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Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass

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Abstract

The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the case of fixed rotors and periodically changing moments of inertia.

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Notes

  1. Instead of Eq. (2.4) one can use an equation for the precession angle \(\psi\).

  2. In robotics problems the characteristic times of motion are such that it suffices to numerically show linear stability and, as a rule, no rigorous analytical investigation of the secular (nonlinear) stability is required.

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ACKNOWLEDGMENTS

The authors thank to Prof. A. A. Kilin and I. A. Bizyaev for useful discussions.

Funding

The work of E. M. Artemova (Section 3) was carried out within the framework of the state assignment of the Ministry of Science and Higher Education of Russia (Project FEWS-2020-0009). The work of Yu. L. Karavaev (Introduction and Section 2) was supported by the Russian Science Foundation under grant 18-71-00096. The work of I. S. Mamaev (Section 5) was carried out within the framework of the state assignment of the Ministry of Science and Higher Education of Russia (Project FZZN-2020-0011) The work of E. V. Vetchanin (Section 4) was supported by the Russian Science Foundation under grant 18-71-00111.

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Authors and Affiliations

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

    Elizaveta M. Artemova & Evgeny V. Vetchanin

  2. Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, 426069, Izhevsk, Russia

    Yury L. Karavaev & Ivan S. Mamaev

Authors
  1. Elizaveta M. Artemova
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  2. Yury L. Karavaev
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  3. Ivan S. Mamaev
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  4. Evgeny V. Vetchanin
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Correspondence to Elizaveta M. Artemova, Yury L. Karavaev, Ivan S. Mamaev or Evgeny V. Vetchanin.

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37J60, 37C60

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Artemova, E.M., Karavaev, Y.L., Mamaev, I.S. et al. Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass. Regul. Chaot. Dyn. 25, 689–706 (2020). https://doi.org/10.1134/S156035472006012X

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