A Low-Earth Orbit Satellite Doppler Positioning and Timing Method and System Based on Extended Kalman Filtering

Abstract

This application discloses a low-Earth orbit (LEO) satellite Doppler positioning and timing method and system based on extended Kalman filtering, involving satellite positioning and timing technology. The method includes: acquiring the LEO satellite position and velocity in the Earth-centered Earth-fixed (ECEF) coordinate system and the Doppler observations calculated by the receiver during the same period; establishing the Doppler observation equation; determining the initial receiver position using a combination of coarse and fine grids and the least squares method; initializing the state vector and covariance matrix, and setting the process noise matrix; based on the initialization results, cyclically interpolating the satellite position and velocity to determine the observation noise matrix and linearize the observation matrix, completing state updates and covariance updates; recording the estimated receiver position and clock drift, analyzing the position error and Doppler residuals, and evaluating the timing accuracy. This application is used to complete LEO satellite Doppler positioning error analysis and timing accuracy evaluation, improving the problem of poor accuracy in least squares Doppler positioning and timing, and enhancing the robustness and reliability of the results.

Description

Technical Field

[0001] This application relates to the field of satellite positioning and timing technology, and in particular to a low-Earth orbit Doppler positioning and timing method and system based on extended Kalman filtering. Background Technology

[0002] Low-Earth orbit (LEO) satellite Doppler positioning technology has undergone a development process from the rise of TRANSIT systems to its replacement by GNSS, and finally its revival by giant LEO constellations. Due to the massive scale, high-power signals, and doppler characteristics of modern LEO constellations, this technology has once again become a research hotspot. Although LEO satellite Doppler positioning faces challenges such as non-navigational signal design and weak spatiotemporal references, its strategic value in GNSS-denied environments is increasingly prominent through advanced signal processing algorithms and positioning strategies, and it is gradually becoming a key component of the next-generation PNT (Positioning and Navigation) system.

[0003] Currently, low-Earth orbit (LEO) satellite Doppler positioning methods often employ algorithm optimization or the addition of other auxiliary means. Patent No. CN119001788 B discloses a LEO satellite Doppler positioning method and device. This method constructs a damping term based on satellite ephemeris information to estimate the initial state and uses weighted filtering based on satellite velocity and line-of-sight angle to reduce convergence time and ensure positioning and velocity stability in dynamic environments. Patent No. CN 121142586 A discloses a positioning method based on LEO satellite opportunistic signal Doppler measurement. This method designs a joint estimation scheme combining a cubic Kalman smoother and an expectation-maximization algorithm. Under conditions of unknown noise statistical characteristics, it achieves accurate estimation of terminal position and Doppler frequency shift. While ensuring high accuracy and robustness, the computational overhead is more controllable, thus significantly improving positioning reliability in complex environments.

[0004] The least squares method is used to locate Doppler observations, which has disadvantages such as poor epoch correlation, susceptibility to outliers, and low positioning accuracy. During the least squares method for time synchronization, the independent estimation of each epoch results in no noise attenuation and large frequency estimation errors, leading to poor time synchronization accuracy. The extended Kalman filter is sensitive to the initial position accuracy of the receiver, and a large initial position error will cause the filter to fail to converge. Summary of the Invention

[0005] This application provides a low-orbit satellite Doppler positioning and timing method and system based on extended Kalman filtering, which completes positioning error analysis and timing accuracy evaluation, improves the problem of poor accuracy in least squares Doppler positioning and timing, and enhances the robustness and reliability of the results.

[0006] This application provides a low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering, including: Set the receiver to stand still, acquire the position and velocity of the low-orbit satellite in the geocentric-ground-fixed coordinate system ECEF, and the Doppler observation values ​​calculated by the receiver at the same time, and establish the Doppler observation equation; The initial position of the receiver is determined by using a combination of coarse and fine grids and the least squares method. Based on the established Doppler observation equation and the initial value of the receiver position, the state vector and covariance matrix are initialized, and the process noise matrix is ​​set. Based on the initialization results, satellite position and velocity interpolation is performed epoch-by-epoch to determine the observation noise matrix and linearize the observation matrix. Kalman gain is calculated to complete state update and covariance update. Record the estimated receiver position and clock drift, analyze the position error and Doppler residual, and evaluate the timing accuracy.

[0007] This application also provides a low-Earth orbit satellite Doppler positioning and timing system based on extended Kalman filtering, including a processor and a memory. The memory stores a computer program, and when the computer program is executed by the processor, it implements the steps of the low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described above.

