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Rotations in a Non-Orthogonal Frame

[+] Author Affiliations
Shengnan Lu, Xilun Ding

Beihang University, Beijing, China

Gregory S. Chirikjian

Johns Hopkins University, Baltimore, MD

Paper No. DETC2018-85862, pp. V05BT07A082; 10 pages
doi:10.1115/DETC2018-85862
From:
  • ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
  • Volume 5B: 42nd Mechanisms and Robotics Conference
  • Quebec City, Quebec, Canada, August 26–29, 2018
  • Conference Sponsors: Design Engineering Division, Computers and Information in Engineering Division
  • ISBN: 978-0-7918-5181-4
  • Copyright © 2018 by ASME

abstract

This paper is concerned with describing the space of matrices that describe rotations in non-orthogonal coordinates. In scenarios such as in crystallography, conformational analysis of polymers, and in the study of deployable mechanisms and rigid origami, non-orthogonal reference frames are natural. For example, non-orthogonal vectors in the direction of atomic bonds in a molecule, the lattice coordinates of a crystal, or the directions of links in a mechanism are intrinsic. In these cases it is awkward to impose an artificial orthonormal reference frame rather than choosing one that is defined by the geometry of the object being studied. With these applications in mind, we fully characterize the space of all possible non-orthogonal rotations. We find that in the 2D case, this space is a three-dimensional subset of the special linear group, SL(2, R), which is itself a three-dimensional Lie group. In the 3D case we find that the space of nonorthogonal rotations is a seven-dimensional subspace of SL(3, R), which is an eight-dimensional Lie group. In the 2D case we use the Iwasawa decomposition to fully characterize the solution. In the 3D case we parameterize this seven-dimensional space by conjugating elements of the rotation group SO(3) by elements of a discrete family of of four-parameter subgroups of GL(3, R), and using this we derive an inversion formula to extract classical orthogonal rotations from those expressed in non-orthogonal coordinates.

Copyright © 2018 by ASME

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