Advertisement

Covariant Decomposition of the Three-Dimensional Rotations

  • Clementina D. Mladenova
  • Danail S. Brezov
  • Ivaïlo M. Mladenov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8918)

Abstract

The main purpose of this paper is to provide an alternative representation for the generalized Euler decomposition (with respect to arbitrary axes) obtained in [2,3] by means of vector parameterization of the Lie group SO(3). The scalar (angular) parameters of the decomposition are explicitly written here as functions depending only on the contravariant components of the compound vector-parameter in the basis, determined by the three axes. We also consider the case of coplanar axes, in which the basis needs to be completed by a third vector and in particular, two-axes decompositions.

Keywords

Rotations vector-parameterization group decompositions Lie algebras Lie groups 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bongardt, B.: Geometric Characterization of the Workspace of Non-Orthogonal Rotation Axes. J. Geom. Mechanics 6, 141–166 (2014)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Brezov, D., Mladenova, C., Mladenov, I.: Vector Decompositions of Rotations. J. Geom. Symmetry Phys. 28, 67–103 (2012)MATHMathSciNetGoogle Scholar
  3. 3.
    Brezov, D., Mladenova, C., Mladenov, I.: Some New Results on Three-Dimensional Rotations and Pseudo-Rotations. In: AIP Conf. Proc., vol. 1561, pp. 275–288 (2013)Google Scholar
  4. 4.
    Brezov, D., Mladenova, C., Mladenov, I.: Quarter Turns and New Factorizations of Rotations. Comptes Rendus de l’Académie Bulgare des Sciences 66, 1105–1114 (2013)MathSciNetGoogle Scholar
  5. 5.
    Davenport, P.: Rotations About Nonorthogonal Axes. AIAA Journal 11, 853–857 (1973)CrossRefMATHGoogle Scholar
  6. 6.
    Mladenova, C.: Group Theory in the Problems of Modeling and Control of Multi-Body Systems. J. Geom. Symmetry Phys. 8, 17–121 (2006)MATHMathSciNetGoogle Scholar
  7. 7.
    Mladenova, C., Mladenov, I.: Vector Decomposition of Finite Rotations. Rep. Math. Phys. 68, 107–117 (2011)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kuipers, J.: Quaternions and Rotation Sequences. Geometry, Integrability and Quantization 1, 127–143 (2000)MathSciNetGoogle Scholar
  9. 9.
    Piña, E.: Rotations with Rodrigues’ Vector. Eur. J. Phys. 32, 1171–1178 (2011)CrossRefMATHGoogle Scholar
  10. 10.
    Rull, A., Thomas, F.: On Generalized Euler Angles. Mechanisms and Machine Science 24, 61–68 (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Clementina D. Mladenova
    • 1
  • Danail S. Brezov
    • 2
  • Ivaïlo M. Mladenov
    • 3
  1. 1.Institute of MechanicsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Civil Engineering and GeodesyUniversity of ArchitectureSofiaBulgaria
  3. 3.Institute of BiophysicsBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations