Trajectory Generation for the N-trailer Problem Using Goursat Normal Form

In this paper, we develop the machinery of exterior diierential forms, more particularly the Goursat normal form for a Pfaaan 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 nding 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 diierential systems is dual to the so-called chained form for vector elds 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 polynomials as inputs are presented. The motion planning strategy is therefore to rst 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. 1 1. Introduction In the past few years there has been a great deal of interest in the generation of motion planning algorithms for robots with nonholonomic or non-integrable velocity constraints in cluttered environments. These constraints on the instantaneous velocities that can be achieved arise from the kinematics of the drive mechanisms of the carts. This work has been a departure from the traditional robot motion planning (see for example 7, 16, 19]) which concentrated on understanding the complexity of the computational eeort associated with planning trajectories for robots (with no constraints on their instantaneous velocities) which would avoid both xed and moving obstacles. Unfortunately the motion plans arising from these more traditional methods often required sideways motion of robot carts with wheels, and as was pointed out by Laumond, most mobile robots are not on castors 20, 21]. In this paper, we consider and solve the motion planning problem for a system