[0008] This application uses satellite ephemeris files and Doppler observation files, employs a combination of coarse and fine grids with the least squares method based on clock drift residuals to determine the initial receiver position, and uses the extended Kalman filter method to perform position calculation and clock drift estimation epoch by epoch, outputting the receiver position and clock drift sequence to complete positioning error analysis and timing accuracy assessment. This improves the problem of poor accuracy in least squares Doppler positioning and timing, while also enhancing the robustness and reliability of the results.

[0009] The above description is only an overview of the technical solution of this application. In order to better understand the technical means of this application and to implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of this application more obvious and understandable, the following are specific embodiments of this application. Attached Figure Description

[0010] Various other advantages and benefits will become apparent to those skilled in the art upon reading the following detailed description of preferred embodiments. The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of this application. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings: Figure 1 This is a basic flowchart illustrating the low-orbit satellite Doppler positioning and timing method based on extended Kalman filtering, as described in an embodiment of this application. Detailed Implementation

[0011] Exemplary embodiments of the present disclosure will now be described in more detail with reference to the accompanying drawings. While exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the disclosure to those skilled in the art.

[0012] The purpose of this application is to leverage the advantages of low-Earth orbit (LEO) satellites, such as significant Doppler frequency shift and strong robustness of extended Kalman filtering. This application provides a LEO satellite Doppler positioning and timing method based on extended Kalman filtering, such as... Figure 1 As shown, it includes the following steps: In step S101, the receiver is set to stationary, and the position and velocity of the low-orbit satellite in the Earth-centered Earth-fixed coordinate system (ECEF) and the Doppler observation values ​​calculated by the receiver during the same period are acquired to establish the Doppler observation equation. In some embodiments, step S101 specifically includes: The satellite position under ECEF was calculated based on the low-orbit ephemeris file, and the satellite velocity was calculated using the discrete point center difference method. Assuming the receiver is stationary, the Doppler observation vectors of the visible n satellites at time k are: The Doppler observation equation for the i-th satellite can be expressed as: in: Let be the 4-dimensional state vector to be determined. The receiver position (m) is given. For receiver clock drift (m / s); For the first The carrier frequency (Hz) of each satellite; The speed of light (m / s); Let be the unit line-of-sight vector from the receiver to the satellite, where For the first The orbital position of the satellite, Location of the receiver; For the first The speed of the satellite (m / s); For measuring noise; The range of values ​​is .

[0013] In step S102, the initial value of the receiver position is determined using a combination of coarse and fine grids and the least squares method. In a specific example, firstly, the epoch at which the receiver can simultaneously observe no fewer than four satellites is found, and the corresponding satellite position coordinates are calculated; then, the longitude, latitude, and elevation ranges are determined, the coarse grid step size is set, and the sum of squares of clock drift residuals is calculated by traversing the coarse grid points to find the initial value; finally, based on the initial value obtained from the coarse grid points, the fine grid range and step size are determined, and the sum of squares of clock drift residuals is further calculated to determine the optimal initial value.

[0014] In step S103, the state vector and covariance matrix are initialized according to the established Doppler observation equation and the initial value of the receiver position, and the process noise matrix is ​​set. In step S104, based on the initialization results, satellite position and velocity interpolation is performed epoch-by-epoch to determine the observation noise matrix and linearize the observation matrix. Kalman gain is calculated to complete state update and covariance update. In step S105, the estimated receiver position and clock drift are recorded, the position error and Doppler residual are analyzed, and the timing accuracy is evaluated.

[0015] In some embodiments, determining the initial value of the receiver position using a combination of coarse and fine grids and the least squares method specifically includes: Before calculating the initial position of the receiver, determine the epoch in which at least four satellites can be observed simultaneously, and calculate the position coordinates of all satellites corresponding to the epoch. Set the coarse grid's longitude, latitude, step size, and elevation, and convert the calculated position coordinates into ECEF coordinates. In some specific examples, this step corresponds to determining the initial receiver position using the coarse grid least squares method; the specific coarse grid longitude range is... Step size is Latitude range is Step size is The elevation range is The step size is 1000m. The latitude, longitude, and elevation of the grid points are converted to ECEF coordinates. The Doppler observation equations are linearized, and the clock drift residual observation vector is set. The average value of the clock drift residual is The sum of squares of the residuals of the clock drift is ,in This represents the number of observable satellites in this epoch, and the grid point corresponding to the minimum value among all residual square sums is the initial value of the receiver position.

[0016] The grid point corresponding to the minimum value among all residual squares is used as the initial value of the receiver position.

