Non-cooperative low earth orbit satellite positioning system

Abstract

The application provides a non-cooperative low-orbit satellite positioning system, comprising: a signal receiving layer, configured to receive a non-cooperative low-orbit satellite radiated opportunity signal and perform signal conditioning on the opportunity signal; a data processing layer, configured to perform preprocessing, feature extraction and data association on the conditioned signal data, extract Doppler frequency shift observation values and signal angle of arrival observation values, and calculate noise variances of the two types of observation values; a positioning solution layer, configured to dynamically switch positioning modes according to the number of observed low-orbit satellites, perform orbit determination of the non-cooperative low-orbit satellites and terminal positioning solution; an orbit prediction layer, configured to perform orbit prediction and update according to satellite accurate orbit parameters; and a result output layer, in communication connection with the positioning solution layer and the orbit prediction layer, configured to format and output positioning results and orbit data; and the modified orbit data of the orbit prediction layer is fed back to the positioning solution layer, forming a closed-loop cooperative mechanism.

Description

Technical Field

[0001] This invention relates to the field of satellite positioning technology, specifically to a non-cooperative low-Earth orbit satellite positioning system, which is suitable for low-Earth orbit satellite orbit determination and terminal positioning in scenarios where no satellite cooperation or dedicated navigation signals are required. It can be applied to resilient positioning, navigation and timing (PNT) systems, and is especially suitable for backup positioning in scenarios where GNSS signals are blocked, interfered with or spoofed. Background Technology

[0002] Non-cooperative low-Earth orbit (LEO) satellite positioning refers to a technology that determines satellite orbit and terminal positioning by receiving opportunity signals (SOPs) radiated by satellites without relying on satellites to actively provide navigation signals or requiring any cooperation from satellites. It is a key component in enhancing the resilience of positioning, navigation, and timing (PNT) systems.

[0003] Currently, traditional non-cooperative low-Earth orbit satellite positioning systems have many technical shortcomings, making it difficult to achieve a balance between reliability, hardware cost, and availability, as follows:

[0004] 1. Limited accuracy due to single positioning observation: Existing technologies mostly rely on Doppler frequency shift as the sole observation for positioning calculation. Since non-cooperative low-Earth orbit satellites are not designed for navigation purposes, users cannot obtain satellite pseudorange information, resulting in large positioning errors, usually only reaching the level of ten meters, which is difficult to meet the requirements of high-precision positioning. At the same time, Doppler positioning is highly dependent on initial values. When initial values ​​are missing, the number of iterations increases significantly, further affecting positioning efficiency and accuracy.

[0005] 2. Limited satellite visibility and application scenarios: Low-Earth orbit satellites have low orbital altitude and fast movement speed, resulting in a short visibility period for a single satellite. It is difficult to obtain observation data from multiple satellites in a short period of time. When there are fewer than four observation satellites, instantaneous positioning in a single epoch cannot be achieved. It is necessary to wait for multiple epochs to accumulate integral Doppler, which is not only time-consuming but also affected by satellite-to-ground clock differences.

[0006] 3. Low computational efficiency and insufficient robustness: Traditional positioning methods mostly use nonlinear least squares (NLS) for calculation, which has the disadvantages of high computational complexity and sensitivity to noise. Although some methods attempt to improve accuracy, they require the addition of base stations at known locations or additional sensors, which increases hardware costs. Moreover, in complex environments with unknown noise statistical characteristics, the positioning reliability drops significantly.

[0007] 4. Inadequate existing system architecture: Most positioning systems have not achieved an integrated design of initial orbit determination, precise orbit determination and orbit prediction, and have not dynamically adjusted the positioning strategy according to the number of satellites, resulting in poor system flexibility, inability to adapt to positioning needs under different observation conditions, and difficulty in achieving long-term stable determination of satellite position and velocity.

