Configuration flatness of Lagrangian control systems with less controls than the 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. Thus, if a Lagrangian control system is configuration flat, desired trajectories are easily realized. However, to find a criterion for determining whether a Lagrangian control system is configuration flat or not remains to be an open problem. Previous studies have provided a necessary condition and a sufficient condition for a Lagrangian control system with n degrees of freedom and n-1 controls to be configuration flat. The main purpose of this paper is to extend the theory so as to be applicable to a wide class of Lagrangian control systems. To be specific, 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