Ultrawideband (UWB) is well-suited for indoor positioning due to its high resolution and good penetration through objects. The observation model of UWB positioning is nonlinear. As one of nonlinear filter algorithms, extended Kalman filter (EKF) is widely used to estimate the position. In practical applications, the dynamic estimation is subject to the outliers caused by gross errors. However, the EKF cannot resist the effect of gross errors. The innovation will become abnormally large and the performance and the reliability of the filter algorithm are inevitably influenced. In this study, a robust EKF (REKF) method accompanied by hypothesis test and robust estimation is proposed. To judge the validity of model, the global test based on Mahalanobis distance is implemented to assess whether the test statistical term exceeds the threshold for outlier detection. To reduce and eliminate the effects of the individual outlier, the robust estimation using scheme III of the Institute of Geodesy and Geophysics of China (IGGIII) based on local test of the normalized residual is performed. Meanwhile, three kinds of stochastic models for outliers are expressed by modeling the contaminated distributions. Furthermore, the simulation and measurement experiments are performed to verify the effectiveness and feasibility of the proposed REKF for resisting the outliers. Simulation experiment results are given to demonstrate that the outliers following all the three kinds of contaminated distributions can be detected. The proposed REKF can effectively control the influences of the outliers being treated as systematic errors and large variance random errors. When the outliers come from the thick-tailed distribution, the robust estimation does not play a role, and the REKF are equivalent to the EKF method. The measured experiment results show that the outliers will be generated in the nonline-of-sight environment whose impact is abnormally serious. The robust estimation can provide relatively reliable optimized residuals and control the influences of the outliers caused by gross errors. We can believe that the proposed REKF is effective to resist the effects of outliers and improves the positioning accuracy compared with least-squares (LS) and EKF method. Moreover, the adaptive filter and ranging error model should be considered to compensate the state model errors and ranging systematic errors respectively. Then, the measurement outliers will be detected more correctly, and the robust estimation will be used effectively.

Single-frequency GPS/BeiDou navigation satellite system (BDS) real-time kinematic (RTK) and inertial navigation system (INS) integration has wide range of application prospects due to the global deployment of GPS along with the rapid development of BDS. The instantaneous single-frequency ambiguity resolution will be significantly improved by the combined GPS/BDS and INS configuration. Owing to road conditions and an inertial measurement unit (IMU) on the carrier not being rigidly mounted, biased measurements in the IMU will occasionally emerge, leading to biased INS predictions. However, bias or inaccuracy from INS-predicted position can prevent the successful resolution of the whole set of ambiguities. This paper proposes the use of a positional polynomial fitting (PPF) constraint to compensate for the epochs with abnormal INS predictions. The aid from PPF is provided at two levels, i.e., at the ambiguity resolution (AR) level and at the solution level. In order to further increase the availability of ambiguity-fixed positioning solutions, a partial ambiguity resolution (PAR) strategy is introduced when full ambiguity resolution (FAR) fails. A field vehicular experiment was performed to show the validity of the proposed PPF-aided method by comparing different schemes regarding different INS-aided satellite system configurations, different AR strategies, and whether the PPF-aided method was adopted. The results show that the most attractive scheme is to combine the PAR with the PPF-aided dual-constellation and INS integration.

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.

Ultrawideband (UWB) is considered as a promising technology in indoor positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, iterative algorithms are often considered for solving the nonlinear problem. With a rough initial value, we can obtain the optimal solution 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. However, in the UWB indoor positioning, since the location information of the user is rather limited, it is not easy to get a good initial value for iteration. Besides, the positioning system is prone to become ill-posed. These factors make the Gauss–Newton method not easy to converge to a global optimal solution or even diverge, especially under ill-conditioned positioning configuration. Furthermore, the linearization of the positioning functional model results in biased least-squares estimation. The Gauss–Newton method only includes the first-order Taylor expansion of distance equations. The bias comes from neglected higher order terms. In this study, the closed-form Newton method which considers the high order partial derivatives is introduced and proposed. The simulation and measurement experiments are implemented to analyze the performance of the closed-form Newton method in UWB positioning. Both the initial value factor and geometric configuration factor are discussed. Experimental results are given to demonstrate that the closed-form Newton method can converge to the global optimal solution with better convergence and higher efficiency regardless of whether the initial iteration value is reasonable or not. Meanwhile, the closed-form Newton method can improve the accuracy of positioning results, particularly when the anchors are not uniformly and symmetrically distributed, or the ranging error is relatively large. The study shows that closed-form of the Newton method has better convergence and positioning performance than the Gauss–Newton method, especially under ill-conditioned positioning configuration or relatively low measurement precision, even though it adds a little computational cost.

Ultra-wideband is a promising technology in indoor positioning due to its accurate time resolution and good penetration. Since the positioning model is non-linear, iterative methods are often considered for solving the localisation problem. However, the positioning system is prone to become ill-posed. The iterative methods cannot easily converge to a global optimal solution. In this paper, the convergence property of four non-linear iterative methods is analytically reviewed under ill-conditioned configuration. For the iteration, three types of initial values are selected. Experimental results are given to demonstrate that although the barycentre method can converge correctly, it is inefficient with too many iterations. In addition, with a good initial value, the Gauss–Newton method can converge effectively, and it sometimes converges to a false local optimisation solution when selecting a bad initial value. Moreover, both the regularised Gauss–Newton method and closed-form Newton method can converge to the global optimum effectively with fewer iterations. This study shows that the closed-form Newton method has higher efficiency of convergence than the other methods. Meanwhile, to make complete use of measurements available to improve the accuracy, the result of non-iterative method is generally used as the initial value of the iterative method.

China is a country in which landslide disasters occur frequently, and slope movement has a direct influence on the landslide, making the analysis of slope movement and forecasting an important research direction. Currently, there is relatively limited research in the field of slope deformation analysis models. The research target is based on the reservoir slope, and the specific research targets are the Dingdong trough Zhengjiadagou slopes. From the perspective of hydrological surveying and mapping, a slope deformation analysis model is constructed. The displacement control factor is the main analysis object, while rainfall and reservoir water level are the main influencing factors. The internal structure of the slope is considered a characteristic factor affecting slope deformation and sliding. The results indicate that the displacement control factor of slope sliding under the influence of gravity is related to the variation of reservoir water level, with the reservoir water level being the main influencing factor. The effect of rainfall on slope deformation is not significant. Using the models of metabolic GM (1, 1), new information GM (1, 1), and GM (1, 1) to predict the deformation trend of the three slopes, of which the average error is 5.45%, 5.85%, and 7.48%. The results indicate that constructing a reservoir slope analysis model from the perspective of hydrological surveying provides a visual advantage in studying slope deformation issues using the model analysis approach, and the metabolism GM (1, 1) model in the slope deformation trend prediction accuracy is higher. This paper provides theoretical guidance and instance reference for the slope instance research.