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.

中文翻译：

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作为Chaplygin球问题的推广，机器人在平面上滚动的不同模型

本文解决了带有内部旋转框架的球形壳体的滚动问题，转子固定在该框架上。假定整个系统的质心在壳体的几何中心。对于橡胶滚动模型和经典滚动模型，表明，如果框架和转子的旋转角速度恒定， ，则存在一个非惯性坐标系（附加在框架上），其中运动方程式并不明确地依赖于时间。所得的运动方程式保留了角动量矢量的模拟，并且在形式上类似于Chaplygin球的方程式。因此，该问题减少到研究二维庞加莱图。橡胶滚动模型的情况进行了详细分析。庞加莱地图的数值研究表明，存在着混沌的轨迹，包括与一个奇怪的吸引子相关的轨迹。另外，对静止状态下的运动情况进行了分析，其中问题减少到研究球体上的矢量场S ²。