# Trajectory generation for the N-trailer problem using Goursat normal form

### D. Tilbury, R. M. Murray and S. S. Sastry

*IEEE T. Automatic Control*, 40: (5) 802-819

In 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 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.

- ERL memo: http://www.cds.caltech.edu/~murray/preprints/tms93-erl.pdf
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