EXPLICIT SOLUTIONS OF THE KINETIC AND POTENTIAL MATCHING CONDITIONS OF THE ENERGY SHAPING METHOD

SERGIO GRILLO; MARCELA ZUCCALLI; LEANDRO SALOMONE

Journal of Geometric Mechanics

AIMS

Año: 2021

1941-4889

In the context of underactuated Hamiltonian systems defined bysimple Hamiltonian functions, the matching conditions of the energy shapingmethod split into two decoupled subsets of equations: the kinetic and potentialequations. The unknown of the kinetic equation is a metric on the configura-tion space of the system, while the unknown of the potential equation are thesame metric and a positive-definite function around some critical point of theHamiltonian function. In this paper, assuming that a solution of the kineticequation is given, we find conditions (in the smooth category) for the existence ofpositive-definite solutions of the potential equation and, moreover, we presenta procedure to construct, up to quadratures, some of these solutions. In orderto illustrate such a procedure, we consider the subclass of systems with onedegree of underactuation, where we find in addition a concrete formula for thegeneral solution of the kinetic equation. As a byproduct, new global and localexpressions of the matching conditions are presented in the paper.