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Regular and Chaotic Dynamics

, Volume 24, Issue 5, pp 560–582 | Cite as

Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem

  • Ivan A. BizyaevEmail author
  • Alexey V. BorisovEmail author
  • Ivan S. MamaevEmail author
Sergey Chaplygin Memorial Issue
  • 14 Downloads

Abstract

This paper addresses the problem of the rolling of a spherical shell with a frame rotating inside, on which rotors are fastened. It is assumed that the center of mass of the entire system is at the geometric center of the shell.

For the rubber rolling model and the classical rolling model it is shown that, if the angular velocities of rotation of the frame and the rotors are constant, then there exists a noninertial coordinate system (attached to the frame) in which the equations of motion do not depend explicitly on time. The resulting equations of motion preserve an analog of the angular momentum vector and are similar in form to the equations for the Chaplygin ball. Thus, the problem reduces to investigating a two-dimensional Poincaré map.

The case of the rubber rolling model is analyzed in detail. Numerical investigation of its Poincaré map shows the existence of chaotic trajectories, including those associated with a strange attractor. In addition, an analysis is made of the case of motion from rest, in which the problem reduces to investigating the vector field on the sphere S2.

Keywords

nonholonomic mechanics Chaplygin ball rolling without slipping and spinning strange attractor straight-line motion stability limit cycle balanced beaver-ball 

MSC2010 numbers

37J60 37C10 

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Notes

Funding

The work of I. A. Bizyaev (Section 2 and Section 4) was supported by the Russian Science Foundation (project 18-71-00110). The work of A. V. Borisov and I. S. Mamaev was supported by the RFBR Grant No. 18-29-10051 mk and was carried out at MIPT under project 5–100 for state support for leading universities of the Russian Federation. The work of A. V. Borisov (Section 1 and Appendix A) was supported by the Russian Science Foundation (project 15-12-20035).

