Cooperative localization revisited: error bound, scaling, and convergence

Cooperative localization, where sensors exchange the information with each other to determine their locations, has received considerable attention. In this work, we study the cooperative localization in order to investigate several fundamental properties that have not been well addressed so far. We formulate the cooperative localization in a general setting, where a relative or absolute location map is obtained, depending on the number of anchors. The (relative or absolute) location map is the output of an optimization problem, where the objective function is given as a norm of a space where a vector composed of distances between sensors is defined. We show that several error bounds and the estimation bias of the cooperative localization can be obtained by simple arguments (e.g. by using triangle inequality) without specifying the detail of the objective function. Next, we theoretically and numerically verify that the cooperative localization has a preferable scaling property such that the estimation becomes more accurate as sensors are more densely deployed. Finally, we consider the problem that the objective functions used in the cooperative localization are usually multimodal and have a number of local optima and saddle points. We show that the gradient descent algorithm starting from a random prior (initial estimates) often fails to find the optimal solution when the distance measurements between some pair of sensors are not available. We propose a new prior, called shortest-path-distance-based prior, which is very powerful for obtaining accurate estimates even when the distances between some sensor pairs are not measurable.