On the relationship between the energy shaping and the Lyapunov constraint based methods

SERGIO GRILLO; LEANDRO SALOMONE; MARCELA ZUCCALLI

Journal of Geometric Mechanics

AIMS

Año: 2017

1941-4889

In this paper, we make a review of the controlled Hamiltonians (CH) method and its related matchingconditions, focusing on an improved version recently developed by D.E. Chang. Also, we review the generalideas around the Lyapunov constraint based (LCB) method, whose related partial differential equations (PDEs)were originally studied for underactuated systems with only one actuator, and then we study its PDEs for anarbitrary number of actuators. We analyze and compare these methods within the framework of DifferentialGeometry, and from a purely theoretical point of view. We show, in the context of underactuated systemsdefined by simple Hamiltonian functions, that the LCB method and the Chang´s version of the CH method areequivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we showthat the Chang´s improvement of the energy shaping method is precisely the LCB method. As a by-product,coordinate-free and connection-free expressions of Chang´s matching conditions are obtained.