Reformulation of Matching Equation in Potential Energy Shaping

Stabilization of an underactuated mechanical system may be accomplished via energy shaping. Interconnection and damping assignment passivity-based control is an approach based on total energy shaping by assigning desired kinetic and potential energy to the system. This method requires solving a partial differential equation (PDE) related to the potential energy shaping of the system. In this short article, we focus on reformulating this PDE in order to be solved easier. For this purpose, two sufficient conditions are proposed such that under the satisfaction of one of them, it is possible to merely solve the homogeneous part of potential energy PDE. Besides, the class of systems that simply verify these conditions are discussed. The results are applied to two benchmark systems.