Leader-Following Graph-Based Distributed Formation Control Jos´ e Rodrigues, Dario Figueira, Carlos Neves, Isabel Ribeiro Abstract— This paper presents the distributed formation control of a multi-agent system using graph theory with emphasis on consensus and cooperation issues. The focal point is to achieve and maintain a formation from any initial condition, with and without a leader that the entire formation must follow. Our analysis framework is based on tools from algebraic graph theory, matrix theory and control theory. We present a brief derivation of multi-agent consensus in continuous-time and the corresponding iterative form stated in discrete-time, because while the real scenario is continuous, the implementation that we simulate is discrete. Based on the discrete-time algorithm, we propose a solution to obtain and uphold consensus when there is a leader to command the entire network. Simulation results are presented, indicating the capabilities and limitations of the algorithms. I. I NTRODUCTION This paper presents consensus based algorithms for the coordination of a networked multi-agent system that aims at achieving and preserving a formation amongst themselves. A. Formation control Networked Systems have lately been the focus of scien- tific attention due to the boom in computation speed and reliable communications. This provided a solid base for the development of several applications like formation flight control [1], [2], satellite clustering [3], and the control of groups of unmanned vehicles [1], [4], [5]. Advantages of interconnected multi agent systems over conventional systems include reduced cost, increased effi- ciency, performance, reconfigurability, robustness, and new capabilities. A team of smaller robots to perform the same task of a larger single robot is at a distinct advantage in case of a malfunction. In one case the team of decentralized units will adapt to the loss of a team member and continue cooperating to accomplish the given task, on the other case the single robot is surely doomed as well as its given mission. Also, a space radar based on satellite clusters [6] is estimated to cost three times less than currently avail- able systems, increase geolocation accuracy by a factor of 500, offer two-orders-of-magnitude smaller propulsion requirement, and be able to track moving targets through formation flight. Undirected Graphs have been often picked to represent formations due to the instinctive way they describe the interconnection topology of a formation, e.g. in [6] and [7]. Moreover, directed graphs have been chosen to reflect the control structure [8], the constraint feasibility [9], the infor- mation flow [10], to quantify error propagation [11] and to reflect leader following inter-agents control specifications throughoutly scrutinized [12], [13], [14]. The authors are with the Institute for Systems and Robotics at Instituto Superior T´ ecnico (IST), Av. Rovisco Pais, 1049-001 Lisboa, Portugal. Contact author: Jos´ e Rodrigues. E-mail: jerasman@gmail.com. The problem of coordination in multi-agent systems can be characterized naturally by a finite representation of the configuration space, namely by using graph-theoretic models to describe the local interactions in the formation, where nodes symbolize the physical entities (agents) and the edges represent virtual entities that support the infor- mation flow between the nodes. B. Graph-based models to control a formation This paper is mainly based on the notable results that have arisen since 2001. The groundwork on stating and solving consensus problems in networked dynamic systems appeared in [15] and [16], results that were later used in [17] and [18]. The issue of reaching an agreement without computing any objective functions was initially addressed in [19] and later extended in [20], [21]. These main results, which have a well described summary in [22] by Olfati-Saber et al., are the base that supports the development in this work. The problem of reaching a formation based on graph theory was already solved by Fax and Murray [16]. This theory consists on given an arbitrary initial position make the agents reach a consensus on a final common point. Then, a bias value is introduced, adding the feature that the final positions of the agents will not be a common point but a formation given by a desired geometric topology. This framework, presented in Section II, consists in an introduction to the main problem discussed on this paper that consists on adding a leader to command the network and maintain the formation while performing the leader motion. In the context of this paper, a formation is defined by relative positions between vehicles in a network inter- connected by inter-vehicle communications. Multi-vehicle systems are an important category of networked systems due to their commercial and military applications. There are two broad approaches to handle distributed formation control: i) representation of formations as rigid struc- tures [7], [23] and ii) representation of formations using the vectors of relative positions of neighboring vehicles and the use of consensus-based controllers with input bias [22]. In this paper we discuss this latter approach. We explore graph-based models to control a desired formation, representing the interactions and the flow of information between the multiple agents in the graph. Graph and control theory support the formulation of the problem and help propose an elegant solution for the cases addressed in this paper. C. Paper Organization In Section II, we address the problem of reaching consensus in a distributed network. We present important theory results known from the literature. Section III solves the problem on reaching consensus in the presence of a Proc. Robotica'2008 978-972-96895-3-6 71