A Geometric Framework for Stabilization by Energy Shaping: Sufficient Conditions for Existence of Solutions

Abstract

We present a geometric formulation for the energy shaping problem. The central objective is the initiation of a more systematic exploration of energy shaping with the aim of de- termining whether a given system can be stabilized using energy shaping feedback. We investigate the partial differential equations for the kinetic energy shaping problem using the formal theory of partial differential equations. The main contribution is sufficient conditions for integrability of these partial differential equations. We couple these results with the integrability results for potential energy shaping (25). This gives some new avenues for answering key questions in energy shaping that have not been addressed to this point. 1. Introduction. In Brockett's 1977 paper (9) it was observed that there were structural aspects of mechanical systems that made them attractive as a class of control problems. In this paper he mentioned differential geometry as the common mathematical structure between control theory and analytical mechanics. He inves- tigated the Lagrangian and Hamiltonian formulations for mechanical systems and considered the interplay of the mechanical and control theoretic structures. One interesting control problem is the following: given a mechanical system with an unstable equilibrium at a point q0, stabilize the system using feedback. One of the recent developments in the stabilization of equilibria is the energy shaping method. The key idea concerns the construction of a feedback for which the closed-loop system possesses the structure of a mechanical system. A feedback so obtained is called an energy shaping feedback and the procedure by which it is obtained is called energy shaping. In the classical notion of energy shaping, the assumed method consists of two stages: shaping the kinetic energy of the system—so-called kinetic energy shaping—and changing the potential energy of the system—so-called potential energy shaping. If such an energy shaping feedback exists, then for stability one has to ensure that the Hessian of the closed-loop potential energy is positive definite. The cart-pendulum, as a mechanical system with one degree of underactuation, is one of the systems that has been stabilized using the energy shaping method (14, 26). The system has the upright equilibria as saddle points and potential energy