Remarks on ISS and formation control

For a leader/follower formation of autonomous mobile agents, the concept of input-to-state stability has been proposed recently as a valid tool to relate the influence of the motion of the leader to the formation error. The ISS approach is quite promising, as it allows great flexibility in the formation topology, and yields computable gains relating the formation error to external inputs and disturbances. While suitable when the dynamics of the agents are modeled as linear systems, the ISS framework is not directly applicable when non-holonomic vehicle models are considered, as the necessary condition of global asymptotic stability of the unforced systems is not met. The aim of this paper is to clarify some of the issues and the obstruction in applying ISS-methods to the control of formations of non-holonomic vehicles, and to offer some remedies. The analysis of the behavior of the vehicle dynamics relative to the steady-state of the formation plays a crucial role in identifying the source of the problem. The proposed solution, based on a local version of ISS originally given by Teel, employs saturated controls to enforce ISS of the internal dynamics in an input-output linearization scheme. Computable bounds on appropriate norms of the formation error as function of external inputs are given explicitly. Simulation results on an illustrative example are presented and discussed.