TRAJECTORY GENERATION FOR THE N-TRAILER PROBLEM USING GOURSAT NORMAL-FORM

n this paper, we develop the machinery of exterior differential forms, more particularly the Goursat normal form for a Pfaffian system, for solving nonholonomic motion planning problems, i.e., motion planning f or systems with nonintegrable velocity constraints. We use this techni que to solve the problem of steering a mobile robot with n trailers, W e present an algorithm for finding a family of transformations which w ill convert the system of rolling constraints on the wheels of the rob ot with n trailers into the Goursat canonical form. Two of these trans formations 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 traject ories for chained-form systems using sinusoids, piecewise constants, a nd polynomials as inputs are presented. The motion planning strategy i s therefore to first convert the N-trailer system into Goursat form, u se this to find the chained-form coordinates, plan a path for the corr esponding chained-form system, and then transform the resulting trajec tory back into the original coordinates. Simulations and frames of mov ie animations of the N-trailer system for parallel parking and backing into a loading dock using this strategy are included.