Transverse linearization for mechanical systems with passive links, impulse effects, and friction forces

We consider a class of mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom. In addition to control forces, we take into account viscous and Coulomb friction forces and impacts with the environment modeled as impulsive updates of the states. We assume that a motion planning task is solved and a feasible forced periodic motion is described in terms of piece-wise smooth virtual holonomic constraints. The main contribution is an analytical method for computing coefficients of an impulsive linear control system, solutions of which approximate dynamics transversal to the preplanned trajectory. This linear system is shown to be useful for stability analysis and for design of feedback controllers orbitally stabilizing forced periodic motions in the hybrid mechanical system. As an illustration, we apply the obtained theoretical results providing a rigorous proof of orbital exponential stability of the periodic tumbling motion for a model of a descending strip of paper in a still air.