A robust GNSS positioning method based on kernel density estimation in non-line-of-sight transmission environment

Abstract

A robust GNSS positioning method based on kernel density estimation in non-line-of-sight (NLS) transmission environments is disclosed. The method includes: calculating pseudorange observations of visible satellites; calculating the pseudorange error vector using the least squares method; estimating the probability density function and its derivative of the pseudorange error vector using the kernel density function; estimating the cost function of the robust navigation solution using the probability density function and its derivative; correcting the estimated values ​​of receiver coordinate changes using the cost function of the robust navigation solution to obtain an updated navigation and positioning solution; and determining convergence based on the receiver's navigation and positioning solution. This invention addresses the significant positioning errors caused by single-satellite NLS transmission. When NLS transmission occurs at higher elevation angles, this method outperforms the weighted least squares method based on carrier-to-noise ratio and elevation angle. Furthermore, this invention effectively suppresses positioning errors caused by multiple satellites in NLS transmission scenarios.

Description

Technical Field

[0001] This invention belongs to the field of satellite navigation technology, and specifically relates to a robust GNSS positioning method based on kernel density estimation in a non-line-of-sight transmission environment. Background Technology

[0002] Global Navigation Satellite System (GNSS) provides high-precision, real-time location information to users worldwide, and is therefore widely used in both military and civilian fields. Pseudorange positioning is the most common positioning method. GNSS receivers measure the pseudorange between themselves and satellites, and use this pseudorange to further calculate the receiver's position information. Common navigation calculation methods include the least squares (LS) method and the Kalman filter method. GNSS receivers can obtain high-precision positioning solutions under good operating conditions. However, in complex environments such as cities and canyons, the line-of-sight (LOS) transmission path between the satellite and the user receiver is blocked or reflected, inevitably resulting in non-line-of-sight (NLOS) transmission. NLOS causes deviations in pseudorange measurements, which in turn severely affects the receiver's positioning solution.

[0003] Numerous research findings have been made on reducing the impact of NLOS and improving positioning accuracy in complex environments. Among these, the weighted least squares positioning solution method is a widely studied and applied approach. Its core idea is to weight the role of pseudorange observations in the positioning solution based on their quality, thereby reducing the influence of NLOS-affected observations. Commonly used weighting models include those based on satellite elevation angle, carrier-to-noise ratio (C / N0), and a combined C / N0 and satellite elevation angle weighting model. While weighted least squares positioning methods can mitigate positioning errors caused by NLOS to some extent, the aforementioned signal quality weights do not fully reflect the actual situation of the received signal; for example, low elevation angles and low C / N0 signals do not necessarily correspond to NLOS signals. Summary of the Invention

[0004] To address the aforementioned problems, the present invention aims to provide a robust GNSS positioning method based on kernel density estimation in non-line-of-sight transmission environments, which can reduce positioning errors caused by non-line-of-sight satellite signal transmission and improve the accuracy of satellite navigation and positioning.

[0005] To achieve the above objectives, the GNSS robust positioning method based on kernel density estimation in non-line-of-sight environments provided by the present invention includes the following steps performed in sequence:

[0006] 1) Calculate the pseudorange observations of all visible satellites after correction at the current positioning time;

[0007] 2) Based on the pseudorange observations mentioned above, calculate the pseudorange error vector at the current positioning time in the kth iteration using the least squares method;

[0008] 3) Use the kernel density function to estimate the probability density function and its derivative of the pseudorange error vector in the kth iteration above;

[0009] 4) The cost function of robust navigation solution in non-line-of-sight transmission environment is estimated by using the probability density function and its derivative of the pseudorange error vector of the kth iteration.

[0010] 5) Use the estimated cost function to correct the receiver coordinate change calculated in the k-th iteration to obtain the updated navigation and positioning solution;

[0011] 6) Determine the convergence of the updated navigation and positioning solution. If the result is yes, directly output the receiver position coordinates and receiver clock error at the current positioning time. Otherwise, return to step 2) and continue to perform the (k+1)th iteration solution.