[0008] Furthermore, while Global Navigation Satellite Systems (GNSS) are currently the mainstream positioning method, they suffer from drawbacks such as high infrastructure investment costs, susceptibility to signal interference, spoofing, or obstruction. Their positioning performance also significantly degrades in scenarios such as urban canyons, indoor environments, and complex electromagnetic environments. Non-cooperative low-Earth orbit (LEO) satellite positioning serves as an effective alternative, but these technical limitations restrict its large-scale application and promotion. Therefore, developing a non-cooperative LEO satellite positioning system and method that can solve these problems and balance accuracy, efficiency, and cost has become a pressing technical challenge in this field. Summary of the Invention

[0009] To address the shortcomings of existing technologies, such as low positioning accuracy, limited observation range, poor computational efficiency, restricted satellite visibility, and insufficient system flexibility of non-cooperative low-Earth orbit satellites, the core objective of this invention is to provide a non-cooperative low-Earth orbit satellite positioning system and method, with the specific objectives as follows:

[0010] 1. Solve the problems of single observation and low positioning accuracy: By fusing two types of observations, Doppler frequency shift and signal angle of arrival, and combining them with an adaptive noise optimization algorithm, the positioning accuracy is improved, and the positioning error is controlled within the meter level. At the same time, the dependence of iterative calculation on the initial value is reduced, the number of calculations is reduced, and the positioning efficiency is improved.

[0011] 2. Overcoming satellite visibility limitations and expanding application scenarios: Designing a multi-mode positioning strategy, dynamically adjusting the positioning algorithm based on the number of observed low-orbit satellites, enabling effective positioning even when there are fewer than four satellites, without waiting for multiple epochs to accumulate data, achieving a balance between instantaneous and long-term positioning.

[0012] 3. Improve computational efficiency and system robustness: Construct an integrated system architecture that integrates initial trajectory determination, precise trajectory determination and trajectory prediction functions, and design an efficient trajectory calculation algorithm to reduce computational complexity; at the same time, realize adaptive learning of noise covariance to improve the system's anti-interference ability and positioning reliability in complex environments.

[0013] 4. Reduce hardware costs and improve system flexibility: No additional base stations or sensors are required; positioning can be achieved simply by receiving satellite opportunistic signals from a single ground station, simplifying the system structure and reducing hardware costs. At the same time, it adapts to different observation conditions, enabling dynamic switching of positioning strategies and improving the system's versatility and flexibility.

[0014] 5. Achieve long-term stable determination of the orbits of non-cooperative low-Earth orbit satellites: Through initial orbit coarse calculation, multi-epoch data association and precise iterative optimization, combined with the orbit prediction model, the long-term accurate determination of satellite position and velocity can be achieved, with positioning accuracy comparable to the accuracy of the two-line orbital element (TLE) updated by the North American Aerospace Defense Command (NORAD).

[0015] The first aspect of this invention proposes a non-cooperative low-Earth orbit satellite positioning system, comprising:

[0016] A signal receiving layer is used to receive opportunistic signals radiated by non-cooperative low-Earth orbit satellites and to perform signal conditioning on the opportunistic signals;

[0017] The data processing layer, which is communicatively connected to the signal receiving layer, is used to preprocess, extract features and correlate data on the conditioned signal data, extract Doppler frequency shift observations and signal angle of arrival observations, and calculate the noise variance of the two types of observations.

[0018] The positioning calculation layer, which is communicatively connected to the data processing layer, is used to dynamically switch the positioning mode according to the number of observed low-Earth orbit satellites and perform orbit determination and terminal positioning calculation for non-cooperative low-Earth orbit satellites.

[0019] The orbit prediction layer, which is communicatively connected to the positioning calculation layer, is used to predict and update the orbit based on the precise orbital parameters of the satellite.

[0020] The result output layer is communicatively connected to the positioning calculation layer and the trajectory prediction layer, and is used to output the positioning results and trajectory data in a formatted manner.

[0021] The corrected orbit data from the orbit prediction layer is fed back to the positioning calculation layer, forming a closed-loop collaborative mechanism.

[0022] Existing non-cooperative low-Earth orbit satellite positioning systems generally suffer from a fragmented architecture. Initial orbit determination, precise orbit determination, and orbit prediction are often performed by independent modules, lacking data interaction and feedback mechanisms between modules. This results in poor system flexibility and difficulty in maintaining consistent positioning accuracy. To address this deficiency, this invention breaks through the traditional module stacking design approach and constructs an integrated closed-loop collaborative architecture encompassing signal reception, data processing, positioning calculation, orbit prediction, and result output.

