Distributed Finite-Time Cooperative Localization for Three-Dimensional Sensor Networks

This paper addresses the distributed localization problem for a network of sensors placed in a three-dimensional space, in which sensors are able to perform range measurements, i.e., measure the relative distance between them, and exchange information on a network structure. While most existing studies primarily develop localization algorithms under the assumption that the entire sensor network is localizable, the problem of determining whether a sensor is localizable has received limited attention. However, neglecting this preliminary step can significantly hamper the accuracy of localization algorithms due to error propagation from unlocalizable sensors in iterative localization procedures. To address this research gap, we focus on two key challenges: i) deriving rigorous theoretical results and developing algorithms to verify sensor localizability, and ii)designing an efficient distributed localization algorithm that utilizes these localizability results. Specifically, we start by deriving a necessary and sufficient condition for sensor localizability using barycentric coordinates. Then, building on this theoretical result, we design a distributed localizability verification algorithm, in which we propose and employ a novel distributed finite-time algorithm for sum consensus. Finally, we develop a distributed localization algorithm based on conjugate gradient method and derive theoretical guarantees on its performance, ensuring finite-time convergence. The efficiency of our algorithm compared to the existing ones from the literature and its capability to handle scenarios with moderate levels of noise in the measurements are further demonstrated through numerical simulations.