Motor Parameterization

Abstract

In this paper, we consider several parameterizations of rigid transformations using motors in 3-D conformal geometric algebra. In particular, we present parameterizations based on the exponential, outer exponential, and Cayley maps of bivectors, as well as a map based on a first-order approximation of the exponential followed by orthogonal projection onto the group manifold. We relate these parameterizations to the matrix representations of rigid transformations in the 3-D special Euclidean group. Moreover, we present how these maps can be used to form retraction maps for use in manifold optimization; retractions being approximations of the exponential map that preserve the convergence properties of the optimization method while being less computationally expensive, and, for the presented maps, also easier to implement.

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Correspondence to Lars Tingelstad.

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This work was partially funded by the Norwegian Research Council, 237896 SFI Offshore Mechatronics.

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Tingelstad, L., Egeland, O. Motor Parameterization. Adv. Appl. Clifford Algebras 28, 34 (2018). https://doi.org/10.1007/s00006-018-0850-2

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Keywords

  • Motion parameterization
  • Exponential map
  • Outer exponential map
  • Cayley map
  • Retractions

Mathematics Subject Classification

  • Primary 99Z99
  • Secondary 00A00