Calibration-free TDOA self-localisation

We present an approach for the localisation 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 synchronised, 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 nonlinear optimisation problem of TDOA using a physical spring-mass simulation. We present a geometrical representation of the error function, which is modelled by physical springs. By iterative relaxation of the springs, the error function is minimised. The approach is tested in numerous simulations, whereby our algorithm shows a smaller tendency to get stuck in local minima than a nonlinear 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 an RC car with a precision in the range of few centimetres.