Regular and Chaotic Dynamics

, Volume 20, Issue 6, pp 716–728 | Cite as

Spherical robot of combined type: Dynamics and control

  • Alexander A. Kilin
  • Elena N. Pivovarova
  • Tatyana B. Ivanova


This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.


spherical robot control nonholonomic constraint combined mechanism 

MSC2010 numbers

37J60 70E18 70F25 


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  1. 1.
    Halme, A., Schönberg, T., and Wang, Y., Motion Control of a Spherical Mobile Robot, in Proc. of the 4th IEEE Internat. Workshop on Advanced Motion Control (Mie, Japan, 1996): Vol. 1, pp. 259–264.Google Scholar
  2. 2.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 1972.Google Scholar
  3. 3.
    Bizyaev, I. A., Nonintegrability and Obstructions to the Hamiltonianization of a Nonholonomic Chaplygin Top, Dokl. Math., 2014, vol. 90, no. 2, pp. 631–634; see also: Dokl. Akad. Nauk, 2014, vol. 458, no. 4, pp. 398–401.CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571–579.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bizyaev, I.A., Bolsinov, A.V., Borisov, A.V., and Mamaev, I. S., Topology and Bifurcations in Nonholonomic Mechanics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2015, vol. 25, no. 10, 21 pp.Google Scholar
  7. 7.
    Borisov, A.V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.CrossRefMathSciNetGoogle Scholar
  8. 8.
    Borisov, A. V. and Mamaev, I. S., Rolling of a Non-homogeneous Ball Over a Sphere Without Slipping and Twisting, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 153–159.CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Borisov, A. V., Fedorov Yu.N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: an Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics-Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.CrossRefGoogle Scholar
  11. 11.
    Borisov, A. V., Jalnine, A.Yu., Kuznetsov, S.P., Sataev, I.R., and Sedova J.V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Model of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512–532.CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272–275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192–195.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Chase, R. and Pandya, A., A Review of Active Mechanical Driving Principles of Spherical Robots, Robotics, 2012, vol. 1, no. 1, pp. 3–23.CrossRefGoogle Scholar
  14. 14.
    Crossley, V.A., A Literature Review on the Design of Spherical Rolling Robots, Pittsburgh,Pa., 2006. 6 pp.Google Scholar
  15. 15.
    Karavaev, Yu. L. and Kilin, A.A., The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ylikorpi, T. and Suomela, J., Ball-Shaped Robots, in Climbing and Walking Robots: Towards New Applications, H. Zhang (Ed.), Vienna: InTech, 2007, pp. 235–256.Google Scholar
  17. 17.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., How To Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Svinin, M., Bai, Y., and Yamamoto, M., Dynamic Model and Motion Planning for a Pendulum-Actuated Spherical Rolling Robot, in Proc. of the 2015 IEEE Internat. Conf. on Robotics and Automation (ICRA), pp. 656–661.Google Scholar
  19. 19.
    Ivanova, T. B. and Pivovarova, E. N., Dynamics and Control of a Spherical Robot with an Axisymmetric Pendulum Actuator, arXiv:1511.02655 (2015).Google Scholar
  20. 20.
    Zhan, Q., Motion Planning of a Spherical Mobile Robot, in Motion and Operation Planning of Robotic Systems, G. Carbone, F. Gomez-Bravo (Eds.), Cham: Springer, 2015, pp. 361–381.Google Scholar
  21. 21.
    Gajbhiye, S. and Banavar, R. N., Geometric Modeling and Local Controllability of a Spherical Mobile Robot Actuated by an Internal Pendulum, Int. J. Robust Nonlinear Control, 2015.Google Scholar
  22. 22.
    Borisov, A. V. and Mamaev, I. S., Two Non-holonomic Integrable Problems Tracing Back to Chaplygin, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 191–198.CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Morinaga, A., Svinin, M., and Yamamoto, M., A Motion Planning Strategy for a Spherical Rolling Robot Driven by Two Internal Rotors, IEEE Trans. on Robotics, 2014, vol. 30, no. 4, pp. 993–1002.CrossRefGoogle Scholar
  24. 24.
    Fantoni, I. and Lozano, R., Non-Linear Control for Underactuated Mechanical Systems, London: Springer, 2002.CrossRefGoogle Scholar
  25. 25.
    Hamel, G., Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. Math. u. Phys., 1904, vol. 50, pp. 1–57.zbMATHGoogle Scholar
  26. 26.
    Borisov, A.V. and Mamaev, I. S., Dynamics of a Rigid Body: Hamiltonian Methods, Integrability, Chaos, 2nd ed., Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  • Alexander A. Kilin
    • 1
  • Elena N. Pivovarova
    • 1
  • Tatyana B. Ivanova
    • 2
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.M. T. Kalashnikov Izhevsk State Technical UniversityIzhevskRussia

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