Unit dual quaternion directed graph and formation control

We first study the multi-agent formation control problem in a directed graph. The relative congurations are expressed by unit dual quaternions (UDQs). We call such a weighted directed graph a unit dual quaternion directed graph (UDQDG). We show that a desired relative conguration scheme is reasonable in a UDQDG if and only if for any cycle in this directed graph, the product of relative congurations of the forward arcs, and inverses of relative congurations of the backward arcs, is equal to 1. We then show that a desired relative conguration scheme in a directed connected graph is reasonable if and only if the dual quaternion Laplacian is similar to the unweighted Laplacian of the directed graph. Then for a reasonable desired relative conguration scheme, we build the relationship between the desired formation and the eigenvector corresponding to the zero eigenvalue. A numerical method and a control law are presented. We then study general weighted directed graphs (WDG). Ordinary graphs, gain graphs, signed directed graphs, complex weighted directed graphs and UDQDGs are special cases of WDGs. A general theory of WDG is presented.