[0012] In step 1), the method for calculating the corrected pseudorange observations of all visible satellites at the current positioning time is as follows:

[0013] First, use the signal reception time t. u (t) and signal transmission time t s Multiplying the difference (t-τ) by the speed of light yields the initial pseudorange observation:

[0014]

[0015] Where n represents the nth visible satellite; c represents the speed of light; and t represents the signal reception time. u (t) represents the receiver clock difference δt between GNSS time t and the GNSS time lead. u The sum of (t) t u (t)=t+δt u (t); signal transmission time t s (t-τ) represents the satellite clock difference δt between the GNSS time t-τ and the time ahead of the GNSS time. s The sum of (t-τ) t s (t-τ)=t-τ+δt s (t-τ); τ represents the actual propagation time of the GNSS signal from the visible satellite to the receiver, including the propagation time required for the signal to cross the geometric distance between the visible satellite and the receiver, as well as the ionospheric delay I. (n) (t) and tropospheric delay T (n) (t);

[0016] Substituting the receiver clock bias, satellite clock bias, geometric distance, ionospheric delay, and tropospheric delay into equation (1), the initial pseudorange observation expression for the nth visible satellite is:

[0017]

[0018] Where r(t-τ, t) represents the geometric distance between the position of the visible satellite at time t-τ and the position of the receiver at time t. This represents the total error not directly reflected in equation (2); satellite clock bias Ionospheric delay I (n) (t) and tropospheric delay T (n) (t) can be obtained from ephemeris information to correct the initial pseudorange observation. The expression for the corrected pseudorange observation can be simplified as follows:

[0019]

[0020] The receiver clock bias δt at this time u Let r be the length quantity, representing the geometric distance r between the nth visible satellite and the receiver. (n) Represented as:

[0021]

[0022] Where x = [x, y, z] T The x-coordinate represents the receiver's position coordinates. (n) =[x (n) y (n) , z (n) ] T This represents the position coordinates of the nth visible satellite.

[0023] In step 2), the method for calculating the pseudorange error vector at the current positioning time in the k-th iteration using the least squares method based on the aforementioned pseudorange observations is as follows:

[0024] First, a system of linear equations for pseudorange positioning is constructed. GNSS positioning actually involves solving the following nonlinear equations:

[0025]

[0026] Each equation in equation (5) corresponds to a corrected pseudorange observation of a visible satellite, where the receiver position coordinates are [x, y, z]. T and receiver clock difference δt u The unknown to be solved is N, which is the number of visible satellites involved in the positioning; the nonlinear positioning equations of equation (5) are applied to [x k-1 ,δt u，k-1 ] TPerforming a first-order Taylor expansion at this point yields the linearized matrix equation shown below:

[0027] G[Δx, Δy, Az, Δδt] u ] T =b (6)

[0028] Where, [Δx, Δy, Δz] T Δδt represents the change in the receiver's position coordinates between two adjacent observation times. u The geometric matrix G represents the change in receiver clock bias as follows:

[0029]

[0030] in, Let r represent the geometric distance r of the nth visible satellite. (n) The values ​​of the first-order partial derivatives of the function with respect to x, y, and z in the (k-1)th iteration; each component in the actual pseudorange residual vector b represents the corrected pseudorange observation. Subtract the geometric distance r between the receiver and the corresponding visible satellite calculated in the (k-1)th iteration. (n) (x l-1 ) and receiver clock difference Δδt u，k-1 sum:

[0031]

[0032] Then, by using the least squares method to solve equation (6), the least squares solution is expressed as:

[0033]

[0034] in, To obtain the estimated value of the change in receiver position coordinates, The estimated value of the receiver clock bias variation is given; then, the estimated pseudorange residual vector is obtained based on the estimated least squares solution.

[0035]

[0036] Finally, the pseudorange error vector is determined using the actual pseudorange residual vector and the estimated pseudorange residual vector:

[0037]

[0038] In step 3), the method for estimating the probability density function and its derivative of the pseudorange error vector of the k-th iteration using the kernel density function is as follows:

[0039] First, determine the kernel function used for kernel density estimation. A commonly used kernel function is the Gaussian impulse kernel function, whose expression is:

[0040]

[0041] The mathematical expression for kernel density estimation is:

[0042]

[0043] Among them, V j Let f(v) be the known j-th sample value; A be the number of known samples; K be the selected Gaussian kernel function; h be the bandwidth of the kernel function, i.e., the standard deviation of the Gaussian kernel function; f(v) be the calculated probability density function; and v be the independent variable when calculating the probability density function.

[0044] The pseudorange error vector estimated based on the kernel density estimation principle The probability density function is expressed as:

[0045]

[0046] in, Let be the i-th pseudorange error value in the pseudorange error vector; let the bandwidth of the kernel function be set to . in This represents the median of N pseudorange error values. Let ε be the probability density function of the estimated pseudorange error vector, and let ε be the pseudorange error independent variable used to calculate the probability density function. Further, by differentiating the probability density function of the pseudorange error vector, we obtain its derivative function.

