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

TL;DR

This paper provides a method to explicitly solve the kinetic and potential matching conditions in energy shaping for underactuated Hamiltonian systems, facilitating the design of stabilizing controllers.

Contribution

It introduces a procedure to integrate the matching conditions up to quadratures, including explicit solutions and conditions for positive-definite solutions, especially for systems with one degree of underactuation.

Findings

Explicit solutions for potential equations are derived.

Integrability and positivity conditions are established.

A concrete formula for the kinetic equation in certain cases is provided.

Abstract

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.