[0023] The core technological value of this architecture lies in two aspects: First, the corrected orbit data from the orbit prediction layer can be fed back to the positioning calculation layer, enabling the positioning calculation to utilize the latest orbit prediction information for correction and avoiding positioning drift caused by the accumulation of orbit extrapolation errors. Second, the global closed-loop correction mechanism set in the result output layer (triggering recalculation when the error exceeds a threshold) allows the system to automatically heal itself when positioning accuracy declines, without manual intervention. This two-way feedback closed-loop design fundamentally solves the shortcomings of existing systems' open-loop operation and continuous error accumulation, achieving long-term stable tracking of non-cooperative low-Earth orbit satellites.

[0024] Based on the above, the signal receiving layer includes:

[0025] The antenna array, employing a multi-element directional antenna, is used to receive opportunistic signals from non-cooperative low-Earth orbit satellites from different directions, enabling preliminary detection of the signal's angle of arrival.

[0026] The low-orbit satellite signal receiver, employing a wideband receiver, is used to acquire, track, and synchronize the opportunistic signals, extract the Doppler frequency shift information of the signals, and cooperate with the antenna array to collect signal angle-of-arrival correlation data.

[0027] The signal conditioning module is used to filter, amplify, and denoise the received satellite signals, conditioning them to a standard format that meets the requirements of data processing.

[0028] This invention faces a key contradiction in designing the signal receiving layer: on the one hand, it needs to extract sufficiently rich signal features to support high-precision positioning; on the other hand, it must control hardware costs and deployment complexity. In existing technologies, obtaining signal angle-of-arrival information typically requires complex multi-antenna arrays and sophisticated RF front-ends, resulting in high hardware costs.

[0029] A combination of a multi-element directional antenna and a wideband receiver allows for the simultaneous acquisition of both Doppler frequency shift and angle of arrival (AHA) information without the need for additional sensors. The Doppler frequency shift is extracted via the receiver's frequency tracking loop, while the AHA is calculated using the phase difference between different elements of the antenna array. The ingenuity of this design lies in its "two birds with one stone" approach: the same hardware outputs both types of observations simultaneously, enriching the sources of positioning information while avoiding additional hardware overhead. The signal conditioning module's filtering, amplification, and noise reduction processes ensure reliable signal quality even in complex electromagnetic environments, laying a solid data foundation for subsequent high-precision positioning.

[0030] Based on the above, the data processing layer includes:

[0031] The preprocessing module is used to remove outliers, smooth data, correct time delay errors, and import preset satellite two-line orbit data files from the raw signal data.

[0032] The feature extraction module is used to estimate Doppler frequency shift information through fast Fourier transform, calculate signal angle of arrival information through the phase difference of the antenna array, and calculate the noise variance of the two types of observations.

[0033] The data association module is used to associate multi-epoch satellite signal data received by a single ground station, match the same satellite signals at different epochs, and integrate multi-epoch Doppler observations with signal angle of arrival observations.

[0034] A core challenge facing non-cooperative low-Earth orbit satellite positioning is that satellite signals are not designed for navigation purposes, and therefore pseudorange information cannot be directly obtained like GNSS. Instead of attempting to modify satellite signals, it's better to fully utilize the information inherent in the existing signals—Doppler shift reflects the relative radial motion between the satellite and the receiver, and the signal angle of arrival reflects the satellite's spatial orientation relative to the receiver; the two are naturally complementary.

[0035] However, the noise characteristics of the two types of observations are quite different: the noise of Doppler shift mainly comes from frequency tracking error and thermal noise, while the noise of the signal angle of arrival mainly comes from the phase measurement error of the antenna array and multipath effects. Simply fusing the two types of observations with equal weight may actually reduce positioning accuracy. Therefore, a noise variance calculation function was specifically designed in the feature extraction module, providing key noise parameters for subsequent adaptive fusion solutions. This design reflects a profound understanding of data quality perception—not only knowing "what was seen," but also "how accurately it was seen."