[0047] In step 4), the method for estimating the cost function of robust navigation solution in non-line-of-sight transmission environment using the probability density function and its derivative of the pseudorange error vector of the k-th iteration is as follows:

[0048] First, determine the loss function for robust navigation calculation in non-line-of-sight transmission environments:

[0049]

[0050] Given the probability density function of the pseudorange error vector At that time, the robust navigation solution for receiver coordinate changes and receiver clock error changes can be obtained by minimizing the loss function of pseudorange error:

[0051]

[0052] in, This represents the observation error of the nth visible satellite; Representing vectors The dimension of G; (nm)Let G represent the element in the nth row and mth column of the geometric matrix G; by differentiating equation (15), we can obtain:

[0053]

[0054] in, The cost function for robust navigation is calculated from the pseudorange error vector estimated in step 3). probability density function and its derivative To determine:

[0055]

[0056] In step 5), the method for correcting the receiver coordinate change calculated in the k-th iteration using the estimated cost function to obtain the updated navigation and positioning solution is as follows:

[0057] When non-line-of-sight transmission exists, the cost function calculated using the robust navigation method described above is first applied. Corrected least squares solution for receiver position coordinate change:

[0058]

[0059] in, This represents the estimated change in the corrected receiver position coordinates. The correction factor μ is defined as follows: Cost function The first derivative of ;

[0060] Then, based on the change in receiver position coordinates, the updated navigation and positioning solution for the kth iteration is obtained:

[0061]

[0062] Where, x′ k =[x′ k y′ k , z′ k ] T δt represents the receiver position coordinates calculated in the k-th iteration after the update. u，k This represents the receiver clock error calculated in the k-th iteration; before the first iteration, the initial receiver position coordinates at the current positioning time are usually set to x0 = [x0, y0, z0]. T and the initial value of receiver clock error δt u，0 Set to 0.

[0063] In step 6), the convergence of the updated navigation and positioning solution is determined. If the result is yes, the receiver position coordinates and receiver clock error at the current positioning time are directly output; otherwise, the process returns to step 2) to continue the (k+1)th iteration.

[0064] The change in receiver position coordinates calculated in this iteration was detected. length If the length of the change in position coordinates is less than a preset threshold, it can be considered that the positioning accuracy requirement is met, and the next iteration will not be performed. The receiver position coordinates and receiver clock error at the current positioning time will be directly output; otherwise, return to step 2) to continue the k+1th iteration calculation.

[0065] The robust localization method based on kernel density estimation in non-line-of-sight transmission environments provided by this invention has the following advantages:

[0066] 1) When a single satellite has non-line-of-sight transmission, causing a large error in GNSS positioning, the performance of the method of this invention is better than the weighted least squares method based on carrier-to-noise ratio and elevation angle when there is non-line-of-sight transmission at a high elevation angle.

[0067] 2) When there is a large error in GNSS positioning caused by non-line-of-sight transmission of multiple satellites, the method of the present invention can also effectively suppress the positioning error caused by non-line-of-sight transmission. Attached Figure Description

[0068] Figure 1 The flowchart of the GNSS robust positioning method based on kernel density estimation in non-line-of-sight environments provided by the present invention is shown.

[0069] Figure 2 This is a geometric distribution map of visible satellites.

[0070] Figure 3 Positioning errors in the ENU direction, which do not exist in the Northeast-Northeast coordinate system, are not present.

[0071] Figure 4 The positioning error in the ENU direction of satellite 30 under the Northeast Sky Coordinate System is due to non-line-of-sight transmission.

[0072] Figure 5 (a) Comparison of errors in the E direction under the following conditions: no non-line-of-sight transmission, non-line-of-sight transmission of satellite 30, weighted by carrier-to-noise ratio and elevation angle, and robust method based on kernel density estimation.

[0073] Figure 5 (b) Comparison of errors in the N-direction positioning method when there is no non-line-of-sight transmission, when there is non-line-of-sight transmission from satellite 30, and when the positioning method is based on carrier-to-noise ratio and elevation angle weighting and kernel density estimation robust method.

[0074] Figure 5 (c) Comparison of errors in the U-direction positioning method when there is no non-line-of-sight transmission, when there is non-line-of-sight transmission from satellite 30, and when the positioning method is based on carrier-to-noise ratio and elevation angle weighting and kernel density estimation robust method.

[0075] Figure 6 The positioning error in the ENU direction of Satellite 5 under the Northeast Sky Coordinate System is due to non-line-of-sight transmission.

