Improved Kalman Filter Based Accurate Pseudo-Range Positioning Method for BeiDou

In response to the issue of poor performance in traditional Beidou pseudorange positioning, this paper proposes a Beidou pseudorange precise positioning method that combines weighted least squares and adaptive Kalman filtering based on moving window covariance estimation. This method utilizes the fast convergence speed of the weighted least squares and the high accuracy of Kalman filtering. On this basis, the Kalman filter is modified with a moving window to ensure the accuracy and global convergence of the positioning process. Through experimental simulation and comparison, the effectiveness of this algorithm is demonstrated, showing its ability to improve positioning accuracy and meet certain positioning requirements.


Introduction
With the rapid development of the Beidou Satellite Navigation System (BDS) positioning technology, its extensive application across various fields has increased.The demand for positioning accuracy is no longer just about obtaining approximate location information, but rather achieving more precise positioning coordinates, high-timeliness positioning information, and a broader range of positioning coverage.The receivers for Beidou pseudorange positioning are simple and have low environmental requirements, making them highly practical in real-world applications.Therefore, realizing a precise Beidou pseudorange positioning method has great practical value and significance.
Global Navigation Satellite System (GNSS) pseudorange single point positioning is widely used due to its simple algorithm, low cost, and flexibility.Literature [1] proposed a multi-mode GNSS iterative combined positioning method based on a priori elevation angle weighting model to obtain precise target locations; Literature [2] compared the pseudorange single point positioning accuracy of different methods under various conditions, showing that the Helmert variance component estimation model based on elevation angle yields the best results.These methods have improved the accuracy of Beidou pseudorange positioning to some extent, but they are easily affected by external environmental interference and have limited effectiveness.Literature [3] improved the pseudorange observations by combining them with Hatch filtering and selecting appropriate sliding windows for filtering effects, but due to the inherent limitations of traditional Hatch filtering, the improvement in accuracy was limited.Literature [4] combined parameter optimal estimation with Hatch filtering to mitigate the

Beidou pseudorange single-point positioning principle
In satellite positioning, the ranging code is affected by the synchronization error between the satellite clock and the receiving end, which leads to the signal delay phenomenon when the receiving end receives the signal.Pseudorange single-point positioning is to calculate the specific coordinates of the receiving end through the pseudorange between multiple satellites and the receiving end, and its principle is shown in Figure 1.
In Equation (1), represents the geometric distance between the receiver and satellite j; , , denotes the three-dimensional coordinates of satellite j in the Earth-Fixed Coordinate System (ECEF); stands for the pseudorange observation; and c signifies the speed of light.
Among them, 、 、 、 is a known quantity, and errors like 、 、 、 、 need to be corrected to improve positioning accuracy.The result after correction becomes a known quantity.The rearranged formula is: In equation ( 2), the 、、、 are all unknown quantities to be solved, therefore, at least four satellite coordinates are required for the solution.

Weighted Least Squares (WLS) Calculation
In pseudorange positioning calculations, the most commonly used method is the least squares method.It employs iterative cycles of matrix calculations to determine state increments, thus obtaining optimized values.However, due to its heteroscedasticity issue, the ultimate positioning accuracy is subpar.The Weighted Least Squares (WLS) method effectively addresses the heteroscedasticity issue and is characterized by its insensitivity to initial positions and rapid convergence.Therefore, this paper utilizes WLS to determine the receiver's location.The computation process of Weighted Least Squares is detailed in literature [7], which will not be elaborated upon here.
Using WLS to calculate the receiver coordinates and the receiver clock bias, the position information is updated.After determining whether the position has converged, the positioning solution for the next epoch is computed.Once all epoch calculations are completed, the coordinates and clock bias obtained will be used as initial values for time updates and measurement updates in the next steps.

Adaptive moving window Kalman filtering
Standard Kalman Filter (KF) calculations are performed when the state and observation noises (Q and R respectively) are known, overlooking situations in real-world positioning where the noise statistical characteristics are uncertain due to environmental factors.To address this, this paper introduces an adaptive Kalman filtering method based on moving window covariance estimation.This method uses prediction residuals from the previous few epochs to estimate the noise covariance matrix, and a convergence factor is incorporated to prevent filter divergence caused by inaccurate residual information.The modification of its covariance is as follows: Let the observation new interest vector be: The estimate of the observation covariance matrix R for the k-th epoch is: In the above equation, P V k represents the covariance matrix of the innovation vector, and m stands for the window length, i.e., the observation innovation information from the previous m epochs.Let the state residual vector be: Therefore, we obtain the state covariance array of the first k state covariance array for the first calendar element.
In the equation, P ΔX k denotes the covariance matrix of the residual vector, and n signifies the window length, i.e., the state residual information from the previous n epochs.
To prevent divergences in the filter caused by anomalies or disturbances in the system, an adaptive factor k is introduced: In the above formula, P V k is the predicted residual covariance matrix, and N represents the number of predicted residual vectors from all k epochs prior.When the model is free from anomalies, the adaptive factor is set to 1. Substituting Eq. ( 7) into Eq.( 4) yields the new interest covariance array after adaptive factor adjustment: This paper enhances the positioning accuracy and shortens convergence time by utilizing an improved Kalman filtering algorithm.Initially, the receiver's position is determined using the Weighted Least Squares method, which is then taken as the initial state position for the Kalman filter.Subsequently, positioning calculations are performed by integrating the adaptive moving window algorithm

