Robust Tracking of a Mobile Beacon using Time Differences of Arrival with Simultaneous Calibration of Receiver Positions

Abstract-Localization based on time differences of arrival (TDOA) has turned out to be a promising approach when neither receiver positions nor the positions of signal origins are known a priori. In this paper, we consider calibration-free tracking of a mobile beacon using TDOA, i.e., the positions of the receivers are not given. We propose a probabilistic formulation using a particle filter to simultaneously localize the signal beacon and the receivers. Our method is robust against measurement outliers and incorrect initialization. This is achieved through a probabilistic sensor model for TDOA data which explicitly considers the measurement uncertainty and takes into account disproportional errors caused by measurement outliers. For the reliable initialization of the particle filter, we apply an iterative optimization approach to multiple subsets of TDOA data, where the best solution is implicitly selected by appropriate weighing of the sensor model. We verify the robustness of our approach in extensive experiments in a spacious indoor environment by an ultrasound beacon moving on various trajectories. We demonstrate that our approach ensures a proper initialization of the particle filter and provides accurate position estimates for the signal beacon and the receivers even in case of measurement outliers. Compared to position references of an optical motion capture system we achieve mean position errors below 5 centimeters

Polynomial Time Approximation Algorithms for Localization based on Unknown Signals

Abstract. We consider the problem of anchor-free self-calibration of receiver locations using only the reception time of signals produced at unknown locations and time points. In our settings the receivers are synchronized, so the time differences of arrival (TDOA) of the signals arriving at the receivers can be calculated. Given the set of distinguishable time points for all receivers the task is to determine the positions of the receivers as well as the signal sources. We present the first polynomial time approximation algorithms for the minimum problem in the plane, in which the number of receivers is four, respectively the number of signals is three. For this, we first consider the problem that the time points of m signals are jittered by at most some > 0. We provide an algorithm which tests whether n given receiver positions are feasible with respect to m unknown sender positions with a run-time of O(nm 2 ) and we provide an algorithm with run-time O(nm log m) which tests the feasibility of m given sender positions for n unknown sender positions. Using these tests, we can compute all possible receiver and signal source positions in time O √ 2/ 2n−3 n 2 m , respectively O √ 2/ 2m−3 nm log m

Self-Localization based on Ambient Signals

Abstract. We present an approach for the localization of passive nodes in a communication network using ambient radio or sound signals. In our settings the communication nodes have unknown positions. They are synchronized but do not emit signals for localization and exchange only the time points when environmental signals are received, the time differences of arrival (TDOA). The signals occur at unknown positions and times, but can be distinguished. Since no anchors are available, the goal is to determine the relative positions of all communication nodes and the environmental signals. The Ellipsoid TDOA method introduces a closed form solution assuming the signals originate from far distances. The TDOA characterize an ellipse from which the distances and angles between three network nodes can be inferred. The approach is tested in numerous simulations and in indoor and outdoor settings where the relative positions of mobile devices are determined utilizing only the sound produced by assistants with noisemakers.

Minimal Solvers for Unsynchronized TDOA Sensor Network Calibration

Calibration of network nodes using only time differences of arrival (TDOA) measurements opens up for interesting applications in wireless ad-hoc sensor networks, e.g. finding the positions of cell phones by only ambient sounds or radio. We present two novel approaches for the problem of self-calibration of network nodes using only TDOA when both receivers and transmitters are unsynchronized. We consider the previously unsolved minimum problem of far field localization in three dimensions, which is to locate four receivers by the signals of nine unknown transmitters, for which we assume that they originate from far away. The first approach, the Ellipsoid TDOA method, is a geometric representation based on the fact that the time differences between four receivers characterize an ellipsoid. We calculate by linear least-squares regression the ellipsoid from the observed measurements of nine or more transmitters, by which the constellation of receivers is characterized. In the second approach we propose using linear algebra techniques on the matrix of unsynchronized TDOA measurements, enabling us to solve a set of linear equations for a parametrization of the unknowns. This approach is extended to more than four receivers and nine transmitters in a straightforward manner. In extensive experiments we evaluate and compare both approaches and analyze specific failure modes of the algorithms. Here, we demonstrate that the algorithms are robust to moderate Gaussian measurement noise and that the far field assumption is reasonable if the distance between transmitters and receivers is at least four times the distance between the receivers. In an indoor experiment using sound we reconstruct the microphone positions up to a mean error of 5 cm