After having obtained robustness against external disturbances, we wanted to extend this work to formations. Our goal was to take a desired formation structure (which might change during operation), and have the formation centre follow the trajectory.
In order to determine the distance of the individual vehicles relative to the desired formation centre (on the trajectory), we needed a reference frame attached to the trajectory. A common way to do so is using a Serret-Frenet frame, but-as pointed out in computer engineering literature (see [CDC'10] for relevant references)-this frame has some drawbacks for our intended use:
* the orientation of the frame is not uniquely defined along straight lines
* the orientation of the frame abrubtly switches at inflection points
The first point makes it impossible to follow lawn-mower patterns, which consist of long, straight lines, and are often used to systematically scan the ocean floors. The second point would change the desired position of a vehicle from one side of the trajectory to the other; for a formation this might easily lead to collisions.
A way to overcome the problems of Serret-Frenet frames, is the use of geodesic reference frames, as discussed in the work of Hanson (please see the references in [CDC'10]). This method can be used for any curve with a non-zero first derivative (here: the along-path speed should be non-zero), as opposed to the need of non-zero first and second derivatives for Serret-Frenet frames (both the along-path speed and acceleration should be non-zero; hence straight lines are excluded). The resulting behaviour is shown in the movie below.
The desired heading of the vehicles can be defined in two ways:
* along the tangent of the main trajectory
* along the tangent of the individual, parallel trajectory
The first method results in a formation for which all vehicles have the same heading, while the second method makes the vehicles move in a more natural way. In the movie, the latter method is shown.