Distributed Formation Control Of Multi-agent Systems Based On Complex Laplacian

In nature, some animals live in groups. For example, some insects such as ants and bees, live in "usocial" groups, in which different members carry out specialized jobs to help maintain the colony. These animals realize efficient and ordered group behaviors based on local interaction. Inspired from natural phenomena, researchers except to accomplish difficult tasks with multiple simple and low-cost agents. Formation control, as one of the fundamental cooperations in multi-agent systems, has experienced a rapid growing since1990s. It has a broad range of applications such as search and rescue operations in hazardous environments, ocean data retrieval and sam-pling, and surveillance/combat tasks, in which autonomous underwater, ground or aerial vehicles have to collaborate together in a formation to perform coordinated tasks.This paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape based on fixed topology and dynamic topology.For the case of fixed network topology in a group of agents, we solve the following two problems for both undirected topology and directed topology.1. Under what kind of topology can a formation shape be realized?2. How to stabilize a group of agents to a formation shape with distributed control based on local relative positions? Concerning the first question, we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex Laplacian satisfying certain rank condition and that a formation shape can be realized al-most surely if and only if the graph modeling the inter-agent specification of the formation shape is2-rooted, a new type of connectivity in graph theory. Concerning the second question, a dis-tributed and linear control law is developed based on the complex Laplacian specifying the target formation shape, and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed-loop system at desired locations are given to ensure not only globally asymptotic stability but also other performance specifications such as robustness and fast convergence rate. For the case of dynamic topology in a group of agents, we study a more general setup with directed topology. By assuming that each robot can only access local information of relative posi-tions of its neighbors, the graph modeling the sensing information flow among the robots is thus directed and may change over time, which lead to a challenging situation for formation control. Under the premise that the topology switches over the family of2-rooted graphs, dwell time and average dwell time conditions are derived to ensure global convergence of the robots to any for-mation shape in the presence of time-varying topologies. This is the first time that a solution is derived for formation control problems under dynamic topology.Both simulations based on Matlab and experiments based on the robots ROVIOs are provided to validate the theoretic analysis.