Lie-group modeling and simulation of a spherical robot, actuated by a yoke–pendulum system, rolling over a flat surface without slipping

The present paper aims at introducing a mathematical model of a spherical robot expressed in the language of Lie-group theory. Since the main component of motion is rotational, the space $\mathrm{SO}\left(3\right)3$ of three-dimensional rotations plays a prominent role in its formulation. Because of friction to the ground, rotation of the external shell results in translational motion. Rolling without slipping implies a constraint on the tangential velocity of the robot at the contact point to the ground which makes it a non-holonomic dynamical system. The mathematical model is obtained upon writing a Lagrangian function that describes the mechanical system and by the Hamilton minimal-action principle modified through d’Alembert virtual work principle to account for non-conservative control actions as well as frictional reactions. The result of the modeling appears as a series of non-holonomic Euler–Poincaré equations of dynamics plus a series of auxiliary equations of reconstruction and advection type. A short discussion on the numerical simulation of such mathematical model complements the main analytic-mechanic development.