Calibration-free TDOA self-localisation

We present an approach for the localization of passive receiver nodes in a communication network. The only source of information is the time when environmental sound or ultrasound signals are received. The discrete signals occur at unknown positions and times, but they can be distinguished. The clocks of the receivers are synchronized, so the time differences of arrival (TDOA) of the signals can be computed. The goal is to determine the relative positions of all receiver nodes and implicitly the positions and times of the environmental signals. Our proposed approach, the Cone Alignment algorithm, solves iteratively a non-linear optimization problem of time differences of arrival by using a physical spring-mass simulation. We present a geometrical representation of the error function, which is modeled by physical springs. By iterative relaxation of the springs the error function is minimized. The approach is tested in numerous simulations, where our algorithm shows a smaller tendency to get stuck in local minima than a non-linear least-squares approach using gradient descent. In experiments in a real-world setting we demonstrate and evaluate a tracking system for a moving ultrasound beacon without the need to initially calibrate the positions of the receivers. Using our algorithm, we estimate the trajectory of a moving model train and of a RC car with a precision in the range of few centimeters.