Geometric Kinematic Control of a Spherical Rolling Robot

TL;DR

This paper presents a geometric framework for controlling a spherical robot with internal wheels, demonstrating fiber controllability, deriving geometric phases, and solving an optimal control problem explicitly.

Contribution

It introduces a geometric approach using shape space and fiber structures, proving controllability and providing explicit solutions for control and optimization.

Findings

System is fiber controllable, enabling full configuration reachability.

Derived explicit expressions for geometric phase and holonomy.

Solved the optimal control problem explicitly, showing complete integrability.

Abstract

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.