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On Cayley’s Factorization of 4D Rotations and Applications

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Abstract

A 4D rotation can be decomposed into a left- and a right-isoclinic rotation. This decomposition, known as Cayley’s factorization of 4D rotations, can be performed using the Elfrinkhof–Rosen method. In this paper, we present a more straightforward alternative approach using the corresponding orthogonal subspaces, for which orthogonal bases can be defined. This yields easy formulations, both in the space of\({4 \times 4}\) real orthogonal matrices representing 4D rotations and in the Clifford algebra\({\mathcal{C}_{4,0,0}}\). Cayley’s factorization has many important applications. It can be used to easily transform rotations represented using matrix algebra to different Clifford algebras. As a practical application of the proposed method, it is shown how Cayley’s factorization can be used to efficiently compute the screw parameters of 3D rigid-body transformations.

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ArticleOpen access28 April 2016

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References

  1. Angeles J.: Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms. Springer, Berlin (2006)

    MATH  Google Scholar 

  2. Baylis, WE.; Hadi, S.: Rotations in n-dimensions as spherical vectors. Applications of Geometric Algebra in Computer Science and Engineering, AGACSE 2001, Ed. Springer, Chapter 6, pp. 79–89 (2002)

  3. Cayley A.: On certain results relating to quaternions. Philos. Mag.26, 141–145 (1845)

    Google Scholar 

  4. Cayley, A.: Recherches ultérieures sur les déterminants gauches. In: The Collected Mathematical Papers Of Arthur Cayley, pp. 202–215. Cambridge Uniniversity Press, Cambridge (article 137) (1891)

  5. Davidson J.K., Hunt K.H.: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  6. Dorst, L.; Fontjine, D.; Mann, S.: Geometric Algebra for Computer Science, Ed. Morgan Kaufmann, Burlington (2007)

  7. Ge Q.J.: On the matrix algebra realization of the theory of biquaternions. Mech. Synth. Anal.70, 425–432 (1994)

    Google Scholar 

  8. Hitchcock, FL.: Analysis of rotations in euclidean four-space by sedenions, J. Math. Phys. Univerisity of Massachusetts, 9(3), 188–193 (1930)

  9. Hsiung, CY.; Mao, GY.: Linear Algebra. Allied Publishers (1998)

  10. Hunt, M.; Mullineux, G.; Cripps, RJ.; Cross, B.: Characterizing isoclinic matrices and the Cayley factorization. Proc. IMechE Part C J. Mech. Eng. Sci. 0(0) 1–7 (2015)

  11. Mebius, JE.: Applications of quaternions to dynamical simulation, computer graphics and biomechanics, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands (1994)

  12. Pertti L.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  13. Rosen N.: on the general Lorentz transformation. J. Math. Phys.9, 181–187 (1930)

    Article MATH  Google Scholar 

  14. Thomas F.: Approaching dual quaternions from matrix algebra. IEEE Trans. Robot.30(5), 1037–1048 (2014)

    Article  Google Scholar 

  15. van Elfrinkhof, L.: Eene eigenschap van de orthogonale substitutie van de vierde orde. Handelingen van het zesde Nederlandsch Natuur- en Geneeskundig Congres (Acts of the sixth Dutch nature and medical congress), pp. 237–240. Delft (1897)

  16. Weiner J.L., Wilkens G.R.: “Quaternions and rotations inE4”. Am. Math. Mon.112(1), 69–76 (2005)

    Article MathSciNet MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Idaho State University, Pocatello, Idaho, 83209, USA

    Alba Perez-Gracia

  2. Institut de Robòtica i Informàtica Industrial (CSIC-UPC), Llorens Artigas 4-6, 08028, Barcelona, Spain

    Federico Thomas

Authors
  1. Alba Perez-Gracia
  2. Federico Thomas

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Correspondence toAlba Perez-Gracia.

Additional information

This work was partially supported by the Spanish Ministry of Economy and Competitiveness through Project DPI2014-57220-C2-2-P and the USA National Science Foundation under Grant No. 1208385. The content is solely the authors’ responsibility.

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Perez-Gracia, A., Thomas, F. On Cayley’s Factorization of 4D Rotations and Applications.Adv. Appl. Clifford Algebras27, 523–538 (2017). https://doi.org/10.1007/s00006-016-0683-9

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