Nilpotent normal form for non-chained driftless systems: Characterization of the cross-chained form

In this paper, we deal with a class of nonholonomic driftless systems whose controllability Lie algebra is structurally different from that of chained form systems. We propose cross-chained systems as a class of these system. By extending the idea of Goursat´s normal form used in the case of chained systems, we introduce a nilpotent normal form for cross-chained systems. We then give a necessary and sufficient condition, with a detailed proof, for the existence of diffeomorphic coordinate and input transformation to obtain the normal form.