Improved Time of Arrival measurement model for non-convex optimization

The quadratic system provided by the Time of Arrival technique can be solved analytically or by nonlinear least squares minimization. An important problem in quadratic optimization is the possible convergence to a local minimum, instead of the global minimum. This problem does not occur for Global Navigation Satellite Systems (GNSS), due to the known satellite positions. In applications with unknown positions of the reference stations, such as indoor localization with self-calibration, local minima are an important issue. This article presents an approach showing how this risk can be significantly reduced. The main idea of our approach is to transform the local minimum to a saddle point by increasing the number of dimensions. In addition to numerical tests, we analytically prove the theorem and the criteria that no other local minima exist for nontrivial constellations.