References

  1. 1.
    Artes, J. C., Llibre, J., and Schlomiuk, D., The Geometry of Quadratic Differential Systems with a Weak Focus of Second Order, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2006, vol. 16, no. 11, pp. 3127–3194.MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balseiro, P. and García-Naranjo, L. C., Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems, Arch. Ration. Mech. Anal., 2012, vol. 205, no. 1, pp. 267–310.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., Dynamics of the Chaplygin Ball on a Rotating Plane, Russ. J. Math. Phys., 2018, vol. 25, no. 4, pp. 423–433.MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bizyaev, I. A., Borisov, A. V., Kozlov, V. V., and Mamaev, I. S., Fermi-Like Acceleration and Power-Law Energy Growth in Nonholonomic Systems, Nonlinearity, 2019, vol. 32, no. 9, pp. 3209–3233.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., Exotic Dynamics of Nonholonomic Roller Racer with Periodic Control, Regul. Chaotic Dyn., 2018, vol. 23, no. 7–8, pp. 983–994.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration, Regul. Chaotic Dyn., 2017, vol. 22, no. 8, pp. 955–975.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bolotin, S. V., The Problem of Optimal Control of a Chaplygin Ball by Internal Rotors, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 559–570.MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Geometrisation of Chaplygin’s Reducing Multiplier Theorem, Nonlinearity, 2015, vol. 28, no. 7, pp. 2307–2318.MathSciNetzbMATHGoogle Scholar
  9. 9.
    Borisov, A. V. and Fedorov, Yu. N., On Two Modified Integrable Problems in Dynamics, Mosc. Univ. Mech. Bull., 1995, vol. 50, no. 6, pp. 16–18; see also: Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, no. 6, pp. 102–105.MathSciNetzbMATHGoogle Scholar
  10. 10.
    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.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Borisov, A. V., Ivanova, T. B., Kilin, A. A., and Mamaev, I. S. Nonholonomic Rolling of a Ball on the Surface of a Rotating Cone, Nonlinear Dynam., 2019, vol. 97, no. 2, pp. 1635–1648.Google Scholar
  12. 12.
    Borisov, A. V., Kilin, A. A., Karavaev, Y. L., and Klekovkin, A. V., Stabilization of the Motion of a Spherical Robot Using Feedbacks, Appl. Math. Model., 2019, vol. 69, pp. 583–592.MathSciNetGoogle Scholar
  13. 13.
    Borisov, A. V., Kilin, A. A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.MathSciNetzbMATHGoogle Scholar
  14. 14.
    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.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Borisov, A. V., Kilin, A. A., and Mamaev, I. S., How to Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, no. 3–4, pp. 258–272.MathSciNetzbMATHGoogle Scholar
  16. 16.
    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.MathSciNetzbMATHGoogle Scholar
  17. 17.
    Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics, De Gruyter Stud. Math. Phys., vol. 52, Berlin: De Gruyter, 2018.Google Scholar
  18. 18.
    Borisov, A. V. and Mamaev, I. S., Symmetries and Reduction in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 5, pp. 553–604.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Borisov, A. V. and Mamaev, I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Borisov, A. V. and Mamaev, I. S., The Dynamics of the Chaplygin Ball with a Fluid-Filled Cavity, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 490–496.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Borisov, A. V. and Mamaev, I. S., Obstacle to the Reduction of Nonholonomic Systems to the Hamiltonian Form, Dokl. Phys., 2002, vol. 47, no. 12, pp. 892–894; see also: Dokl. Akad. Nauk, 2002, vol. 387, no. 6, pp. 764–766.MathSciNetGoogle Scholar
  23. 23.
    Borisov, A. V. and Mamaev, I. S., Chaplygin’s Ball Rolling Problem Is Hamiltonian, Math. Notes, 2001, vol. 70, no. 5–6, pp. 720–723; see also: Mat. Zametki, 2001, vol. 70, no. 5, pp. 793–795.MathSciNetzbMATHGoogle Scholar
  24. 24.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., Dynamical Systems with Non-Integrable Constraints: Vaconomic Mechanics, Sub-Riemannian Geometry, and Non-Holonomic Mechanics, Russian Math. Surveys, 2017, vol. 72, no. 1, pp. 1–32; see also: Uspekhi Mat. Nauk, 2017, vol. 72, no. 5(437), pp. 3–62.MathSciNetzbMATHGoogle Scholar
  25. 25.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regul. Chaotic Dyn., 2016, vol. 21, no. 4, pp. 455–476.MathSciNetzbMATHGoogle Scholar
  26. 26.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Jacobi Integral in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 383–400.MathSciNetzbMATHGoogle Scholar
  27. 27.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277–328.MathSciNetzbMATHGoogle Scholar
  28. 28.
