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?