Stabilization of an autonomous rolling sphere navigating in a labyrinth arena: A geometric mechanics perspective

Several concepts and results in geometric mechanics are used to analyze and control the locomotion system of an unconventional robot encapsulated in a sphere shell, assumed to roll without slipping on the floor and internally equipped with a set of inertia gyros as indirect driving devices. Lie group symmetries intrinsic to this problem, i.e., invariance of the system’s Lagrangian and velocity distribution to some group of motions, allows the reduction of the equations of motion. This system whose motion ability is based on angular momentum conservation is established as a controllable nonholonomic system for which the attitude/position cannot be stabilized by smooth feedback laws. Pursuing the reduction process permits us to design a feedback law extensible to both kinematic and dynamic levels of actuation, enabling the robot to execute finite-time reorientation and repositioning maneuvers while confined to move in corridor-like domains. The derivation of the underlying nonlinearity contents via the geometric approach helps the analysis not to rely on a specific choice of coordinates and allows taking profit of the vector structure of the equations for further investigations.