Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration

Authors:
Anna Yershova;Swati Jain;Steven M. Lavalle;Julie C. Mitchell
Affiliations:
Duke University, Durham, NC 27707, USA;Duke University, Durham, NC 27707, USA;University of Illinois, Urbana, IL 61801, USA;University of Wisconsin-Madison, Madison, WI 53706,USA
Venue:
International Journal of Robotics Research
Year:
2010

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Abstract

The problem of generating uniform deterministic samples over the rotation group, SO(3), is fundamental to computational biology, chemistry, physics, and numerous branches of computer science. We present the best-known method to date for constructing incremental, deterministic grids on SO(3); it provides: (1) the lowest metric distortion for grid neighbor edges, (2) optimal dispersion-reduction with each additional sample, (3) explicit neighborhood structure, and (4) equivolumetric partition of SO(3) by the grid cells. We also demonstrate the use of the sequence on motion planning problems.