On Agreement Problems with Gossip Algorithms in absence of common reference frames Mauro Franceschelli and Andrea Gasparri Abstract— In this paper a novel approach to the problem of decentralized agreement toward a common point in space in a multi-agent system is proposed. Our method allows the agents to agree on the relative location of the network centroid respect to themselves, on a common reference frame and therefore on a common heading. Using this information a global positioning system for the agents using only local measurements can be achieved. In the proposed scenario, an agent is able to sense the distance between itself and its neighbors and the direction in which it sees its neighbors with respect to its local reference frame. Furthermore only point-to-point asynchronous communications between neighboring agents are allowed thus achieving robustness against random communication failures. The proposed algorithms can be thought as general tools to locally retrieve global information usually not available to the agents. I. I NTRODUCTION In the last decade networked multi-agent systems have drawn the attention of a large part of the control systems community. Most of the attention has been devoted to the de- velopment of decentralized motion coordination algorithms, for a representative example see [1], [2], [3], [4], [5]. In this framework, coordination algorithms have been developed making use of either absolute position information or relative distance measurements between agents [6], [7], [8], [9] to perform the most various tasks, for instance rendezvous [10], leader following [11], attitude control [12] and many others [13], [14], [15]. Many of these algorithms, dealing with decentralized motion coordination problems, assume that the agents have access to absolute position information (GPS) and thus have a common global reference frame that makes it easy to interpret the information passed by other agents. Even when in multi agent systems the agents are not supposed to know their absolute position, many times they are assumed to have a common attitude reference to exchange information that can be achieved by using a compass and gravity as common reference for their coordinate system. For space applications another technological solution is to use a frame of fixed stars to have a common reference. In all these instances several technological countermeasures have to be undertaken for the implementation of coordination algorithms increasing the total costs of the single agents. On the other hand, algo- rithms that use only relative distance measurements tend to A. Gasparri is with the Department of Computer Science and Automation, University of “Roma Tre”, Via Vasca Navale 79, Roma 00146, Italy e-mail: gasparri@dia.uniroma3.it . M. Franceschelli is with the Dept. of Electrical and Electronic Engineering, University of Cagliari, Piazza D’Armi, 09123 Cagliari, Italy e-mail: mauro.franceschelli@diee.unica.it . Authors are listed in alphabetical order. achieve only low complexity tasks due to extremely difficult coordination problems in absence of reference frames. We believe that having a common reference frame, or agreement on some common fixed points in space, greatly simplifies the necessary coordination algorithms and in- creases their effectiveness. Nevertheless, not relaying on external systems like GPS could significantly advance the technological feasibility of mobile swarms of agents, reduc- ing their dependence on the global positioning system in the low level control loops. In this paper a novel approach to the problem of de- centralized agreement toward a common point in space in a multi-agent system is proposed. The proposed method allows the agents to agree on the network centroid, on a common reference frame, on a common heading. Using this information a global positioning system for the agents using only local measurements can be achieved. Furthermore only point-to-point asynchronous communications between neigh- boring agents are allowed thus achieving robustness against random communication failures. The proposed algorithms can be thought as general tools to locally retrieve global information usually not available to the agents. In this way, any assumption on the absence of a common reference frame could be relaxed and therefore, simpler algorithms could be developed. II. BACKGROUND ON GOSSIP ALGORITHMS OVER NETWORKS Let the network of agents be described by a time-varying graph G(t)= {V,E(t)}, where V = {v i : i =1,...,n} is the set of nodes (agents) and E(t)= {e ij =(v i ,v j )} is the set of edges (connectivity) representing the point-to-point communication channel availability at time t. A position p i ∈ R d in the d th dimensional space is associated to each node v i ∈ V , with i =1,...,n. In particular, an edge representing a connection between two agents exists if and only if the distance between these agents is less then or equal to their sensing radius r, namely E(t)= {e ij : ‖p i (t) − p j (t)‖≤ k, i = j }, where ‖·‖ d is the Euclidean norm in R d . Therefore, a generic couple of agents {i, j } is able to sense ‖p i −p j ‖ reciprocally. In addition, each agent has a local reference frame defined by an orthonormal basis of vectors in R d fixed on it and, is able to determine the direction in which neighbors are sensed, strictly with respect to its own local reference frame.