Motor Parameterization

  • Lars TingelstadEmail author
  • Olav Egeland
Open Access
Part of the following topical collections:
  1. T.C. : Geometric Algebra for Computing, Graphics and Engineering with Yu Zhaoyuan, Guest E-i-C


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.


Motion parameterization Exponential map Outer exponential map Cayley map Retractions 

Mathematics Subject Classification

Primary 99Z99 Secondary 00A00 


  1. 1.
    Absil, P.-A., Baker, C.G., Gallivan, K.A.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7(3), 303–330 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22(3), 359–390 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baker, C.G.: Riemannian Manifold Trust-Region Methods with Applications to Eigenproblems. PhD thesis, College of Arts and Sciences, Florida State University (2008)Google Scholar
  5. 5.
    Bayro-Corrochano, E.: Motor algebra approach for visually guided robotics. Pattern Recognit. 35(1), 279–294 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bayro-Corrochano, E., Zhang, Y.: The motor extended kalman filter: a geometric approach for rigid motion estimation. J. Math. Imaging Vis. 13(3), 205–228 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boumal, N., Mishra, B., Absil, P.-A., Sepulchre, R.: Manopt, a matlab toolbox for optimization on manifolds. J. Mach. Learn. Res. 15, 1455–1459 (2014)zbMATHGoogle Scholar
  8. 8.
    Celledoni, E., Owren, B.: Lie group methods for rigid body dynamics and time integration on manifolds. Comput. Methods Appl. Mech. Eng. 192(3), 421–438 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chasles, M.: Notes sur les Propriétés Générales du Système de Deux Corps Semblades entr’eux. Bulletin de Sciences Mathematiques, Astronomiques Physiques et Chimiques, pp. 321–326 (1830)Google Scholar
  10. 10.
    Clifford, W.K.: Preliminary sketch of biquaternions. Proc. Lond. Math. Soc. 4, 361–395 (1873)MathSciNetGoogle Scholar
  11. 11.
    Davidson, J.K., Hunt, K.H.: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press Inc., New York (2004)zbMATHGoogle Scholar
  12. 12.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann Publishers Inc., San Francisco (2007)Google Scholar
  14. 14.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer series in computational mathematics. Springer, Berlin (2006)zbMATHGoogle Scholar
  15. 15.
    Helmstetter, J.: Exponentials of bivectors and their symplectic counterparts. Adv. Appl. Clifford Algebras 18(3), 689–698 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numer. 9, 215–365 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Li, H.: Parameterization of 3D Conformal Transformations in Conformal Geometric Algebra. Springer, London, pp. 71–90 (2010)Google Scholar
  18. 18.
    Lounesto, P.: Cayley transform, outer exponential and spinor norm. In: Proceedings of the Winter School “Geometry and Physics”. Circolo Matematico di Palermo, pp. 191–198 (1987)Google Scholar
  19. 19.
    Lounesto, P.: Clifford Algebras and Spinors. London Mathematical Society lecture note series, 2nd edn. Cambridge University Press, Cambridge (2001)CrossRefzbMATHGoogle Scholar
  20. 20.
    Ma, Y., Košecká, J., Sastry, S.: Optimization criteria and geometric algorithms for motion and structure estimation. Int. J. Comput. Vis. 44(3), 219–249 (2001)CrossRefzbMATHGoogle Scholar
  21. 21.
    McCarthy, J.M., Soh, G.S.: Geometric design of linkages. Springer, New York, NY (2011)Google Scholar
  22. 22.
    Riesz, M.: Clifford Numbers and Spinors (Chapters I–IV). Springer, Dordrecht, pp. 1–196 (1993)Google Scholar
  23. 23.
    Sarkis, M., Diepold, K.: Camera–Pose estimation via projective Newton optimization on the manifold. IEEE Trans. Image Process. 21(4), 1729–1741 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Selig, J.M.: Cayley maps for se(3). In: 12th International Federation for the Promotion of Mechanism and Machine Science World Congress, p. 6 (2007)Google Scholar
  25. 25.
    Selig, J.M.: Exponential and Cayley maps for dual quaternions. Adv. Appl. Clifford Algebras 20(3), 923–936 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Smith, S.T.: Geometric Optimization Methods for Adaptive Filtering. PhD thesis, Harvard University (1993)Google Scholar
  27. 27.
    Tingelstad, L., Egeland, O.: Automatic multivector differentiation and optimization. Adv. Appl. Clifford Algebras 27(1), 707–731 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Tingelstad, L., Egeland, O.: Motor estimation using heterogeneous sets of objects in conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2035–2049 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Townsend, J., Koep, N., Weichwald, S.: Pymanopt: a python toolbox for optimization on manifolds using automatic differentiation. J. Mach. Learn. Res. 17(137), 1–5 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Valkenburg, R., Dorst, L.: Estimating Motors from a Variety of Geometric Data in 3D Conformal Geometric Algebra. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice, pp. 25–45. Springer, London (2011)CrossRefGoogle Scholar
  31. 31.
    Vandereycken, B.: Riemannian and multilevel optimization for rank-constrained matrix problems. PhD thesis, Department of Computer Science, KU Leuven (2010)Google Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial Engineering, Faculty of EngineeringNTNU, Norwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations