Stabilization of systems with one degree of underactuation with energy shaping, a geometric approach

TL;DR

This paper presents a geometric method for stabilizing mechanical systems with one underactuated degree of freedom using energy shaping, including cases where the closed-loop metric is not positive-definite.

Contribution

It provides a complete geometric solution to the energy shaping stabilization problem for such systems, extending applicability to non-positive-definite metrics.

Findings

Any linearly controllable simple mechanical system with one underactuation can be stabilized by energy shaping.

The method can stabilize systems even with non-positive-definite closed-loop metrics.

An example demonstrates stabilization failure with positive-definite metrics, highlighting the method's generality.

Abstract

A geometric formulation for stabilization of systems with one degree of underactuation which fully solves the energy shaping problem for these system is given. The results show that any linearly controllable simple mechanical system with one degree of underactuation is stabilizable by energy shaping, possibly via a closed-loop metric which is not necessarily positive-definite. An example of a system with one degree of underactuation is provided for which the stabilization by energy shaping method is not achievable using a positive-definite closed-loop metric.