Criterion for the Existence of a Separated Coordinate Representation for Underactuated Mechanical Systems

Mechanical rigid body systems typically are modeled by a system of coupled second order nonlinear differential equations, where the right hand side describes the effect of the system's input, i. e., the external generalized forces. The components of the input are assigned to the scalar differential equations via a matrix which in general depends on the configuration. On the other hand, in the literature predominates the assumption that a separated coordinate representation exists such that a decomposition of the equations of motion into a non-actuated and a fully actuated subsystem is possible. This decomposition disburdens the systems analysis and the controller design but the conditions of its existence have not yet obtained much attention. The present contribution proposes a criterion, whether or not such a choice of separated coordinates is possible. The criterion is an application of the famous Frobenius theorem and will be discussed both in differential form and in vector field formulation. Its application is shown by three simple examples.