Summary: We develop the machinery of exterior differential forms, more particularly the Goursat normal form for a Pfaffian system, for solving nonholomorphic motion planning problems, i.e., motion planning for systems with nonintegrable velocity constraints. We use this technique to solve the problem of steering a mobile robot with \(n\) trailers. We present an algorithm for finding a family of transformations which will convert the system of rolling constraints on the wheels of the robot with \(n\) trailers into the Goursat canonical form. Two of these transformations are studied in detail. The Goursat normal form for exterior differential systems is dual to the so-called chained-form for vector fields that has been studied previously. Consequently, we are able to give the state feedback law and change of coordinates to convert the \(n\)-trailer system into chained form. Three methods for planning trajectories for chained-form systems using sinusoids, piecewise constants, and polynomials as inputs are presented. The motion planning strategy is therefore to first convert the \(n\)-trailer system into Goursat form, use this to find the chained-form coordinates, plan a path for the corresponding chained-form system, and then transform the resulting trajectory back into the original coordinates. Simulations and frames of movie animations of the \(n\)-trailer system for parallel parking and backing into a loading dock using this strategy are included.