[0036] The data association module addresses another practical problem: low-Earth orbit satellites move at high speeds and have short visibility periods, meaning that signals received by a single ground station are often discrete, multi-epoch observation segments. Correctly associating signals from different epochs belonging to the same satellite is a prerequisite for orbit determination. The inventors, through a multi-epoch data association and matching mechanism, have achieved effective integration of observation data and elimination of invalid data, providing reliable data support for orbit determination.

[0037] Based on the above, the positioning solution layer includes:

[0038] The initial orbit determination module is used to perform a three-step algorithm to coarsely calculate the initial orbit of the satellite: First, the coarse orbit of the satellite is estimated using single-epoch Doppler observations through an improved search method; second, data matching and verification are completed by combining multi-epoch Doppler observations; third, the initial orbit of the satellite is determined by combining the coarse orbit data and multi-epoch observations using the least squares method.

[0039] The precise orbit determination module is used to construct a satellite orbit dynamics model based on the initial orbit data, introduce a perturbation model to correct orbit errors, and adopt a joint estimation scheme of cubic Kalman smoother and expectation-maximization algorithm to adaptively optimize the noise covariance, and iteratively optimize the initial orbit to obtain the precise orbit parameters of the satellite.

[0040] The positioning calculation module is used to dynamically switch the positioning mode according to the number of low-orbit satellites observed: when the number of satellites is greater than or equal to a preset threshold, the terminal coarse positioning information is generated based on the signal angle of arrival information and the precise satellite orbit information and used as the initial value for the Doppler positioning iterative calculation. The final positioning result of the terminal is obtained through iterative optimization; when the number of satellites is less than the preset threshold, the terminal positioning result is directly calculated by fusing the signal angle of arrival, Doppler frequency shift and satellite orbit information.

[0041] This invention achieves three innovations at the positioning solution level, which respectively address the three core bottlenecks of non-cooperative low-Earth orbit satellite positioning.

[0042] The first innovation: a three-step initial orbit determination algorithm. In non-cooperative scenarios, the initial orbit information of a satellite is completely unknown, and Doppler positioning algorithms are highly sensitive to initial values—the greater the initial value deviation, the more iterations are required, and it may even fail to converge. This invention designs a three-step strategy of coarse calculation, verification, and fine solution: the first step uses single-epoch Doppler data to quickly estimate the coarse orbit (not seeking precision, but convergence); the second step uses multi-epoch data to cross-verify the coarse orbit (eliminating erroneous assumptions); and the third step uses the least squares method to perform a fine solution near the coarse orbit. This coarse-to-fine, progressive strategy effectively reduces the dependence of orbit determination on initial values, enabling rapid convergence even without prior orbit information.

[0043] The second innovation: the CKS+EM joint estimation scheme. This is one of the most technically profound innovations of this invention. Traditional Kalman filtering requires prior knowledge of the statistical characteristics of the noise (i.e., the noise covariance). However, in non-cooperative low-Earth orbit satellite positioning scenarios, the statistical characteristics of the observed noise are unknown and time-varying—multipath effects in urban environments, ionospheric scintillation, receiver thermal noise, and other factors can all lead to dynamic changes in noise characteristics. If a fixed noise covariance is used, the filtering performance will be severely degraded or even diverge. This invention creatively combines the Expectation-Maximization (EM) algorithm with the Cubic Kalman Smoother (CKS): the EM algorithm is responsible for learning the noise covariance parameters from the observed data during the iteration process, while CKS is responsible for optimally estimating the satellite state given the noise parameters. The two alternate—E-step estimation of the state, M-step update of the noise parameters—until convergence. This joint estimation scheme enables the system to adaptively adjust in unknown, time-varying noise environments, greatly improving the positioning robustness in complex environments. The cubic Kalman smoother (CKS) was chosen over the extended Kalman filter (ECF) because it does not require the calculation of the Jacobian matrix, has better numerical stability and estimation accuracy for strongly nonlinear systems, and is well-suited to the highly nonlinear characteristics of low-Earth orbit satellite orbit dynamics.

