Configuration Flatness of Lagrangian Systems Underactuated by One Control

Lagrangian control systems that are differentially flat with flat outputs that depend only on configuration variables are said to be configuration flat. We provide a complete characterization of configuration flatness for systems with n degrees of freedom and n-1 controls whose range of control forces only depends on configuration and whose Lagrangian has the form of kinetic energy minus potential. The method presented allows us to determine if such a system is configuration flat and, if so, provides a constructive method for finding all possible configuration flat outputs. Our characterization relates configuration flatness to Riemannian geometry. We illustrate the method with two examples.