Kernel Methods for Ill-Posed Range-Based Localization Problems

A kernel regression technique for the ill-posed range-based localization problem is proposed. We introduce a generic kernel design method which relies on a probabilistic modeling of the respective physical environment in terms of a class of stochastic differential equations. The combination of dynamic modeling with physical and stochastic interpretability leads to a unique solution in a Reproducing Kernel Hilbert Space and thus eliminates the need for a time-discrete representation of the localization solution. By design of the problem formulation, an extended Kalman filter naturally evolves as an iterative optimization method. As a practical example, we tackle an ill-posed localization problem that stems from a preinstalled sensor network of unknown geometry, providing nonsynchronous range information to a moving beacon. To obtain a unique solution for the beacon positions and the network geometry, we apply the proposed kernel regression technique and kernel design method. The proposed approach eventually leads to a least squares optimization problem that can be interpreted as a maximum a posteriori estimation problem. While the methodology is not restricted to localization problems only, its validity is shown by a demonstrator which is comprised of a mobile robot and a network of multiple nonmoving beacons that provide time-of-arrival measurements of ultra wide band signals.