[0044] The third innovation: a dual-mode adaptive positioning strategy. Poor visibility of low-Earth orbit (LEO) satellites is a recognized pain point in the industry—the visibility window for a single LEO satellite is typically only a few minutes, and at any given time, fewer than four satellites can be observed simultaneously. Traditional methods can only wait for data to accumulate over multiple epochs. This invention proposes a completely different approach: instead of passively waiting for the number of satellites to reach a certain threshold, it actively adapts to different satellite count conditions. When the number of satellites is ≥4, the system employs a two-step method: coarse positioning using the angle of arrival (Angle of Arrival) + fine positioning using Doppler iteration. First, an approximate position is obtained using the Angle of Arrival (Angle of Arrival) (as the initial value for iteration), and then fine iteration is performed using the Doppler observation equations, ensuring both accuracy and convergence. When the number of satellites is <4, the system does not wait or give up, but instead integrates the Angle of Arrival, Doppler frequency shift, and satellite orbit information to construct a positioning equation set for direct solution. Although the positioning accuracy in the fewer-satellite mode is slightly lower than in the more-satellite mode, it achieves instantaneous response capability with any satellite available, completely solving the defect of traditional methods being completely inoperable when the number of satellites is insufficient.

[0045] Based on the above, the orbit prediction layer includes:

[0046] The orbit prediction model employs an improved Runge-Kutta integrator, using precise satellite orbit parameters and a dynamic model to predict satellite orbit parameters over a future period, and utilizes the correlation between orbits in adjacent time intervals to reduce redundant calculations.

[0047] The orbit update module is used to receive new observation data in real time and periodically correct the predicted orbit to eliminate orbit drift errors. When the changes in satellite orbit parameters exceed the preset threshold, a recalculation process is triggered.

[0048] This invention faces a trade-off between efficiency and accuracy when designing the orbit prediction layer: high-precision orbit integrators (such as higher-order Runge-Kutta methods) have high computational costs, while low-Earth orbit (LEO) satellites move at high speeds and require high-frequency predictions. If integration starts from scratch for each prediction, the computational burden is too heavy. However, the orbits of LEO satellites exhibit strong correlation over short periods, and the orbital state changes between adjacent time intervals are continuous and smooth. Based on this observation, this invention employs an improved Runge-Kutta integrator—specifically, a variable-step fourth-order Runge-Kutta (RK4) integrator—combined with an interval recursive hot-start strategy. By utilizing the correlation between orbits in adjacent time intervals and using the endpoint of the previous interval as the initial value for integration in the next interval, redundant computation is significantly reduced while maintaining integration accuracy, lowering the computational cost by more than two orders of magnitude. Simultaneously, a dual-step error prediction mechanism based on local truncation errors in orbital dynamics is used to dynamically adjust the integration step size, replacing the traditional fixed step size, further improving the balance between integration efficiency and accuracy.

[0049] The orbit update module addresses another core issue in orbit prediction: model errors and perturbation errors cause predicted orbits to gradually deviate from the true orbit (i.e., orbit drift). This invention employs a dual-protection mechanism of periodic correction and threshold triggering. On one hand, it periodically corrects the predicted orbit with new observation data to eliminate accumulated drift; on the other hand, it automatically triggers recalculation when orbital parameter changes exceed a preset threshold, ensuring the long-term accuracy of orbit prediction. This mechanism, combined with a closed-loop architecture, forms a positive cycle of prediction-correction-repreneurial prediction.

[0050] A second aspect of this invention proposes a non-cooperative low-Earth orbit satellite positioning method, comprising the following steps:

[0051] Step S1: Configure signal reception parameters, import two lines of satellite orbit data files, orbit dynamics model parameters, and positioning calculation thresholds;

[0052] Step S2: Receive opportunistic signals radiated by non-cooperative low-Earth orbit satellites through the antenna array, acquire, track, and synchronize the signals, and perform filtering, amplification, and noise reduction processing.

[0053] Step S3: Perform outlier removal and smoothing on the original signal data, estimate Doppler frequency shift information through fast Fourier transform, calculate signal angle of arrival information through antenna array phase difference, calculate noise variance of the two types of observations, and correlate multi-epoch signal data.

[0054] Step S4: The initial orbit of the satellite is determined by a three-step algorithm, including estimating the coarse orbit of the satellite using single-epoch Doppler observations, completing data matching and verification by combining multi-epoch Doppler observations, and solving the initial orbit parameters of the satellite using the least squares method.

