K(n) Formation and Resizing

Abstract

In Chaps. 3, 4 and 5, we provided global stabilization, local stabilization, and stabilization of directed persistent formations under gradient-based control laws. However, as shown in Fig. 4.5, it is not possible to stabilize a general formation to the desired configuration under the gradient control laws. There have been some efforts to improve the performance of a formation system by adding a perturbation term into the gradient control law. But, as commented in Trinh et al. (Automatica 77:393–396, 2018 [10]), there is no clear clue for an improvement of a formation system even with perturbation terms. Hence, it is generally considered that the gradient control laws may be effective for local stabilizations. However, the gradient control laws still can be a good solution for certain circumstances. This chapter provides some advanced results in gradient control laws. As the first result, in Sect. 12.1 we provide a gradient control law enhanced by virtual variables for ensuring a global stabilization of general complete graphs in general dimensional space. However, note that since the control law proposed in Sect. 12.1 uses virtual variables that should be communicated between neighboring agents, it has a drawback in terms of computations and communications over the traditional gradient control laws. As the second results, in Sect. 12.2 we present a gradient control law for resizing the formation in the distance-based setup, and also solve a scaling problem under the bearing-based setup. Since a resizing of formation can be achieved by controlling the distance of an edge, it is convenient for reorganizing the positions of agents for some applications. For example, when a group of unmanned aerial vehicles needs to rescale the size of formation for surveillance or for gas detection, it can be done by simply changing the distance between the leader and the first follower (see Lemma 9.3).