Dynamic formation control over directed networks using graphical Laplacian approach

This paper investigates the dynamic formation control problem for multi-agent systems over directed networks, in which a desired spatial shape is time-varying instead of fixed one as usually assumed in the literature. Inspired by the fact that the existing approaches, including absolute positions based, relative positions based, inter-agent distances based, and inter-agent bearings based, for specifying a formation shape are not invariant under all three transformations: translations, rotations, and scalings, a novel specification for formation shapes is proposed that is invariant under translations, rotations, and scalings in two- and three-dimensional spaces, and thus more intrinsic. In doing so, a new notion, called matrix-valued Laplacian, for graphs is introduced in detail along with some useful properties. It is demonstrated that the matrix-valued Laplacian provides much flexibility. Subsequently, two controllers are designed for guaranteeing the achievement of a dynamic formation shape. It is proved that arbitrary dynamic geometric shape can be reached by the designed controllers. Finally, two numerical examples are provided for demonstrating the effectiveness of the theoretical results