Article Contents
Article Contents

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

S. Grillo and L. Salomone thank CONICET for its financial support

• In the context of underactuated Hamiltonian systems defined by simple Hamiltonian functions, the matching conditions of the energy shaping method split into two decoupled subsets of equations: the kinetic and potential equations. The unknown of the kinetic equation is a metric on the configuration space of the system, while the unknown of the potential equation are the same metric and a positive-definite function around some critical point of the Hamiltonian function. In this paper, assuming that a solution of the kinetic equation is given, we find conditions (in the $C^{\infty}$ category) for the existence of positive-definite solutions of the potential equation and, moreover, we present a procedure to construct, up to quadratures, some of these solutions. In order to illustrate such a procedure, we consider the subclass of systems with one degree of underactuation, where we find in addition a concrete formula for the general solution of the kinetic equation. As a byproduct, new global and local expressions of the matching conditions are presented in the paper.

Mathematics Subject Classification: Primary: 93D05, 93D20; Secondary: 93C10.

 Citation:

• Figure 1.  Planar inverted double pendulum

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