In order to convert the system to this multi-input chained form, we use dynamic state feedback. We draw some motivation from the very simple example of a kinematic unicycle and the relationships of the angular velocities therein, and we show how the dynamic state feedback that we use corresponds to adding, in front of the steerable cars, a chain of virtual axles which diverges from the original chain of trailers.
We briefly discuss how some of the methods which have been proposed for steering and stabilizing two-input chained form systems can be generalized to multi-chained systems. For concreteness, we also present two different example systems: a fire truck (three axles) and a five-axle, two-steering system. Simulation results for a parallel-parking maneuver for the five-axle system are included in the form of margin movies.
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.