[0055] Step S5: Construct a satellite orbit dynamics model based on the initial orbit parameters, introduce a perturbation model to correct orbit errors, and use a joint estimation scheme of cubic Kalman smoother and expectation-maximization algorithm to adaptively optimize the noise covariance. Iterate the initial orbit calculation until the convergence threshold is met, and output the accurate satellite orbit parameters.

[0056] Step S6: Dynamically switch the positioning mode according to the number of observed low-orbit satellites. When the number of satellites is greater than or equal to a preset threshold, perform Doppler iterative precise positioning. When the number of satellites is less than the preset threshold, perform multi-source information fusion direct positioning.

[0057] Step S7: Based on the satellite's precise orbital parameters, an improved Runge-Kutta integrator is used to predict the orbit, and new observation data is received in real time to correct the predicted orbit.

[0058] Step S8: Format the output of the positioning results and track data. If the positioning accuracy or track prediction error exceeds the preset threshold, return to step S3 to reprocess and calculate the data.

[0059] This method clearly defines the logical timing and dependencies between each step. A key point to note is the closed-loop correction mechanism in step S8—when the positioning accuracy or orbit prediction error exceeds a preset threshold, the system automatically returns to step S3 to reprocess and calculate the data. This mechanism echoes the closed-loop architecture of the system, ensuring its self-correcting capability from a methodological perspective. Traditional positioning methods often perform a one-time calculation, outputting the result immediately upon completion, lacking error awareness and automatic recalculation capabilities. This invention, by monitoring positioning accuracy and orbit prediction errors in real time, automatically triggers recalculation when accuracy deteriorates, achieving long-term stable operation of the system. Furthermore, the two-line satellite orbit data file (TLE) imported in step S1 is only used to provide initial reference (such as the coarse orbit stage of a three-step algorithm). The final accurate orbit determination relies entirely on measured Doppler and angle-of-arrival observation data, which is the essential characteristic of non-cooperative positioning—it does not depend on the active cooperation of satellites.

[0060] Based on the above, in step S6, when the number of observed satellites is greater than or equal to a preset threshold, the terminal coarse positioning result is calculated based on the signal angle of arrival information and the satellite's precise orbit information. The coarse positioning result is used as the initial value for the Doppler positioning iterative solution and substituted into the Doppler observation equation to obtain the terminal's final positioning result through iterative optimization.

[0061] The reason why Doppler positioning is sensitive to initial values ​​is fundamentally because the Doppler observation equation is nonlinear. Iterative optimization algorithms (such as the Gauss-Newton method) are prone to getting trapped in local optima or failing to converge when the initial value deviates significantly from the true solution. Traditional methods often randomly select initial values ​​or simply use the result of the previous epoch as the initial value, which leads to a sharp increase in the number of iterations when the initial value is missing. This invention cleverly utilizes the signal angle of arrival (AHA) information to generate high-quality iterative initial values—although the accuracy of the AHA is not as high as that of the Doppler iteration result, it can provide a roughly correct spatial orientation, sufficient to lock the iterative initial value near the true solution. Using this coarse positioning result as the initial value for Doppler iteration ensures the convergence of the iteration (the initial value is already within the convergence region) and reduces the dependence on the initial value (precise prior position information is not required). This two-step strategy of determining the initial value with the AHA and refining it with Doppler is one of the core methodological innovations of this invention at the positioning solution level.

[0062] Based on the above, in step S6, when the number of observed satellites is less than a preset threshold, the signal angle of arrival information, Doppler frequency shift information and satellite precise orbit information are fused to construct a set of positioning equations, and the terminal positioning result is directly obtained by solving them.

[0063] Positioning problems with fewer than four satellites: When there are fewer than four satellites, traditional least-squares positioning methods cannot solve the problem due to the insufficient number of equations. However, the observed angle of arrival (AHA) of the signal itself can provide directional constraints—even with only one satellite, its AHA can determine that the terminal is located on a ray with the receiver as the vertex and the satellite's direction as the center; Doppler shift provides constraints on the rate of change of distance. Combining these with known precise satellite orbit information, a solvable set of positioning equations can be constructed when there are fewer than four satellites. Although the solution accuracy is lower than the iterative optimization results under multi-satellite conditions, it represents a breakthrough from scratch—in scenarios where traditional methods are completely ineffective, this invention can still output positioning results. This has significant practical value for backup positioning in scenarios with limited GNSS signals, such as urban canyons and indoor environments.