Improved pseudorange localization solution process
Error information from Beidou can be acquired from RENIX files and navigation messages, with correction formulas derived from error sources and their mitigation.Building upon the minimization of errors during satellite signal propagation, this paper chiefly focuses on designing algorithms for the pseudorange single-point positioning calculation process, aiming to enhance positioning accuracy.The specific solution steps for pseudorange single-point localization are: (1) Use the RINEX file from the satellite to list the pseudorange equations and set initial filter values.(2) Apply the Weighted Least Squares method to compute the initial position of the receiver until convergence.If not converged, return to step (2) for continued calculations.(3) Use the position and clock bias information derived from the Weighted Least Squares calculations as the initial state values for the Kalman filter, followed by the initialization of the covariance matrix.(4) Establish the Kalman filter system equations, perform adaptive moving window estimates on the covariance matrices for the state and observation vectors, and then proceed with the prediction and update processes.( 5) Determine if all epochs have been calculated; if not, return to step (4) for further computations.

Error Comparison Under Different Calculation Schemes
To validate the positioning efficacy of the method proposed in this paper, Beidou data in RINEX format is processed using MATLAB.Given an observation file containing 2880 epochs of data, this paper employs five different methods: Weighted Least Squares, Kalman Filtering, Optimized Adaptive Kalman Filtering, a combined approach of Weighted Least Squares and Kalman Filtering (WLS-KF), and WLS-AKF to compute coordinates along the X, Y, and Z axes.Initial data for the experiment is presented in Table 1.
Table 1.Experimental initial value setting.

Curation metadata 2880
Periodic Pulse Reverse (T/ms) 30 Matrix R Unit matrix Matrix Q Unit matrix

Window m 5 Window n
In this paper, the above five methods are solved for the received position respectively, and the results are shown in Table 2.By comparing the five experimental schemes through Table 2, the average positioning errors in different directions and the number of epochs at which convergence occurs provide a more intuitive reflection of the advantages and disadvantages of the algorithms.As can be observed from the table, the average error of the method proposed in this paper has been significantly reduced compared to the first three methods.The positioning accuracy has improved by 50% compared to the WLS+KF method, and the convergence speed has been reduced from 100 epochs in the AKF method to 20 epochs.Overall, the WLS+AKF method shows the best positioning performance.

Static Positioning Comparison
Acquiring the spatial position coordinates of the receiver end under different algorithm models, the comparison results of the above five schemes are shown in Table 3. From Table 3, it is evident that the positioning accuracy of WLS is the lowest, while the WLS-AKF method proposed in this paper has the highest accuracy.The experimental results confirm that the combination of WLS and the adaptive moving window KF can significantly enhance positioning precision.

Dynamic Positioning Comparison
Assuming the receiver starts from a stationary position, the stationary coordinates are transformed to Cartesian coordinates (100,200).During movement, noise is minimal, and it's approximated that the receiver moves horizontally at a constant speed of 2m/s and vertically at a constant speed of 20m/s.This process is simulated.
In the dynamic positioning comparison experiment, only the combinations of Weighted Least Squares with standard Kalman filtering (WLS-KF) and windowed adaptive Kalman filtering (WLS-AKF) are compared.The results are presented in Table 4. From Table 4, after WLS+KF filtering, the average positioning error in the horizontal coordinate is 2.26m, and in the vertical coordinate, it's 2.69m.After WLS+AKF filtering, the average positioning error in the horizontal coordinate is 1.42m, and in the vertical coordinate, it's 1.36m.It can be seen that the method proposed in this paper yields better dynamic positioning results.

Conclusion
This article mainly studies the Beidou pseudorange single point positioning algorithm and proposes a combination of the weighted least squares method and adaptive Kalman filtering based on moving window covariance estimation.The positioning results obtained by the weighted least squares method are used as the initial values for subsequent algorithm improvements.Then, the improved Kalman filtering is used for positioning calculation, which solves the problems of large positioning error when using the weighted least squares method alone and slow convergence process when using Kalman filtering, ensuring the accuracy and global convergence of the positioning process.Finally, the experimental results show that the convergence speed and positioning accuracy of this algorithm have been significantly improved.Lastly, from the static and dynamic positioning results, it can be seen that this algorithm has strong applicability in determining initial positions in high-precision positioning.

Figure 1 .
Figure 1.Pseudo-distance single point positioning schematic.Pseudorange single-point localization in satellite j The observation equation at the i moment of time the observation equations are as follows: ρ j = r j + ct r − ct sj + ct ionj + ct itoj + ct mpj r j = x sj − x 2 + y sj − y 2 + z sj − z 2

Table 2 .
Comparison of experimental schemes.

Table 3 .
Comparison of spatial positioning errors.