Decomposition of the problem of controlling a mobile one-wheel robot with an unperturbed gyrostabilized platform

Abstract

The design of the mobile robot under consideration is based on the analysis of the classical problem of rolling a heavy disc that carries a flywheel and whose rotation axis Cz is perpendicular to the disc plane and passes through its center of mass C on an absolutely rough horizontal plane [1] (see Fig. 1). The nonholonomic mechanical system “disc + flywheel” has four degrees of freedom and is a gyrostat. Its position is uniquely specified by six generalized coordinates: Euler angles ψ, θ, and φ between the trihedron rigidly bound to the principal central axes of inertia of the disc and stationary trihedron OX*Y*Z*, angle γ of the flywheel rotation with respect to the disc about the Cz axis, and coordinates x and y of the projection of the disc’s center of mass in the plane OX*Y*, where the disc rolls. The problem involves three cyclic coordinates ψ, θ, and φ and the energy integral. Thus, the analytical analysis of the equations of motion of a disc with a flywheel reduces to integrating one second-order linear differential equation of the form