Stabilization of steady motions for systems with redundant coordinates

The stability and stabilization problem of steady motions for mechanical systems with nonlinear geometric constraints is considered. The steady state information is assumed to be incomplete. Redundant coordinates, Routh's variables and Shul'gin's equations of motion are used. The set of cyclical coordinates is divided into two parts for impulses (Routh variables) and controlled coordinates (Lagrange variables). The rest of coordinates is assumed to be uncontrolled. The characteristic equation for the perturbed motion has zero roots. Its number is equal to the number of impulses plus the number of redundant coordinates. The stabilization theorem is proved for three variants of the measurement vector. The control law and the observing system coefficients can be determined by solving the Krasovskiy linear-quadratic problems for the controlled subsystem. This system does not depend on the critical variables (redundant coordinates and impulses). The stability of the complete nonlinear system follows from the reduction to Lyapunov's special case and Malkin's stability theorem under time-varying perturbations.