Jory Denny CSCE 643

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

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

3. 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

4. Why not set up a simple grid like in R2 or R3 • Difficult to visualize • Basically RP3 but with antipodal points identified • Metric Distortion • Like a world map distorts how Greenland looks

5. 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

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

7. 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

8. 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)

9. Quaternions • Parameterization for rotations • Let x=(x1, x2, x3, x4) 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

10. 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)

11. 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

12. 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

13. 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)

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

15. Axis-Angle Representation • Rotation, θ, about some unit axis, n = (n1, n2, n3), ||n||=1 • From Quaternions • X = (cos(θ/2), sin(θ/2)n1, sin(θ/2)n2, sin(θ/2)n3)

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

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

19. 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

20. 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

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

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

23. 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

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

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

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

28. 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 m1 are the grid and base resolution for the circle • T2 and m2 are the grid and base resolution for the sphere

29. Choosing the Base Resolution • 2π/m1 = sqrt(4π/m2); 2π is the circumference of the circle, 4π is the surface area of the sphere

30. 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

31. The Sequence • Start with the base ordering, for each successive m points (m = m1*m2) 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]

32. Analysis

33. Visualization of the Results

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

35. 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

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

37. 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)

38. 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

39. Any questions? Thank you