# Trajectory Generation for the N-Trailer Problem Using Goursat Normal Form

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

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/ERL M93/12

1993

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. planning problems with non-integrable velocity constraints. We apply this technique to solving the problem of steering a mobile robot with n trailers. We present an algorithm for finding a family of transformations which will display the given system of rolling constraints on the wheels of the robot with n trailers in 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 we have studied in our earlier work. 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 steering chained form systems using sinusoids, piecewise constants and poly- nomials as inputs are presented. The motion planning strategy is therefore to first convert the N-trailer system into chained form, steer the corresponding chained form system, 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 also included.

BibTeX citation:

@techreport{Tilbury:M93/12, Author = {Tilbury, D. and Murray, R. and Sastry, S. Shankar}, Title = {Trajectory Generation for the N-Trailer Problem Using Goursat Normal Form}, Institution = {EECS Department, University of California, Berkeley}, Year = {1993}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2281.html}, Number = {UCB/ERL M93/12}, Abstract = {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. planning problems with non-integrable velocity constraints. We apply this technique to solving the problem of steering a mobile robot with n trailers. We present an algorithm for finding a family of transformations which will display the given system of rolling constraints on the wheels of the robot with n trailers in 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 we have studied in our earlier work. 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 steering chained form systems using sinusoids, piecewise constants and poly- nomials as inputs are presented. The motion planning strategy is therefore to first convert the N-trailer system into chained form, steer the corresponding chained form system, 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 also included.} }

EndNote citation:

%0 Report %A Tilbury, D. %A Murray, R. %A Sastry, S. Shankar %T Trajectory Generation for the N-Trailer Problem Using Goursat Normal Form %I EECS Department, University of California, Berkeley %D 1993 %@ UCB/ERL M93/12 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/2281.html %F Tilbury:M93/12