Summary: We propose a constructive procedure that transforms locally, even at singular configurations, the kinematics of a car towing trailers into Kumpera-Ruiz normal form. This construction converts the non-holonomic motion planning problem into an algebraic problem (the resolution of a system of polynomial equations), which we illustrate by steering the two-trailer system in a neighbourhood of singular configurations. We show also that the \(n\)-trailer system is a universal local model for all Goursat structures and that all Goursat structures are locally nilpotentizable.