Stability of autonomous vehicle formations using an ISS small-gain theorem for networks

We consider a formation of vehicles moving on the two dimensional plane. The movement of each vehicle is described by a system of ordinary differential equations with inputs. The formation is maintained using autonomous controls that are designed to maintain fixed relative distances and orientations between vehicles. Moreover this formation should track a given trajectory on the plane. The vehicles can measure the relative distances and angles to their neighbors. These values are the inputs from one system to another. With the help of a general ISS small-gain theorem for networks we will show that the dynamics of such a formation is stable for the given controls. The notion of local input-tostate stability (local ISS) will be used for this purpose.