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Serre-Frenet Frame in n-dimensions at Regular and Minimally Singular Points

  • Ignacy DulebaEmail author
  • Iwona Karcz-Duleba
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9520)

Abstract

In contemporary robotics more and more complicated systems are considered. To plan their motions, well-known coordinate frames, living in natural, three-dimensional spaces, should be modified to cover multidimensional spaces as well. In this paper an algorithm is proposed to determine the Serre-Frenet frame in high dimensional spaces. The frame is examined at regular points where consecutive derivatives of a given curve, the robot moves along, are independent of each other. An interpolation procedure is provided when a minimally singular points appear and dimensionallity of the space spanned by the derivatives drops by one with respect to the regular, full dimensional space.

References

  1. 1.
    Mazur, A., Plaskonka, J.: The Serret-Frenet parametrization in a control of a mobile manipulator of (nh, h) type. In: the 10th IFAC Symposium on Robot Control, Dubrovnik, pp. 405–410 (2012)Google Scholar
  2. 2.
    Duleba, I, Karcz-Duleba, I.: Algorithmics of Serre-Frenet frame in \(R^n\). In: Moreno-Diaz, R., Pichler, F.R., Quesada-Arencibia, A. (eds.) EUROCAST, Las Palmas, pp. 215–216 (2015). (Extended abstract)Google Scholar
  3. 3.
    Griffiths, P.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41(4), 775–814 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gantmacher, F.: Theory of Matrices. AMS Chelsea Publishing, New York (1959)zbMATHGoogle Scholar
  5. 5.
    Yilmaz, S., Turgut, M.: A method to calculate Frenet apparatus of the curves in Euclidean-5 space. World Acad. Sci. Eng. Technol. 19, 771–773 (2008)MathSciNetGoogle Scholar
  6. 6.
    Gallier, J., Xu, D.: Computing exponentials of skew-symmetric matrices and logarithm of orthogonal matrices. Int. J. Robot. Autom. 17(4), 2–11 (2002)zbMATHGoogle Scholar
  7. 7.
    Andrica, D., Rohan, R.-A.: Computing the Rodrigues coefficients of the exponential map of the Lie groups of matrices. Balkan J. Geom. Appl. 18(2), 1–10 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Dobrowolski, P.: Evaluation of the usefulness of exact methods to motion planning in configuration space. Ph.D. thesis, Warsaw University of Technology, Faculty of Electronics and Information Technology (2013)Google Scholar
  9. 9.
    LaValle, S.M.: Planning Algorithms. Cambridge University Press, New York (2006)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Electronics FacultyWroclaw University of TechnologyWroclawPoland

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