[0017] In some embodiments, determining the initial value of the receiver position using a combination of coarse and fine grids and the least squares method further includes: Based on the grid points estimated by the coarse grid, deviation ranges for longitude, latitude, and elevation are set, and the optimal initial value for the receiver position is determined using the least squares method based on the fine grid. Specifically, this step corresponds to determining the initial value for the receiver position using the fine grid least squares method. In this embodiment, it is further assumed that the grid points estimated by the coarse grid are [ The fine grid longitude range is set to []. ], step size is The fine grid latitude range is set to [ ], step size is The elevation range is The step size is 100m. The least squares calculation process is the same for both coarse and fine grids, and the optimal initial value for the receiver position is finally determined, denoted as [ ].

[0018] In some embodiments, the initialization of the state vector and covariance matrix, and the setting of the process noise matrix, based on the established Doppler observation equation and the initial receiver position, include: The state vector to be determined is initialized to... ; covariance matrix P Initial settings are: Process noise matrix Q Represented as: In some of these examples, =0.01, representing the standard deviation of the position process, in m / ; =0.1, representing the standard deviation of the clock drift process, in m / ( ); This indicates the epoch interval, measured in seconds (s).

[0019] The prediction steps are as follows: in Indicates an epochal time.

[0020] In some embodiments, epoch-by-epoch interpolation of satellite position and velocity is performed to determine the observation noise matrix and linearize the observation matrix. The calculation of the Kalman gain to complete state update and covariance update includes: The satellite position and velocity are interpolated using the 9th-order Lagrange method to determine the satellite position and velocity at the target time. Observation noise matrix for: in ; Linearized observation matrix for: in: Calculating the Kalman gain completes the state update and covariance update: The Kalman gain matrix is: The state update matrix is: in: .

[0021] The covariance update matrix is:

[0022] In some embodiments, recording the estimated receiver position and clock drift, analyzing position errors and Doppler residuals, and evaluating timing accuracy includes: Based on the estimated receiver location and clock drift Statistical analysis of receiver position error; Define the estimated receiver position as The actual receiver location is The error components in each direction are The three-dimensional error is ; The estimated receiver position and clock drift are used to calculate the Doppler prediction value. This prediction is then summed by taking the absolute values ​​of the differences between the estimated Doppler position and the observed Doppler values, and the resulting Doppler residual is averaged. The average of the absolute values ​​of the Doppler residuals is:

[0023] In some embodiments, recording the estimated receiver position and clock drift, analyzing position errors and Doppler residuals, and evaluating timing accuracy further includes: Timing accuracy is assessed based on the estimated receiver clock drift: in The accumulated time error over k epochs, This represents the initial time error.

[0024] After removing the linear trend from the accumulated time error, it becomes: in The time error for removing the linear trend at epoch k is given. For time k epoch, , .

[0025] Compared to traditional initial value determination methods, the method in this application employs a combination of coarse and fine grids with the least squares method based on clock drift residuals to determine the optimal initial value for receiver position, resulting in higher accuracy and avoiding filter divergence caused by large initial value errors. This application also establishes a precise positioning and timing method based on extended Kalman filtering using Doppler observations. Compared to least squares Doppler positioning, this method offers higher positioning accuracy, smoother results, and stronger robustness.

[0026] The method in this application uses extended Kalman filtering to estimate clock drift, which can make full use of the advantages of time correlation to continuously estimate clock drift, improve noise suppression capability, smooth frequency error, and effectively improve time synchronization accuracy.

[0027] This application also proposes a low-Earth orbit satellite Doppler positioning and timing system based on extended Kalman filtering, including a processor and a memory. The memory stores a computer program, which, when executed by the processor, implements the steps of the aforementioned low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering.

[0028] It should be noted that, in the embodiments of this application, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element.

[0029] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0030] Through the above description of the embodiments, those skilled in the art can clearly understand that the methods of the above embodiments can be implemented by means of software plus necessary general-purpose hardware platforms. Of course, they can also be implemented by hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium (such as ROM / RAM, magnetic disk, optical disk) and includes several instructions to cause a terminal (which may be a mobile phone, computer, server, air conditioner, or network device, etc.) to execute the methods described in the various embodiments of this application.

[0031] The embodiments of this application have been described above with reference to the accompanying drawings. However, this application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims. All of these forms are within the protection scope of this application.

Claims

1. A low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering, characterized in that, include: Set the receiver to stand still, acquire the position and velocity of the low-orbit satellite in the geocentric-ground-fixed coordinate system ECEF, and the Doppler observation values ​​calculated by the receiver at the same time, and establish the Doppler observation equation; The initial position of the receiver is determined by using a combination of coarse and fine grids and the least squares method. Based on the established Doppler observation equation and the initial value of the receiver position, the state vector and covariance matrix are initialized, and the process noise matrix is ​​set. Based on the initialization results, satellite position and velocity interpolation is performed epoch-by-epoch to determine the observation noise matrix and linearize the observation matrix. Kalman gain is calculated to complete state update and covariance update. Record the estimated receiver position and clock drift, analyze the position error and Doppler residual, and evaluate the timing accuracy.

