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. First, we derive a necessary
and sufficient condition for node 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 we
derive a theoretical guarantee on its performance, which ensures finite-time
convergence to the exact position for all localizable nodes. The efficiency of
our algorithm compared to the existing ones from the state-of-the-art
literature is further demonstrated through numerical simulations.Comment: 39 pages, 7 figures, under revie