Transverse linearization for mechanical systems with several passive degrees of freedom with applications to orbital stabilization

A class of mechanical systems with many unactuated degrees of freedom is studied. An analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics of transverse part of coordinates of an underactuated mechanical system along a feasible motion, is proposed. The procedure is constructive and is based on a particular choice of coordinates in a vicinity of the motion. It allows explicit introduction of the so-called moving Poincare section associated with a finite-time or periodic motion. It is shown that the coordinates admit analytical linearization of transverse part of the system dynamics prior to any controller design. If the motion is periodic, then these coordinates are used for developing feedback controllers. Necessary and sufficient conditions for exponential orbital stabilization of a cycle for underactuated mechanical systems are derived. Two illustrative examples are elaborated in details.