Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem

We develop a method for the stabilization of mechanical systems with symmet ry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian For the uncontrolled sys tem, thereby constructing the controlled Lagrangian. The Fuler-Lagrange equ ations derived from the controlled Lagrangian describe the closed-loop syst em, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods c an be used to find control gains that yield closed-loop stability. In this paper we use kinetic shaping to preserve symmetry and only stabiliz e systems module the symmetry group, In the sequel to this paper (Part II), we extend the technique to include potential shaping and we achieve stabil ization in the full phase space. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendul um on a cart moving in the plane.