[0064] Based on the above, in step S5, the joint estimation scheme of the cubic Kalman smoother and the expectation-maximization algorithm substitutes the observed data into the nonlinear state-space model, estimates the noise covariance parameter iteratively through the expectation-maximization algorithm, and performs optimal estimation of the state by the cubic Kalman smoother, thereby achieving adaptive optimization of the noise covariance.

[0065] The above scheme provides a more detailed methodological constraint on the CKS+EM joint estimation scheme. Its core technical contribution lies in solving the chicken-and-egg problem in nonlinear filtering where the statistical characteristics of noise are unknown—accurate state estimation requires knowledge of the noise covariance; knowing the noise covariance, in turn, requires accurate state estimation. The EM algorithm cleverly overcomes this dilemma through an iterative approach: in the E-step, based on the current noise covariance estimate, CKS is used to smooth the state estimation; in the M-step, based on the state estimate obtained in the E-step, the noise covariance is re-estimated. Each iteration makes the noise covariance estimate closer to the true value, and the state estimation more accurate, until convergence. This alternating optimization and co-evolutionary mechanism enables the system to autonomously learn noise characteristics from observational data without any prior noise information, greatly improving the system's environmental adaptability.

[0066] Based on the above, in step S7, when the change in satellite orbit parameters exceeds a preset threshold, the process of steps S4 to S5 is triggered to recalculate the precise satellite orbit.

[0067] The above scheme defines a threshold-triggered recalculation mechanism for orbit prediction at the methodological level. The orbit prediction model is based on extrapolation from a dynamic model; over time, model errors and perturbation errors gradually accumulate. If only periodic corrections are relied upon, the correction speed may not keep up with the rate of change when significant orbital changes occur (such as satellite orbital maneuvers). The threshold-triggered recalculation mechanism: when a change in satellite orbital parameters exceeds a preset threshold, it no longer relies on incremental corrections but directly triggers the complete recalculation process from steps S4 to S5—from initial orbit determination to precise orbit determination, recalculating from scratch. This dual-mode mechanism of incremental correction + threshold recalculation ensures both the continuity of orbit prediction under normal circumstances (no need for frequent recalculation) and the rapid response capability under abnormal circumstances (immediate recalculation in the event of large changes), providing a crucial guarantee for long-term stable orbit tracking.

[0068] This invention has outstanding substantive features and significant progress compared to the prior art. Specifically, this invention has the following technical advantages:

[0069] 1. Significantly improved positioning accuracy: By fusing two types of observations, Doppler frequency shift and signal angle of arrival, and combining the CKS and EM joint estimation algorithm, the noise covariance is adaptively optimized, which solves the problems of single observation and low accuracy in traditional methods. The positioning accuracy can reach the meter level, and the trajectory determination accuracy is comparable to the TLE accuracy updated by NORAD.

[0070] 2. Overcoming satellite visibility limitations: A dual-mode positioning strategy is designed to achieve instant positioning even when fewer than four satellites are observed, without waiting for the accumulation of long-term data. This expands the system's application scenarios and adapts to positioning needs in complex environments.

[0071] 3. Improved computational efficiency and robustness: The improved orbit calculation algorithm and iterative solution strategy reduce computational complexity; the closed-loop correction mechanism and adaptive noise optimization enhance the system's anti-interference capability and positioning reliability under complex electromagnetic environments and unknown noise statistical characteristics.

[0072] 4. Reduced hardware costs and high versatility: Positioning can be achieved by receiving satellite opportunistic signals through a single ground station, without the need for additional base stations or sensors, which simplifies the system structure and reduces hardware costs; it is compatible with opportunistic signals from various types of low-Earth orbit satellites and can serve as a backup for GNSS systems, making it highly versatile.

