Explicit solutions of the kinetic and potential matching conditions of the energy shaping method

In this paper we present a procedure to integrate, up to quadratures, the matching conditions of the energy shaping method. We do that in the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions. For such systems, the matching conditions split into two decoupled subsets of equations: the kinetic and potential equations. First, assuming that a solution of the kinetic equation is given, we find integrability and positivity conditions for the potential equation (because positive-definite solutions are the interesting ones), and we find an explicit solution of the latter. Then, in the case of systems with one degree of underactuation, we find in addition a concrete formula for the general solution of the kinetic equation. An example is included to illustrate our results.