Advertisement

From Differential Geometry of Curves to Helical Kinematics of Continuum Robots Using Exponential Mapping

  • Stanislao GraziosoEmail author
  • Giuseppe Di Gironimo
  • Bruno Siciliano
Conference paper
Part of the Springer Proceedings in Advanced Robotics book series (SPAR, volume 8)

Abstract

Kinematic modeling of continuum robots is challenging due to the large deflections that these systems usually undergone. In this paper, we derive the kinematics of a continuum robot from the evolution of a three-dimensional curve in space. We obtain the spatial configuration of a continuum robot in terms of exponential coordinates based on Lie group theory. This kinematic framework turns out to handle robotic helical shapes, i.e. spatial configurations with constant curvature and torsion of the arm.

Keywords

Differential geometry Continuum robotics Kinematics 

Notes

Acknowledgments

This work was partially supported by the FlexARM project, which has received funding from the European Commission’s Euratom Research and Training Programme 2014–2018 under the EUROfusion Engineering Grant EEG-2015/21 “Design of Control Systems for Remote Handling of Large Components” and partially by the RoDyMan project, which has received funding from the European Research Council under Advanced Grant 320992.

References

  1. 1.
    Walker, I.D., Choset, H., Chirikjian, G.S.: Snake-like and continuum robots. In: Springer Handbook of Robotics, pp. 481–498. Springer, Cham (2016)CrossRefGoogle Scholar
  2. 2.
    Simo, J.C., Vu-Quoc, L.: A three-dimensional finite-strain rod model. part II: Computational aspects. Comput. Methods Appl. Mech. Eng. 58(1), 79–116 (1986)CrossRefGoogle Scholar
  3. 3.
    Grazioso, S., Sonneville, V., Di Gironimo, G., Bauchau, O., Siciliano, B.: A nonlinear finite element formalism for modelling flexible and soft manipulators. In: 2016 IEEE International Conference on Simulation, Modeling, and Programming for Autonomous Robots, pp. 185–190. IEEE (2016)Google Scholar
  4. 4.
    Grazioso, S., Di Gironimo, G., Siciliano, B.: A geometrically exact model for soft robots: the finite element deformation space formulation. Soft RoboticsGoogle Scholar
  5. 5.
    Andersson, S.B.: Discretization of a continuous curve. IEEE Trans. Rob. 24(2), 456–461 (2008)CrossRefGoogle Scholar
  6. 6.
    Webster III, R.J., Jones, B.A.: Design and kinematic modeling of constant curvature continuum robots: a review. Int. J. Rob. Res. 29(13), 1661–1683 (2010)CrossRefGoogle Scholar
  7. 7.
    Hannan, M.W., Walker, I.D.: Kinematics and the implementation of an elephant’s trunk manipulator and other continuum style robots. J. Field Rob. 20(2), 45–63 (2003)zbMATHGoogle Scholar
  8. 8.
    Jones, B.A., Walker, I.D.: Kinematics for multisection continuum robots. IEEE Trans. Rob. 22(1), 43–55 (2006)CrossRefGoogle Scholar
  9. 9.
    Grzesiak, A., Becker, R., Verl, A.: The bionic handling assistant: a success story of additive manufacturing. Assem. Autom. 31(4), 329–333 (2011)CrossRefGoogle Scholar
  10. 10.
    Mahl, T., Hildebrandt, A., Sawodny, O.: A variable curvature continuum kinematics for kinematic control of the bionic handling assistant. IEEE Trans. Rob. 30(4), 935–949 (2014)CrossRefGoogle Scholar
  11. 11.
    Struik, D.J.: Lectures on classical differential geometry. Courier Corporation, North Chelmsford (2012)zbMATHGoogle Scholar
  12. 12.
    Lynch, K.M., Park, F.C.: Modern Robotics: Mechanics, Planning, and Control. Cambridge University Press, Cambridge (2017)Google Scholar
  13. 13.
    Crouch, P.E., Grossman, R.: Numerical integration of ordinary differential equations on manifolds. J. Nonlinear Sci. 3(1), 1–33 (1993)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Munthe-Kaas, H.: Runge-Kutta methods on lie groups. BIT Numer. Math. 38(1), 92–111 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grazioso, S., Di Gironimo, G., Siciliano, B.: Analytic solutions for the static equilibrium configurations of externally loaded cantilever soft robotic arms. In: 2018 IEEE International Conference on Soft Robotics. IEEE (2018)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Stanislao Grazioso
    • 1
    Email author
  • Giuseppe Di Gironimo
    • 1
  • Bruno Siciliano
    • 2
  1. 1.Dipartimento di Ingegneria IndustrialeUniversità degli Studi di Napoli Federico IINapoliItaly
  2. 2.Dipartimento di Ingegneria Elettrica e Tecnologie dell’InformazioneUniversità degli Studi di Napoli Federico IINapoliItaly

Personalised recommendations