We give a geometric account of kinematic control of a spherical rolling robot controlled by two internal wheels just like the toy robot Sphero. Particularly, we introduce the notion of shape space and fibers to the system by exploiting its symmetry and the principal bundle structure of its configuration space; the shape space encodes the rotational angles of the wheels, whereas each fiber encodes the translational and rotational configurations of the robot for a particular shape. We show that the system is fiber controllable—meaning any translational and rotational configuration modulo shapes is reachable—as well as find exact expressions of the geometric phase or holonomy under some particular controls. We also solve an optimal control problem of the spherical robot, show that it is completely integrable, and find an explicit solution of the problem.

中文翻译：

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球形滚动机器人的几何运动学控制

我们给出了一个由两个内轮控制的球形滚动机器人运动学控制的几何方法，就像玩具机器人Sphero。特别是，我们利用系统的对称性和配置空间的主要束结构，将形状空间和纤维的概念引入系统。形状空间编码车轮的旋转角度，而每种纤维编码特定形状的机器人的平移和旋转配置。我们证明了该系统是光纤可控的-意味着可以实现任何平移和旋转模的形状-并在某些特定控制下找到了几何相位或完整性的精确表示。我们还解决了球形机器人的最优控制问题，证明它是完全可集成的，并找到了该问题的明确解决方案。