2. The low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 1, characterized in that, With the receiver stationary, the position and velocity of the low-orbit satellite in the ECEF coordinate system (Geocentric Earth-Fixed Equivalent System) are acquired, along with the Doppler observations calculated by the receiver during the same period. The Doppler observation equations are then established, including: The satellite position under ECEF was calculated based on the low-orbit ephemeris file, and the satellite velocity was calculated using the discrete point center difference method. Assuming the receiver is stationary, the Doppler observation vectors of the visible n satellites at time k are: The Doppler observation equation for the i-th satellite can be expressed as: in: Let be the state vector to be determined. For receiver location, For receiver clock drift; For the first The carrier frequency of each satellite; The speed of light; Let be the unit line-of-sight vector from the receiver to the satellite, where For the first The orbital position of the satellite, Location of the receiver; For the first The speed of a satellite; For measuring noise; The range of values ​​is .

3. The low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 2, characterized in that, The determination of the initial receiver position using a combination of coarse and fine grids and the least squares method specifically includes: Before calculating the initial position of the receiver, determine the epoch in which at least four satellites can be observed simultaneously, and calculate the position coordinates of all satellites corresponding to the epoch. Set the coarse grid longitude, latitude, step size, and elevation to convert the calculated location coordinates into ECEF coordinates; Linearize the Doppler observation equations and set the clock drift residual observation vector. The average value of the clock drift residual is The sum of squares of the residuals of the clock drift is , where n is the number of satellites observable in this epoch; The grid point corresponding to the minimum value among all residual squares is used as the initial value of the receiver position.

4. The low-orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 2, characterized in that, The determination of the initial receiver position using a combination of coarse and fine grids and the least squares method also includes: Based on the grid points estimated by the coarse grid, the deviation ranges of longitude, latitude, and elevation are set, and the optimal initial value of the receiver position is determined by the least squares method based on the fine grid.

5. The low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 3, characterized in that, Based on the established Doppler observation equations and the initial receiver position, the state vector and covariance matrix are initialized, and the process noise matrix is ​​set as follows: The state vector to be determined is initialized to... ; covariance matrix P Initial settings are: Process noise matrix Q Represented as: in, Indicates the standard deviation of the positional process; Indicates the standard deviation of the clock drift process; Indicates the epoch interval; The prediction steps are as follows: in Indicates an epochal time.

6. The low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 4, characterized in that, The satellite position and velocity interpolation is performed iteratively epoch-by-epoch to determine the observation noise matrix and linearize it. The Kalman gain is then calculated to complete state and covariance updates, including: The satellite position and velocity are interpolated using the 9th-order Lagrange method to determine the satellite position and velocity at the target time. Observation noise matrix for: in ; Linearized observation matrix for: in: Calculating the Kalman gain completes the state update and covariance update: The Kalman gain matrix is: The state update matrix is: in: The covariance update matrix is: 。 7. The low-orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 5, characterized in that, Record the estimated receiver position and clock drift, analyze the position error and Doppler residual, and evaluate the timing accuracy, including: Based on the estimated receiver location and clock drift Statistical analysis of receiver position error; Define the estimated receiver position as The actual receiver location is The error components in each direction are The three-dimensional error is ; The estimated receiver position and clock drift are used to calculate the Doppler prediction value. This prediction is then averaged with the sum of the absolute values ​​of the differences between the estimated Doppler position and the observed Doppler values. The resulting Doppler residual is: The average of the absolute values ​​of the Doppler residuals is: 。 8. The low-Earth orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in claim 5, characterized in that, Recording the estimated receiver position and clock drift, analyzing position errors and Doppler residuals, and evaluating timing accuracy also includes: Timing accuracy is assessed based on the estimated receiver clock drift: in The accumulated time error over k epochs, This represents the initial time error. After removing the linear trend from the accumulated time error, we get: in For time k, delinear trend time error. For time k epoch, , .

9. A low-Earth orbit satellite Doppler positioning and timing system based on extended Kalman filtering, characterized in that, It includes a processor and a memory, wherein the memory stores a computer program, which, when executed by the processor, implements the steps of the low-orbit satellite Doppler positioning and timing method based on extended Kalman filtering as described in any one of claims 1 to 8.

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