Feasibility of Gauss-Newton method for indoor positioning

This paper addresses the feasibility of using iterative least-squares (ILS) methods for indoor positioning, where coordinates are to be determined based on estimated/measured ranges. Special attention is given to the Gauss-Newton method, since it is well suited for solving (small residual) non-linear problems and most commonly applied in positioning. Dealing with non-linear inverse problems, the parameter estimation possibly features local minima next to the sought for global one, and the position estimator is inherently biased. In satellite positioning systems, e.g. GPS, or other large scale systems, an initial guess that leads the iterative method to a global convergence is not hard to obtain, and the bias due to non-linearity is negligible. However, in indoor systems, both effects can become problematic, and therefore extra care needs to be taken to properly apply the ILS methods. Two schemes are proposed in this paper, one to obtain a good initial guess and the other to test the significance of the bias due to non-linearity. The validation of the proposed schemes is supported by simulations as well as by experimental results obtained with an UWB acoustic positioning system, which achieves centimeter level positioning accuracy with LoS propagation and full bandwidth between 3.6 and 12.1 kHz (with equal wavelength compared to the radio signals with the bandwidth between 3.1 to 10.6 GHz allowed for UWB radio communications). The proposed schemes are validated with a number of system setups, differing in the positions of transmitters, the position of the receiver, redundancy, the bandwidth used and therefore the range measurement error.