Calibration-Free Localization using Time Differences of Arrival

The continuous rise of mobile technology in everyday life has led to an increasing demand for location-based services for handheld applications and autonomous systems. In recent time, a shift of interest towards indoor localization can be observed where global navigation satellite systems, such as the Global Positioning System, are mostly not available. The hardware for positioning and navigation based on sound or ultrasound is affordable and precise in the range of centimeters. Localization based on the time differences of arrival (TDoA) relies on the propagation time of such a signal. In conventional hyperbolic TDoA localization the reference positions, i.e. the positions of either receivers or senders of a signal, are known a priori, and the position of a target is subject to locate. In calibration-free TDoA localization, reference positions are calculated during the process of locating the target – or just the references are calculated from unknown signals, eliminating the agonizing need to determine the positions of references by hand. In this thesis, we propose several novel approaches to the domain of calibration-free TDoA localization. Addressing the problem from several points of view, we propose algorithms in four fields – the far-field assumption, local optimization, branch-andbound algorithms, and probabilistic state estimation. The assumption that signals originate from remote locations, the so-called far-field assumption, simplifies the equation system and allows for fast and robust closed-form algorithms. We propose the Ellipsoid TDoA method, which relies on the fact that TDoA measurements from three receivers in the plane form an ellipse. This ellipse is robustly determined by regression, and the distances and angles between the receivers are calculated from the parameters of the ellipse. The Ellipsoid TDoA method is the first algorithm that solves the minimum problem of the TDoA far-field assumption, requiring no synchronization between receivers. We demonstrate the robustness of the algorithm in simulation and in experiments, where we show that the Ellipsoid method is still reliable when the far-field assumption is violated to some extent. Far-field algorithms are limited in generality, especially when measurements are rare. We consider the field of non-linear local optimization where we set focus to the failure cases. Due to the high dimensionality of the calibration-free TDoA problem, iterative optimization tends to not find the global optimum, which is the only acceptable solution. We propose the Cone Alignment algorithm, an iterative mass-spring simulation where signal and receiver positions are represented by physical particles which gather momentum to overcome local minima. In numerical simulations we compare the algorithm to standard optimization approaches, where we demonstrate the superiority of Cone Alignment in finding the solution. To address this intrinsic problem of local optimization, also denoted as the problem of completeness, we propose a polynomial time branch-and-bound algorithm that is a proof to enumerate all solutions of calibration-free TDoA up to an error bound . The algorithm is based on subdivision of a five-dimensional search space into subspaces, by