Limited bene﬿t of cooperation in distributed relative localization

Important applications in robotic and sensor networks require distributed algorithms to solve the so-called relative localization problem: a node-indexed vector has to be reconstructed from measurements of differences between neighbor nodes. In a recent note, we have studied the estimation error of a popular gradient descent algorithm showing that the mean square error has a minimum at a finite time, after which the performance worsens. This paper proposes a suitable modification of this algorithm incorporating more realistic a priori information on the position. The new algorithm presents a performance monotonically decreasing to the optimal one. Furthermore, we show that the optimal performance is approximated, up to a $1 + \epsilon$ factor, within a time which is independent of the graph and of the number of nodes. This bounded convergence time is closely related to the minimum exhibited by the previous algorithm and both facts lead to the following conclusion: in the presence of noisy data, cooperation is only useful till a certain limit.