[0076] Figure 7 (a) Comparison of errors in the E direction under the following conditions: no non-line-of-sight transmission, non-line-of-sight transmission of satellite 5, and robust methods based on carrier-to-noise ratio and elevation angle weighting, and based on kernel density estimation.

[0077] Figure 7 (b) Comparison of errors in the N-direction positioning method when there is no non-line-of-sight transmission, when there is non-line-of-sight transmission from satellite 5, when the method is based on carrier-to-noise ratio and elevation angle weighting, and when the method is based on kernel density estimation.

[0078] Figure 7 (c) Comparison of errors in the U-direction positioning method when there is no non-line-of-sight transmission, when there is non-line-of-sight transmission from satellite 5, when the method is based on carrier-to-noise ratio and elevation angle weighting, and when the method is based on kernel density estimation.

[0079] Figure 8 The positioning error in the ENU direction exists for satellites 30 and 5 under the Northeast Sky Coordinate System when transmitting signals at non-line-of-sight.

[0080] Figure 9 (a) Comparison of errors in the E direction for positioning in the absence of non-line-of-sight transmission, the presence of non-line-of-sight transmission for satellites 30 and 5, and the robust method based on carrier-to-noise ratio and elevation angle weighting and kernel density estimation.

[0081] Figure 9 (b) Comparison of errors in the N-direction positioning method when there is no non-line-of-sight transmission, when there is non-line-of-sight transmission for satellites 30 and 5, and when the method is based on carrier-to-noise ratio and elevation angle weighting and kernel density estimation.

[0082] Figure 9 (c) Comparison of errors in the U-direction positioning method when there is no non-line-of-sight transmission, when there is non-line-of-sight transmission for satellites 30 and 5, and when the method is based on carrier-to-noise ratio and elevation angle weighting and kernel density estimation. Detailed Implementation

[0083] The robust GNSS positioning method based on kernel density estimation in non-line-of-sight environments provided by the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0084] like Figure 1As shown, the GNSS robust positioning method based on kernel density estimation in non-line-of-sight environments provided by this invention includes the following steps performed in sequence:

[0085] 1) Calculate the pseudorange observations of all visible satellites after correction at the current positioning time;

[0086] To calculate the pseudorange observations corrected for all visible satellites at the current positioning time, we first use the signal reception time t. u (t) and signal transmission time t s Multiplying the difference (t-τ) by the speed of light yields the initial pseudorange observation:

[0087]

[0088] Where n represents the nth visible satellite; c represents the speed of light; and t represents the signal reception time. u (t) represents the receiver clock difference δt between GNSS time t and the GNSS time lead. u The sum of (t) t u (t)=t+δt u (t); signal transmission time t s (t-τ) represents the satellite clock difference δt between the GNSS time t-τ and the time ahead of the GNSS time. s The sum of (t-τ) t s (t-τ)=t-τ+δt s (t-τ); τ represents the actual propagation time of the GNSS signal from the visible satellite to the receiver, including the propagation time required for the signal to cross the geometric distance between the visible satellite and the receiver, as well as the ionospheric delay I. (n) (t) and tropospheric delay T (n) (t).

[0089] Substituting the receiver clock bias, satellite clock bias, geometric distance, ionospheric delay, and tropospheric delay into equation (1), the initial pseudorange observation expression for the nth visible satellite is:

[0090]

[0091] Where r(t-τ, t) represents the geometric distance between the position of the visible satellite at time t-τ and the position of the receiver at time t. This represents the total error not directly reflected in equation (2); satellite clock bias Ionospheric delay I (n) (t) and tropospheric delay T (n) (t) can be obtained from ephemeris information to correct the initial pseudorange observation. The expression for the corrected pseudorange observation can be simplified as follows:

[0092]

[0093] The receiver clock bias δt at this time u Let r be the length quantity, representing the geometric distance r between the nth visible satellite and the receiver. (n) Represented as:

[0094]

[0095] Where x = [x, y, z] T The x-coordinate represents the receiver's position coordinates. (n) =[x (n) y (n) , z (n) ] T This represents the position coordinates of the nth visible satellite.

