Stabilization of Systems with One Degree of Underactuation with Energy Shaping: A Geometric Approach

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, even in the absence of gyroscopic force, 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, in the absence of gyroscopic forces, using a positive-definite closed-loop metric.