Distance-based control of cycle-free persistent formations

In this paper, we study distance-based control of cycle-free persistent formations of single-integrator modeled agents in the plane. First, we propose a sequential control law consisting of algebraic calculations and primitive motions for agent groups having cycle-free persistent formations. Second, we prove the local asymptotic stability of cycle-free persistent formations under the well-known gradient law, which can be interpreted as a simultaneous version of the proposed sequential law, based on input-to-state stability. Furthermore, we show that if the leader-agent of a cycle-free persistent formation moves sufficiently slowly, then the formation of the group remains in the neighborhood of the desired formation.