[0096] 2) Based on the pseudorange observations mentioned above, calculate the pseudorange error vector at the current positioning time in the kth iteration using the least squares method;

[0097] To calculate the pseudorange error vector at the current positioning time in the k-th iteration, we first construct a system of linear equations for pseudorange positioning. GNSS positioning actually involves solving the following nonlinear system of equations:

[0098]

[0099] Each equation in equation (5) corresponds to a corrected pseudorange observation of a visible satellite, where the receiver position coordinates are [x, y, z]. T and receiver clock difference δt u The unknown to be solved is N, which is the number of visible satellites involved in the positioning; the nonlinear positioning equations of equation (5) are applied to [x k-1 ,δt u，k-1 ] T Performing a first-order Taylor expansion at this point yields the linearized matrix equation shown below:

[0100] G[Δx, Δy, Δz, Δδt] u ] T =b (6)

[0101] Where, [Δx, Δy, Δz] T Δδt represents the change in the receiver's position coordinates between two adjacent observation times. u The geometric matrix G represents the change in receiver clock bias as follows:

[0102]

[0103] in, Let r represent the geometric distance r of the nth visible satellite. (n)The values ​​of the first-order partial derivatives of the function with respect to x, y, and z in the (k-1)th iteration; each component in the actual pseudorange residual vector b represents the corrected pseudorange observation. Subtract the geometric distance r between the receiver and the corresponding visible satellite calculated in the (k-1)th iteration. (n) (x l-1 ) and receiver clock difference Δδt u，k-1 sum:

[0104]

[0105] Then, by using the least squares method to solve equation (6), the least squares solution is expressed as:

[0106]

[0107] in, To obtain the estimated value of the change in receiver position coordinates, The estimated value of the receiver clock bias variation is given; then, the estimated pseudorange residual vector is obtained based on the estimated least squares solution.

[0108]

[0109] Finally, the pseudorange error vector is determined using the actual pseudorange residual vector and the estimated pseudorange residual vector:

[0110]

[0111] 3) Use the kernel density function to estimate the probability density function and its derivative of the pseudorange error vector in the kth iteration above;

[0112] To calculate the probability density function and its derivative of the pseudorange error vector, we first determine the kernel function used for kernel density estimation. A commonly used kernel function is the Gaussian impulse kernel function, expressed as:

[0113]

[0114] Kernel density estimation is a commonly used nonparametric estimation method for calculating probability density functions. Its mathematical expression is:

[0115]

[0116] Among them, V jLet be the j-th known sample value; A be the number of known samples; K be the selected Gaussian kernel function; h be the bandwidth of the kernel function, i.e., the standard deviation of the Gaussian kernel function; f(v) be the calculated probability density function; and v be the independent variable used in calculating the probability density function. The idea of ​​kernel density estimation originates from histogram-based methods. Its basic principle is: place a Gaussian kernel function with bandwidth h centered on each known sample value, and sum the values ​​of all kernel functions to obtain the overall probability density function.

[0117] The pseudorange error vector estimated based on the kernel density estimation principle described above. The probability density function is expressed as:

[0118]

[0119] in, Let be the i-th pseudorange error value in the pseudorange error vector; let the bandwidth of the kernel function be set to . in This represents the median of N pseudorange error values. Let ε be the probability density function of the estimated pseudorange error vector, and let ε be the pseudorange error independent variable used to calculate the probability density function. Further, by differentiating the probability density function of the pseudorange error vector, we obtain its derivative function.

[0120] 4) The cost function of robust navigation solution in non-line-of-sight transmission environment is estimated by using the probability density function and its derivative of the pseudorange error vector of the kth iteration.

[0121] To obtain the cost function for robust navigation calculation, we first determine the loss function for robust navigation calculation in a non-line-of-sight transmission environment:

[0122]

[0123] Given the probability density function of the pseudorange error vector At that time, the robust navigation solution for receiver coordinate changes and receiver clock error changes can be obtained by minimizing the loss function of pseudorange error:

[0124]

[0125] in, This represents the observation error of the nth visible satellite; Representing vectors The dimension of G; (nm) Let G represent the element in the nth row and mth column of the geometric matrix G; by differentiating equation (15), we can obtain:

[0126]

[0127] in, The cost function for robust navigation is calculated from the pseudorange error vector estimated in step 3). probability density function and its derivative To determine:

[0128]

[0129] 5) Use the estimated cost function to correct the receiver coordinate change calculated in the k-th iteration to obtain the updated navigation and positioning solution;

[0130] To obtain the updated navigation and positioning solution, the cost function of the robust navigation solution described above is first used when non-line-of-sight transmission is present. Corrected least squares solution for receiver position coordinate change:

[0131]

[0132] in, This represents the estimated change in the corrected receiver position coordinates. The correction factor μ is defined as follows: Cost function The first derivative of ;

[0133] Then, based on the change in receiver position coordinates, the updated navigation and positioning solution for the kth iteration is obtained:

[0134]

[0135] Where, x′ k =[x′ k y′ k , z′ k ] T δt represents the receiver position coordinates calculated in the k-th iteration after the update. u，k This represents the receiver clock error calculated in the k-th iteration; before the first iteration, the initial receiver position coordinates at the current positioning time are usually set to x0 = [x0, y0, z0]. T and the initial value of receiver clock error δt u，0 Set to 0.

