Characteristic numbers and normal form for a class of driftless systems in a one-dimension sub-manifold neighborhood

From the works of Murray and Sastry (1993) sufficient conditions to transform a driftless system into a so-called one chained form are well known. Some of these conditions are the involutivity of specific distributions, which may be too restrictive. Thus in this paper, in order to relax theses conditions, we use the higher-order technique introduced by Krener (1984). A quadratic normal form is found for driftless system. Moreover, we give conditions in order to transform, due to homogeneous feedback and diffeomorphism, the system into a chained form at least up to the third order in an approximated state and inputs meaning