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

  • Sergey Chaplygin Memorial Issue
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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.

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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).

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700, Russia

    Ivan A. Bizyaev & Ivan S. Mamaev

  2. Center for Technologies in Robotics and Mechatronics Components, Innopolis University, ul. Universitetskaya 1, Innopolis, 420500, Russia

    Ivan A. Bizyaev

  3. Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034, Russia

    Alexey V. Borisov

  4. Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S. Kovalevskoi 16, Ekaterinburg, 620990, Russia

    Ivan S. Mamaev

Authors
  1. Ivan A. Bizyaev
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  2. Alexey V. Borisov
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  3. Ivan S. Mamaev
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Correspondence to Ivan A. Bizyaev, Alexey V. Borisov or Ivan S. Mamaev.

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Bizyaev, I.A., Borisov, A.V. & Mamaev, I.S. Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem. Regul. Chaot. Dyn. 24, 560–582 (2019). https://doi.org/10.1134/S1560354719050071

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  • DOI: https://doi.org/10.1134/S1560354719050071

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