Configuration flatness of Lagrangian control systems with fewer controls than degrees of freedom

A Lagrangian control system is said to be configuration flat if its states are determined by configuration flat outputs depending on configuration variables only. This paper gives a necessary condition and two types of sufficient conditions for Lagrangian control systems with n degrees of freedom and m controls with m < n to be configuration flat, and further makes a remark on the choice of output variables for configuration flatness. Two examples are given in which the sufficient conditions are utilized for finding configuration flat outputs.