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.
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).
Citation:
Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem”, Regul. Chaotic Dyn., 24:5 (2019), 560–582
\Bibitem{BizBorMam19}
\by Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev
\paper Different Models of Rolling for a Robot Ball on a Plane as a Generalization of the Chaplygin Ball Problem
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 5
\pages 560--582
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This publication is cited in the following 18 articles:
Mariana Costa-Villegas, Luis C. García-Naranjo, “Affine Generalizations of the Nonholonomic Problem of a Convex Body Rolling without Slipping on the Plane”, Regul. Chaot. Dyn., 2025
A. A. Kilin, T. B. Ivanova, “The Integrable Problem of the Rolling Motion
of a Dynamically Symmetric Spherical Top
with One Nonholonomic Constraint”, Rus. J. Nonlin. Dyn., 19:1 (2023), 3–17
A. A. Kilin, T. B. Ivanova, “The Problem of the Rolling Motion
of a Dynamically Symmetric Spherical Top
with One Nonholonomic Constraint”, Rus. J. Nonlin. Dyn., 19:4 (2023), 533–543
Seyed Amir Tafrishi, Mikhail Svinin, Motoji Yamamoto, Yasuhisa Hirata, “A geometric motion planning for a spin-rolling sphere on a plane”, Applied Mathematical Modelling, 121 (2023), 542
Evgeniya A. Mikishanina, “Dynamics of the Chaplygin sphere with additional constraint”, Commun. Nonlinear Sci. Numer. Simul., 117 (2023), 106920–15
E. A. Mikishanina, “Rolling motion dynamics of a spherical robot with a pendulum actuator controlled by the Bilimovich servo-constraint”, Theoret. and Math. Phys., 211:2 (2022), 679–691
Yu. L. Karavaev, “Spherical Robots:
An Up-to-Date Overview of Designs and Features”, Rus. J. Nonlin. Dyn., 18:4 (2022), 709–750
E. A. Mikishanina, “Motion Control of a Spherical Robot with a Pendulum
Actuator for Pursuing a Target”, Rus. J. Nonlin. Dyn., 18:5 (2022), 899–913
Alexander Kilin, Elena Pivovarova, 2021 International Conference “Nonlinearity, Information and Robotics” (NIR), 2021, 1
Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of Rubber Chaplygin Sphere under Periodic Control”, Regul. Chaotic Dyn., 25:2 (2020), 215–236
Alexey V. Borisov, Evgeniya A. Mikishanina, “Two Nonholonomic Chaotic Systems. Part II. On the Rolling of a Nonholonomic Bundle of Two Bodies”, Regul. Chaotic Dyn., 25:4 (2020), 392–400
Elizaveta M. Artemova, Yury L. Karavaev, Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass”, Regul. Chaotic Dyn., 25:6 (2020), 689–706
A. V. Borisov, E. A. Mikishanina, “Dynamics of the Chaplygin Ball with Variable Parameters”, Rus. J. Nonlin. Dyn., 16:3 (2020), 453–462
A. A. Kilin, E. N. Pivovarova, “Neintegriruemost zadachi o kachenii sfericheskogo volchka po vibriruyuschei ploskosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:4 (2020), 628–644
I. A. Bizyaev, I. S. Mamaev, “Separatrix splitting and nonintegrability in the nonholonomic rolling of a generalized Chaplygin sphere”, Int. J. Non-Linear Mech., 126 (2020), 103550
A. V. Borisov, A. V. Tsiganov, “The motion of a nonholonomic Chaplygin sphere in a magnetic field, the Grioli problem, and the Barnett-London effect”, Dokl. Phys., 65:3 (2020), 90–93
Yury Karavaev, Alexander Kilin, Anton Klekovkin, Elena Pivovarova, 2020 International Conference Nonlinearity, Information and Robotics (NIR), 2020, 1
Alexey V. Borisov, Andrey V. Tsiganov, “On the Chaplygin Sphere in a Magnetic Field”, Regul. Chaotic Dyn., 24:6 (2019), 739–754
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