[0136] 6) Determine the convergence of the updated navigation and positioning solution. If the result is yes, directly output the receiver position coordinates and receiver clock error at the current positioning time. Otherwise, return to step 2) and continue to perform the (k+1)th iteration solution.

[0137] The change in receiver position coordinates calculated in this iteration was detected. length If the length of the change in position coordinates is less than a preset threshold, it can be considered that the positioning accuracy requirement is met, and the next iteration will not be performed. The receiver position coordinates and receiver clock error at the current positioning time will be directly output; otherwise, return to step 2) to continue the k+1th iteration calculation.

[0138] The effects of this invention can be further illustrated by the following simulation results.

[0139] Simulation Experiment Description: In the simulation experiment, a 40-second GPS L1 signal containing signals from 8 satellites was simulated. Weighted least squares positioning based on carrier-to-noise ratio and satellite elevation angle, and robust positioning based on kernel density estimation were used under different conditions: low-elevation satellites, high-elevation satellites, and two satellites with NLOS transmission. The positioning errors of the two methods were compared. Signal parameter settings are shown in Table 1.

[0140] Table 1 Signal Parameters

[0141]

[0142]

[0143] Figure 2 This represents the geometric distribution of multiple visible satellites during the experimental simulation.

[0144] Figure 3 There is no positioning error in the non-line-of-sight transmission ENU direction under the Northeast-Northeast coordinate system;

[0145] The horizontal axis represents signal time, and the vertical axis represents the positioning error in each direction in the ENU coordinate system. The figure shows that when there is no non-line-of-sight transmission, the directional errors of traditional least-squares positioning are relatively small.

[0146] Figure 4 The positioning error in the ENU direction of satellite 30 under the Northeast Sky Coordinate System is due to non-line-of-sight transmission.

[0147] The low-elevation satellite 30 experienced a 50m increase in pseudorange observations compared to LOS transmission during the 18-24s period due to NLOS transmission. The ENU direction error between the positioning result and the receiver's true position during this time is as follows: Figure 4 As shown, the positioning error in all directions increases significantly during NLOS transmission.

[0148] Figure 5 The positioning errors of satellite 30 include non-line-of-sight transmission, weighted by carrier-to-noise ratio and elevation angle, and the positioning error based on the method of this invention.

[0149] in Figure 5 (a) Figure 5 (b) and Figure 5 (c) The positioning errors are in the E, N, and U directions, respectively. As can be seen from the figure, both the least squares positioning method based on the combined weighting of carrier-to-noise ratio and elevation angle and the method of this invention suppress the influence of NLOS transmission, and the suppression effect of this invention is better than that of the least squares positioning method based on the combined weighting of carrier-to-noise ratio and elevation angle.

[0150] Figure 6 The positioning error in the ENU direction of Satellite 5 under the Northeast Sky Coordinate System is due to non-line-of-sight transmission.

[0151] Assuming that the high-elevation satellite #5 experiences anomalies in pseudorange observations due to NLOS transmission between the 18th and 24th seconds of signal transmission, resulting in a 50m increase in pseudorange compared to LOS transmission, the ENU direction error between the positioning result and the receiver's true position at this time is as follows: Figure 6 As shown, the positioning error in all directions increases significantly during NLOS transmission. Figure 6 and Figure 4 The comparison shows that when satellites at different positions and elevation angles have pseudorange anomalies of the same size, the final positioning errors in each direction are significantly different. When the same pseudorange anomaly is generated due to NLOS transmission, the satellite with a larger elevation angle will result in a larger positioning error.

[0152] Figure 7 The positioning errors of the No. 5 satellite include non-line-of-sight transmission, weighted by carrier-to-noise ratio and elevation angle, and the positioning method of this invention.

