Convergence Analysis on Iterative Algorithm in Ultra-Wideband Positioning Under Ill-Conditioned Configuration

Since the functional model of UWB positioning is nonlinear, iterative algorithms are often considered for solving the localization problem. With a rough initial value, we can obtain the optimal solution by the way of continuous iteration. However, in the UWB indoor positioning, the positioning system is prone to become ill-posed. It resulting in iterative algorithm cannot easily converge to a global optimal solution. In this paper, the convergence on iterative algorithm is analyzed. First of all, the nonlinear least-squares solution of distance equations in UWB positioning is given, then, four optimization iterative methods are presented. Finally, the four methods are applied to UWB static and dynamic positioning under ill-conditioned positioning configuration, the convergence property of the four methods are compared. For the iteration, three types of initial values are selected, and the three cases represent bad, general and good initial value respectively. Experimental results are given to demonstrate that although the barycenter method can converge correctly, it is inefficient with too more iterations. In addition, with a good initial value, the Gauss-Newton method can converge effectively, its iterations increase a lot with a general initial value, and this method rarely converges successfully, and sometimes converges to a false local optimization solution when selecting a bad initial value. Moreover, both the regularized Gauss-Newton method and closed-form Newton method work and converge to the global optimum effectively under the three types of initial values, and the closed-form Newton method has fewer iterations. The study shows that the closed-form of Newton method has higher efficiency of convergence than the other methods in Ultra-wideband positioning under ill-conditioned configuration.