Post on 14-Jan-2016
description
Generating Uniform Incremental Grids on SO(3) Using the Hopf
Fibration
Anna Yershove, Steven M LaValle, and Julie C. Mitchell
Jory DennyCSCE 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 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
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 R3
• Diffeomorphic to RP3
• RP3 = S3/(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=(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
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)
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
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
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)
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 = (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)
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 ϵ 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 S1 and S2
• 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 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
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
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
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]
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?