Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass
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Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass Elizaveta M. Artemova1* , Yury L. Karavaev2** , Ivan S. Mamaev2*** , and Evgeny V. Vetchanin1**** 1
Udmurt State University, ul. Universitetskaya 1, 426034 Izhevsk, Russia 2 Kalashnikov Izhevsk State Technical University, ul. Studencheskaya 7, 426069 Izhevsk, Russia Received April 15, 2020; revised May 11, 2020; accepted May 12, 2020
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. MSC2010 numbers: 37J60, 37C60 DOI: 10.1134/S156035472006012X Keywords: nonholonomic constraint, rubber rolling, unbalanced ball, rolling on a plane
1. INTRODUCTION Reference [45] addresses the dynamics of a nonholonomic system describing the motion of a balanced spherical robot with variable moments of inertia within the framework of the “rubber” rolling model which suppose the absence of slipping or twisting (about using the model of “rubber” rolling and others nonholonomic models see [5–7, 11, 15, 26, 33, 40–42, 49, 51]). It is shown that this system admits three invariant manifolds corresponding to the plane-parallel motions of the robot. Results of a numerical stability analysis of the system are presented and the existence of instability regions similar to Arnold’s tongues [2, 22] in a neighborhood of resonant points is demonstrated. In addition, in Ref. [45] it is shown that the system is nonconservative and that its phase space may exhibit stable limit cycles attracting tori and strange attractors. In practice, it is very difficult to implement such an idealized model, since technically it is very difficult to ensure absolute balance, and the smallest deviations of the position of the center of mass from the geometric center of the sphere can lead to a significant deviation of the trajectory of motion. Such a problem arose for spherical robots with internal rotors [50] and for spherical robots that moved by changing the position of the center of mass, which was positioned with an error [31, 36]. In addition, a large number of spherical robots move by controlled changes in the position of the center of mass, which is a more efficient method of motion of mobile spherical robots [3, 16, 25, 27–29, 52, 53]. It should be noted that the dynamics of robots with periodic controls has received a lot of attention in the literature. For example, in Refs. [16, 32], periodic controls, namely, rotational oscillations of the rotor, were used to stabilize the motion of a spherical robot of c
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