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