Normal forms for the endpoint map near nice singular curves for rank-two distributions

Given a rank-two sub-Riemannian structure $(M,\Delta)$ and a point $x_0\in M$, a singular curve is a critical point of the endpoint map $F:\gamma\mapsto\gamma(1)$ defined on the space of horizontal curves starting at $x_0$. The typical least degenerate singular curves of these structures are called \emph{regular singular curves}; they are \emph{nice} if their endpoint is not conjugate along $\gamma$. The main goal of this paper is to show that locally around a nice singular curve $\gamma$, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which $F$ writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures.