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Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration.
Anna Yershova1, Swati Jain, Steven M Lavalle
1Duke University, Durham, NC 27707, USA, yershova@cs.duke.edu.
The International Journal of Robotics Research
|July 8, 2010
Summary
We developed a new method for creating uniform samples on the rotation group SO(3). This approach minimizes distortion and optimizes sample distribution, proving useful in computational fields and motion planning.
Area of Science:
- Computational science and applied mathematics.
- Focuses on geometric sampling and group theory.
Background:
- Generating uniform samples on the special orthogonal group SO(3) is crucial for various scientific and engineering disciplines.
- Existing methods face challenges in achieving optimal sample distribution and minimizing geometric distortion.
Purpose of the Study:
- To present a novel, deterministic method for generating uniform samples on SO(3).
- To provide a grid construction that offers superior properties compared to existing techniques.
Main Methods:
- Development of an incremental, deterministic grid generation algorithm for SO(3).
- Analysis of grid properties including metric distortion, dispersion, neighborhood structure, and spatial partitioning.
Main Results:
- The proposed method achieves the lowest metric distortion for grid neighbor edges.
- Demonstrates optimal dispersion-reduction with increasing sample density.
- Provides an explicit neighborhood structure and an equivolumetric partition of SO(3).
Conclusions:
- The presented method is the best-known approach for constructing deterministic grids on SO(3) to date.
- The generated sequences are effective for applications such as motion planning.
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