b e :N->[1,...8]

Analysis

Visualization of the Results

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

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

Outline Introduction Properties of SO(3) Problem Formation Previous Sampling Methods Approach Application: Motion Planning Conclusion

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)

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

Thank you Any questions?