Noisy localization on the sphere: Planar approximation

In real localization systems, noise from hardware and the environment makes it impossible for any algorithms to precisely localize objects, e.g. sensors and targets. Besides that, planar approximation is another source of error when dealing with localization over spherical surfaces, e.g. the surface of the earth, though it is neglected in many algorithms. This work deals with evaluating the error arising from planar approximation in localization problems over spherical surfaces. We characterize the error as arising for two different though related causes, and accordingly introduce concepts of radial error and angular error to account for these. A localization algorithm, based upon a Cayley-Menger Determinantal condition introduced recently for localization problems, is utilized for the analysis, and analytical results are confirmed through a number of simulations. As an end result of the study, we characterize the regions over which a planar approximation will be satisfactory, given an upper bound on the acceptable error it introduces in comparison with treating localization as a task in three-dimensional space.