[0153] in Figure 7 (a) Figure 7 (b) and Figure 7 (c) Positioning errors in the E, N, and U directions, respectively. As shown in the figure, the semi-parametric robust positioning method based on kernel density estimation effectively suppresses the impact of NLOS transmission, while the least squares method based on combined carrier-to-noise ratio (CNR) and elevation angle weighting has a poorer suppression effect. This is because the least squares positioning method based on CNR and elevation angle weighting assigns larger weights to satellites with high elevation angles and high CNR, while assigning smaller weights to satellites with low elevation angles and low CNR. Therefore, when a satellite with a high elevation angle is transmitting NLOS, even if the CNR of that satellite is relatively reduced, its role in positioning is still significant, ultimately leading to a relatively poor effect in suppressing NLOS. The method of this invention effectively improves the NLOS transmission error.

[0154] Figure 8 The positioning error in the ENU direction exists for satellites 30 and 5 under the Northeast Sky Coordinate System when transmitting signals at non-line-of-sight.

[0155] Satellite 5, with a higher elevation angle, and satellite 30, with a lower elevation angle, experienced pseudorange anomalies in their signal measurements during the 18th to 24th seconds due to NLOS transmission. Compared to LOS transmission, these anomalies increased by 50m and 80m respectively. The error between the positioning result and the receiver's true position at this time is as follows: Figure 8 As shown, it can be seen that the positioning error in all directions increases significantly during NLOS transmission.

[0156] Figure 9 The positioning error is due to the presence of non-line-of-sight transmission between two satellites, weighted by carrier-to-noise ratio and elevation angle, and the method of this invention.

[0157] in Figure 9 (a) Figure 9 (b) and Figure 9 (c) Positioning errors in the E, N, and U directions, respectively. As can be seen from the figure, the method of this invention effectively suppresses the impact of NLOS transmission, while the least squares method based on the combined weighting of carrier-to-noise ratio and elevation angle has a relatively poor effect on NLOS suppression.

Claims

1. A robust GNSS positioning method based on kernel density estimation in non-line-of-sight environments, characterized in that: The GNSS robust positioning method includes the following steps performed in sequence: 1) Calculate the pseudorange observations of all visible satellites after correction at the current positioning time; 2) Based on the pseudorange observations mentioned above, the current positioning time is calculated using the least squares method at the [time range]. The pseudorange error vector of the next iteration; 3) Use kernel density function to estimate the above-mentioned... The probability density function and its derivative of the pseudorange error vector in the next iteration; 4) Using the above-mentioned... The probability density function and its derivative of the pseudorange error vector in the next iteration estimate the cost function of robust navigation solution in non-line-of-sight transmission environment. The method is: First, determine the loss function for robust navigation calculation in non-line-of-sight transmission environments: （15）； Given the probability density function of the pseudorange error vector At that time, the robust navigation solution for receiver coordinate changes and receiver clock bias changes is obtained by minimizing the loss function of pseudorange error: （16）； in, ; Indicates the first The observational errors of a single visible satellite; Representing vectors dimensionality; Representing geometric matrices No. OK The elements of the column; obtained by differentiating equation (15): （17）； in, The cost function for robust navigation is calculated from the pseudorange error vector estimated in step 3). probability density function and its derivative To determine: （18）； 5) Correct the first cost function using the estimated cost function described above. The receiver coordinate changes calculated in each iteration yield the updated navigation and positioning solution; The method is: When non-line-of-sight transmission exists, the cost function calculated using the robust navigation method described above is first applied. Corrected least squares solution for receiver position coordinate change: （19）； in, This represents the corrected estimate of the change in receiver position coordinates, with the correction factor as the factor. Defined as , Cost function The first derivative of ; Then, based on the change in receiver position coordinates, the updated navigation and positioning solution for the kth iteration is obtained: （20）； in, Indicates the updated number The receiver position coordinates calculated in the next iteration Indicates the first The receiver clock error is calculated in the next iteration; before the first iteration, the initial values ​​of the receiver position coordinates at the current positioning time are usually set. Initial value of receiver clock difference Set to 0; 6) Determine the convergence of the updated navigation and positioning solution. If the result is yes, directly output the receiver position coordinates and receiver clock error at the current positioning time; otherwise, return to step 2) and continue to the next step. The solution is obtained in the next iteration.

