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

TL;DR

This paper compares the energy shaping and Lyapunov constraint based methods for control of underactuated systems, showing their equivalence and providing a geometric framework for understanding their relationship.

Contribution

It demonstrates the equivalence of the improved controlled Hamiltonians method and the Lyapunov constraint based method within a differential geometric framework.

Findings

Proves the equivalence of the two control methods for simple Hamiltonian systems.

Provides coordinate-free expressions of Chang's matching conditions.

Enhances understanding of control methods through differential geometry.

Abstract

In this paper, we make a review of the controlled Hamiltonians (CH) method and its related matching conditions, focusing on an improved version recently developed by D.E. Chang. Also, we review the general ideas 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 an arbitrary number of actuators. We analyze and compare these methods within the framework of Differential Geometry, and from a purely theoretical point of view. We show, in the context of underactuated systems defined by simple Hamiltonian functions, that the LCB method and the Chang's version of the CH method are equivalent stabilization methods (i.e. they give rise to the same set of control laws). In other words, we show that 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.