    Borisov, A. V., Mamaev, I. S., and Kilin, A. A., The Rolling Motion of a Ball on a Surface. New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201–219.MathSciNetzbMATHGoogle Scholar
  29. 29.
    Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139–168.MathSciNetzbMATHGoogle Scholar
  30. 30.
    Chaplygin, S. A., On the Theory of Motion of Nonholonomic Systems. The Reducing-Multiplier Theorem, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376.MathSciNetzbMATHGoogle Scholar
  31. 31.
    Chaplygin, S. A., On Some Generalization of the Area Theorem with Applications to the Problem of Rolling Balls, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 199–217.MathSciNetzbMATHGoogle Scholar
  32. 32.
    Chaplygin, S. A., On a Pulsating Cylindrical Vortex, Regul. Chaotic Dyn., 2007, vol. 12, no. 1, pp. 101–116.MathSciNetzbMATHGoogle Scholar
  33. 33.
    Chaplygin, S. A., One Case of Vortex Motion in Fluid, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 219–232.MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ehlers, K. M. and Koiller, J., Rubber Rolling: Geometry and Dynamics of 2 — 3 — 5 Distributions, in Proc. IUTAM Symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, Russia, 25–30 August 2006), pp. 469–480.Google Scholar
  35. 35.
    Fassò, F., García-Naranjo, L. C., and Sansonetto, N., Moving Energies As First Integrals of Nonholonomic Systems with Affine Constraints, Nonlinearity, 2018, vol. 31, no. 3, pp. 755–782.MathSciNetzbMATHGoogle Scholar
  36. 36.
    Fassò, F. and Sansonetto, N., Conservation of “Moving” Energy in Nonholonomic Systems with Affine Constraints and Integrability of Spheres on Rotating Surfaces, J. Nonlinear Sci., 2016, vol. 26, no. 2, pp. 519–544.MathSciNetzbMATHGoogle Scholar
  37. 37.
    Fedorov, Yu. N., Motion of a Rigid Body in a Spherical Suspension, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1988, no. 5, pp. 91–93 (Russian).Google Scholar
  38. 38.
    Fedorov, Y. N. and Kozlov, V. V., Various Aspects of n-Dimensional Rigid Body Dynamics, Amer. Math. Soc. Transl. (2), 1995, vol. 168, pp. 141–171.MathSciNetzbMATHGoogle Scholar
  39. 39.
    Golubev, V. V., Chaplygin, Izhevsk: Institute of Computer Science, 2002 (Russian).Google Scholar
  40. 40.
    Hatcher, A., Algebraic Topology, Cambridge: Cambridge Univ. Press, 2002.zbMATHGoogle Scholar
  41. 41.
    Ilin, K. I., Moffatt, H. K., and Vladimirov, V. A., Dynamics of a Rolling Robot, Proc. Natl. Acad. Sci. USA, 2017, vol. 114, no. 49, pp. 12858–12863.MathSciNetzbMATHGoogle Scholar
  42. 42.
    Kilin, A. A., Pivovarova E. N., Qualitative Analysis of the Nonholonomic Rolling of a Rubber Wheel with Sharp Edges, Regul. Chaotic Dyn., 2019, vol. 24, no. 2, pp. 212–233.MathSciNetGoogle Scholar
  43. 43.
    Kilin, A. A. and Pivovarova, E. N., Chaplygin Top with a Periodic Gyrostatic Moment, Rus. J. Math. Phys., 2018, vol. 25, no. 4, pp. 509–524.MathSciNetzbMATHGoogle Scholar
  44. 44.
    Kozlov, V. V., On the Theory of Integration of the Equations of Nonholonomic Mechanics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 191–176.MathSciNetGoogle Scholar
  45. 45.
    Kuleshov, A. S., On the Generalized Chaplygin Integral, Regul. Chaotic Dyn., 2001, vol. 6, no. 2, pp. 227–232.MathSciNetzbMATHGoogle Scholar
  46. 46.
    Li, C., Two Problems of Planar Quadratic Systems, Sci. Sinica Ser. A, 1983, vol. 26, no. 5, pp. 471–481.MathSciNetzbMATHGoogle Scholar
  47. 47.
    Lichtenberg, A. J., Lieberman, M. A., and Cohen, R. H., Fermi Acceleration Revisited, Phys. D, 1980, vol. 1, no. 3, pp. 291–305.MathSciNetzbMATHGoogle Scholar
  48. 48.
    Markeev, A. P., Integrability of the Problem of Rolling of a Sphere with a Multiply Connected Cavity Filled with an Ideal Fluid, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1986, vol. 21, no. 1, pp. 64–65 (Russian).Google Scholar
  49. 49.
    Putkaradze, V. and Rogers, S., On the Dynamics of a Rolling Ball Actuated by Internal Point Masses, Meccanica, 2018, vol. 53, no. 15, pp. 3839–3868.MathSciNetGoogle Scholar
  50. 50.
    Svinin, M., Morinaga, A., and Yamamoto, M., On the Dynamic Model and Motion Planning for a Spherical Rolling Robot Actuated by Orthogonal Internal Rotors, Regul. Chaotic Dyn., 2013, vol. 18, no. 1–2, pp. 126–143.MathSciNetzbMATHGoogle Scholar
  51. 51.
    Tsiganov, A. V., Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane, Regul. Chaotic Dyn., 2019, vol. 24, no. 2, pp. 171–186.MathSciNetGoogle Scholar
  52. 52.
    Tsiganov, A. V., On the Poisson Structures for the Nonholonomic Chaplygin and Veselova Problems, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 439–450.MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Center for Technologies in Robotics and Mechatronics ComponentsInnopolis UniversityInnopolisRussia
  3. 3.Udmurt State UniversityIzhevskRussia
  4. 4.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia

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