Fast distributed consensus with Chebyshev polynomials

Global observation of the environment is a key component in sensor networks and multi-robot systems. Distributed consensus algorithms make all the nodes in the network to achieve a common perception by local interactions between direct neighbors. The convergence rate of these algorithms depends on the network connectivity, which is related to the second largest eigenvalue of the weighted adjacency matrix of the communication graph. When the connectivity is small, a large number of communication rounds is required to achieve the consensus. In this paper we present a new distributed consensus algorithm which uses the properties of Chebyshev polynomials to significantly increase the convergence rate. The algorithm is expressed in the form of a linear iteration and, at each step, the nodes only require to transmit their current state to their neighbors. The difference with respect to previous approaches is that our algorithm is based on a second order difference equation. We provide the analytical expression of the convergence rate and we study in which conditions it is faster than computing the powers of the weighted matrix. This improvement reduces the number of messages between nodes, saving both power and time to the networked system. We evaluate our algorithm in a simulated environment showing the benefits of our approach.