Cascade design for formation control of nonholonomic systems in chained form

This paper presents a constructive method to design a cooperative state and output feedback to steer a group of nonholonomic mobile robots in chained form to form a desired geometric formation shape. The control methodology divides the resulting tracking error dynamics into a cascaded of linear and time-varying subsystems. A basic consensus algorithm is first applied to the linear subsystem which makes the states synchronize exponentially to zero. Once this first linear subsystem has converged, the second cascade can be treated as a linear time-varying subsystem perturbed by a vanishing term from its cascade. A dynamic state and output feedback is constructed to achieve synchronization of the rest of the states. The proof of stability is given using a result from cascade systems. Since time delay appears in many interconnection networks and particularly in cooperative control, its effect on the stability of the closed-loop system is analyzed using Razumikhim theorem. It is shown that the established cooperative controller work well even in the presence of time delay. Numerical simulations are performed on models of car-like mobile robots to show the effectiveness of the proposed cooperative state and output-feedback controllers.