On singularities of flat affine systems with n states and n−1 controls

We study the set of intrinsic singularities of flat affine systems with n−1 controls and n states using the notion of Lie-Bäcklund atlas, previously introduced by the authors. For this purpose, we prove two easily computable sufficient conditions to construct flat outputs as a set of independent first integrals of distributions of vector fields: the first one in a generic case, namely, in a neighborhood of a point where the n−1 control vector fields are independent and the second one at a degenerate point where p−1 control vector fields are dependent of the n−p others, with p>1. After introducing the Γ-accessibility rank condition, we show that the set of intrinsic singularities includes the set of points where the system does not satisfy this rank condition and is included in the set where a distribution of vector fields introduced in the generic case is singular. We conclude this analysis by three examples of apparent singularities of flat systems in generic and nongeneric degenerate cases.