Trajectory tracking of leader–follower formations characterized by constant line-of-sight angles

A group of identical unicycles is controlled by means of local feedback laws that require measurements of the unicycles relative positions and speeds. Vehicle interconnections are considered unilateral and are modeled by means of a directed acyclic graph. Although not needed for the implementation of the controllers, the desired trajectory of each vehicle is derived from the requirement that the formation must rotate with the leader while ensuring that the relative positions and line-of-sight angles between unicycles are time-invariant. Exponential convergence of the actual trajectories to a ball centered on the desired trajectories is obtained by computing the Jacobian of the nonlinear dynamics and using results from contraction theory. Instrumental to this derivation is the subsystem feedback decomposition interpretation of the plant model. In this context, convergence depends on the uniform negative definiteness and strict positive realness of the forward and feedback subsystems. Such properties are obtained provided a set of linear and bilinear matrix inequalities as well as kinematic constraints are satisfied. A numerical example illustrates the convergence property of a leader–follower formation.