The Matching Equations of Energy Shaping Controllers for Mechanical Systems are not Simplified with Generalized Forces*

Total Energy Shaping is a controller design methodology that achieves (asymptotic) stabilization of mechanical systems endowing the closed-loop system with a Lagrangian or Hamiltonian structure with a desired energy function. The success of the method relies on the possibility of solving two partial differential equations (PDE) which identify the kinetic and potential energy functions that can be assigned to the closed-loop. Particularly troublesome is the PDE associated to the kinetic energy which is quasi-linear and inhomogeneous and the solution, that defines the desired inertia matrix, must be positive definite. This task is simplified by the inclusion of gyroscopic forces in the target dynamics, which translates into the presence of a free skew-symmetric matrix in the matching equations that reduces the number of PDE's to be solved. Recently, it has been claimed that considering a more general form for the target dynamic forces, that relax the skew-symmetry condition, further reduces the number of PDE's. The purpose of this paper is to prove that this claim is wrong.