Robust exponential stabilization for the extended chained form via hybrid control

The extended (or 2nd. order) chained form (ECF) is a canonical system used to model a class of 3-DOF mechanical systems with 2 inputs. Stabilization of this system to the origin is far from trivial in view of the well known Brockett\´s necessary condition for asymptotic stabilization by continuous feedback. In this paper we propose a control law, based on a hybrid "feedback/open-loop" strategy, which ensures exponential stability of a dynamic extension of the ECF while exhibiting a degree of robustness to additive disturbance terms taken from a class of vector fields. These disturbances may represent phenomena such as parameter uncertainty or unmodeled dynamics. The approach is illustrated by a numerical simulation of a PPR manipulator with passive third joint and parameter uncertainties.