Bias Analysis of Parameter Estimator Based on Gauss-Newton Method Applied to Ultra-Wideband Positioning

Abstract

Ultra-wideband (UWB) is considered as a promising technology in short-distance indoor wireless positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, the optimal solution is generally estimated by the way of continuous iteration. As an iterative descent method of high efficiency, the Gauss–Newton method is widely used to estimate the position. The nonlinear distance equations are linearized, and the solution can be found iteratively. Therefore, the nonlinear least-squares solution is generally biased even if the observations are normally distributed. In outdoor satellite positioning, the ranging distances are long enough so that the bias caused by nonlinearity is very small. However, in UWB positioning, the relative ranging error is large, and the positioning system is prone to become ill-posed, hence the bias due to nonlinearity is not negligible. In this study, both the statistical factor and geometric factor for bias in the nonlinear least-squares estimator of UWB positioning are discussed. In order to assess whether the linearized model is sufficiently approximate for the positioning estimation, a hypothesis test criterion based on Mahalanobis distance is proposed. The simulation and measurement experiments are performed to analyze the factors affecting the bias in UWB positioning. Experimental results are given to demonstrate that the linearization is valid and the bias in UWB positioning estimation can be neglected for the relatively high measurement precision. Moreover, for a positioning configuration, when the anchors are evenly distributed, the amount of nonlinearity is orthogonal to the ranging space of the design matrix, the UWB positioning estimation tends to be unbiased. Meanwhile, the hypothesis test based on Mahalanobis distance is carried out to determine the validity of the linearized model. When the bias is large for UWB positioning, the bias estimate can be used to correct the estimator to guarantee the unbiasedness for UWB positioning. Furthermore, the correction of parameter estimator bias is more effective in the case of relatively low measurement precision or ill-conditioned configuration.