Differential flatness is a characterization of specific nonlinear systems that allow a linear description in a suitable equivalent space. The equivalence does not necessarily preserve the dimension of the underlying manifold that describes the nonlinear systems. Among complicated systems that present challenging control issues, one can mention cranes, unmanned aerial vehicles, kinematically constrained (nonholonomic) chained systems. These systems are, on the one hand, differentially flat, but they can also be considered as trivial systems in a larger space of coordinates that is quadratically constrained. This point of view leads to the study of trivial ambient spaces that, once restricted on a submanifold, generate complicated dynamics. We characterize mathematically the type of reduction that occurs in the context of differential flatness.
. Application of Legendrian Foliations in Differential Flatness Problems. IEEE CDC 2012, Hawaii, 2012.