Trajectory Generation for the N-Trailer Problem Using Goursat Normal Form. Murray, R., M. IEEE Transactions on Automatic Control, 40(5):802-819, 1995.
Trajectory Generation for the N-Trailer Problem Using Goursat Normal Form [pdf]Paper  abstract   bibtex   
Develops the machinery of exterior differential forms, more\nparticularly the Goursat normal form for a Pfaffian system, for solving\nnonholonomic motion planning problems, i.e., motion planning for systems\nwith nonintegrable velocity constraints. The authors use this technique\nto solve the problem of steering a mobile robot with n trailers. The\nauthors present an algorithm for finding a family of transformations\nwhich will convert the system of rolling constraints on the wheels of\nthe robot with n trailers into the Goursat canonical form. Two of these\ntransformations are studied in detail. The Goursat normal form for\nexterior differential systems is dual to the so-called chained-form for\nvector fields that has been studied previously. Consequently, the\nauthors are able to give the state feedback law and change of\ncoordinates to convert the N-trailer system into chained-form. Three\nmethods for planning trajectories for chained-form systems using\nsinusoids, piecewise constants, and polynomials as inputs are presented.\nThe motion planning strategy is therefore to first convert the N-trailer\nsystem into Goursat form, use this to find the chained-form coordinates,\nplan a path for the corresponding chained-form system, and then\ntransform the resulting trajectory back into the original coordinates.\nSimulations and frames of movie animations of the N-trailer system for\nparallel parking and backing into a loading dock using this strategy are\nincluded

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