2. The GNSS robust positioning method based on kernel density estimation in non-line-of-sight environments according to claim 1, characterized in that: In step 1), the method for calculating the corrected pseudorange observations of all visible satellites at the current positioning time is as follows: First, use the signal reception time. With signal transmission time Multiplying the difference by the speed of light yields the initial pseudorange observation: （1）； in, Indicates the first One visible satellite; Represents the speed of light; signal reception time Indicates GNSS time Receiver clock bias leading GNSS time sum Signal transmission time Indicates GNSS time Satellite clock bias ahead of GNSS time sum ; This represents the actual propagation time of a GNSS signal from a visible satellite to a receiver, including the propagation time required for the signal to cross the geometric distance between the visible satellite and the receiver, as well as the ionospheric delay. and tropospheric delay ; Substituting the receiver clock bias, satellite clock bias, geometric distance, ionospheric delay, and tropospheric delay into equation (1), the first... The initial pseudorange observation expression for the visible satellite is: （2）； in, Indicates time Visible satellite positions and times The geometric distance between the receiver positions, This represents the total error not directly reflected in equation (2); satellite clock bias Ionospheric delay and tropospheric delay The initial pseudorange observation is obtained from ephemeris information. The expression for the corrected pseudorange observation is abbreviated as: （3）； Receiver clock bias at this time For length, the first Geometric distance between visible satellites and receiver Represented as: （4）； in, Indicates the receiver's position coordinates. Indicates the first The position coordinates of the visible satellite.

3. The GNSS robust positioning method based on kernel density estimation in non-line-of-sight environments according to claim 1, characterized in that: In step 2), the current positioning time is calculated using the least squares method based on the aforementioned pseudorange observations. The method for obtaining the pseudorange error vector in the next iteration is: First, a system of linear equations for pseudorange positioning is constructed. GNSS positioning actually involves solving the following nonlinear equations: （5）； Each equation in equation (5) corresponds to a corrected pseudorange observation of a visible satellite, where the receiver position coordinates are... and receiver clock difference For the unknown quantity to be solved, The number of visible satellites participating in the positioning; the nonlinear positioning equations of equation (5) are then applied in... Performing a first-order Taylor expansion at this point yields the linearized matrix equation shown below: （6）； in, This represents the change in the receiver's position coordinates between two adjacent observation times. Geometric matrix representing receiver clock bias variation Represented as: （7）； in, They represent the first Geometric distance of visible satellites right The first-order partial derivative function at the th The value of the next iteration; the actual pseudo-range residual vector Each component in the equation represents the corrected pseudorange observation. Subtract the receiver and the corresponding visible satellite in The geometric distance calculated in the next iteration Clock difference with receiver sum: （8）； Then, by using the least squares method to solve equation (6), the least squares solution is expressed as: （9）； in, To obtain the estimated value of the change in receiver position coordinates, The estimated value of the receiver clock bias variation is given; then, the estimated pseudorange residual vector is obtained based on the estimated least squares solution. （10）； Finally, the pseudorange error vector is determined using the actual pseudorange residual vector and the estimated pseudorange residual vector: （11）。 4. The GNSS robust positioning method based on kernel density estimation in non-line-of-sight environments according to claim 1, characterized in that: In step 3), the above-mentioned first step is estimated using the kernel density function. The method for determining the probability density function and its derivative of the pseudorange error vector in each iteration is as follows: First, determine the kernel function used for kernel density estimation. A commonly used kernel function is the Gaussian impulse kernel function, whose expression is: （12）； The mathematical expression for kernel density estimation is: （13）； in, For the known first Each sample value; The number of known samples; The selected Gaussian kernel function; The bandwidth of the kernel function is the standard deviation of the Gaussian kernel function. The calculated probability density function, This is the independent variable used when calculating the probability density function; The pseudorange error vector estimated based on the kernel density estimation principle The probability density function is expressed as: （14）； in, The first in the pseudorange error vector One pseudorange error value; the bandwidth of the kernel function is set to... ,in , express The median of the pseudorange error values, Let be the probability density function of the estimated pseudorange error vector. Let the pseudorange error be the independent variable used to calculate the probability density function; further, differentiate the probability density function of the pseudorange error vector to obtain its derivative function. .

5. The robust GNSS positioning method based on kernel density estimation in non-line-of-sight environments according to claim 1, characterized in that: In step 6), the convergence of the updated navigation and positioning solution is determined. If the result is yes, the receiver position coordinates and receiver clock error at the current positioning time are directly output; otherwise, the process returns to step 2) to continue. The method for solving this iteration is: The change in receiver position coordinates calculated in this iteration was detected. length If the length of the change in position coordinates is less than a preset threshold, the positioning accuracy requirement is met, and the next iteration is not performed. The receiver position coordinates and receiver clock error at the current positioning time are directly output; otherwise, return to step 2) to continue the process. The solution is obtained in the next iteration.

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