[0073] 5. Achieve long-term stable orbit determination: Integrate initial orbit determination, precise orbit determination and orbit prediction functions, and achieve long-term accurate tracking of satellite position and velocity through multi-history data association and orbit update mechanism to meet the needs of long-term positioning and orbit monitoring. Attached Figure Description

[0074] Figure 1 This is a system architecture diagram of the non-cooperative low-Earth orbit satellite positioning system of the present invention.

[0075] Figure 2 This is a flowchart illustrating the workflow of the non-cooperative low-Earth orbit satellite positioning method of the present invention. Detailed Implementation

[0076] The following detailed description, in conjunction with the accompanying drawings and specific embodiments, provides further details.

[0077] This embodiment provides a non-cooperative low-Earth orbit satellite positioning system, such as Figure 1 As shown, from top to bottom, it is divided into a signal receiving layer, a data processing layer, a positioning calculation layer, a trajectory prediction layer, and a result output layer.

[0078] Specific implementation of the signal receiving layer:

[0079] The antenna array employs a 4-element uniform circular array (UCA) directional antenna with an element spacing of half the signal wavelength to achieve high-precision angle-of-arrival estimation. The low-Earth orbit (LEO) satellite signal receiver utilizes a wideband zero-IF receiver architecture, supporting the reception of various LEO satellite signals (including Starlink and Globalstar) within the 100MHz-3GHz frequency band. The frequency tracking loop employs a phase-locked loop (PLL) structure, achieving Doppler shift estimation accuracy better than 1Hz. The signal conditioning module includes a bandpass filter, a low-noise amplifier, and an analog-to-digital converter; the filter bandwidth is adaptively adjusted according to the target satellite signal bandwidth.

[0080] Specific implementation of the data processing layer:

[0081] The preprocessing module uses the 3σ criterion to remove outliers and employs moving average filtering for data smoothing. Delay error correction is based on correlation peak detection of the signal frame header synchronization word. In the feature extraction module, Doppler frequency shift is achieved through FFT frequency estimation combined with phase-locked loop fine tracking, and the signal angle of arrival is estimated using the phase difference of the received signals from each element of the antenna array combined with a multiple signal classification (MUSIC) algorithm. The azimuth angle (0° with true north as the reference point, rotating clockwise) is calculated as θ = arctan2(e,n), and the elevation angle (0° with the horizontal plane as the reference point, upward as the reference point) is calculated as φ = arcsin(u / ||d||). Noise variance is obtained through statistical calculation of observation residuals. The data association module uses a dual-threshold association criterion based on the signal carrier frequency and angle of arrival (frequency tolerance ±500Hz, angle of arrival tolerance ±5°) to match the same satellite signal at different epochs, achieving a data association accuracy higher than 99%.

[0082] Specific implementation of the location solution layer:

[0083] In the three-step algorithm of the initial orbit determination module, the first step, an improved search method, employs a multi-stage search strategy of hierarchical dimensionality reduction, adaptive step size, and prior constraints—transforming the traditional blind search across the entire space into a hierarchical refined search within a constrained space, reducing computational complexity by more than two orders of magnitude while ensuring convergence probability. The second step, data matching, is based on the criterion of minimizing the residual between orbit recursion and observations. The third step, least squares solution, uses the Levenberg-Marquardt algorithm for iterative optimization. In the precise orbit determination module, the SGP4 perturbation model parameters are adaptively selected based on satellite type and orbital altitude. CKS uses a third-order volume rule for state prediction and updating, and the iteration termination condition of the EM algorithm is set to a change in the log-likelihood function of less than 10. -4 The preset threshold for the positioning solution module is set to 4 (adjustable range 3~5), and the convergence threshold for Doppler positioning iteration is set to a positioning error change of less than 0.1 meters (adjustable range 0.05~1 meters).

[0084] Specific implementation of the orbit prediction layer:

[0085] The orbit prediction model uses a variable-step fourth-order Runge-Kutta (RK4) integrator as its basic format. The integration step size is dynamically adjusted through a dual-step error prediction mechanism, and the prediction time window is set to 7 days. Simultaneously, an interval recursive hot-start strategy is employed, utilizing the correlation between orbits in adjacent time intervals, using the endpoint of the previous interval as the initial value for the integration of the next interval, thus avoiding repeated initialization and redundant calculations. The correction cycle of the orbit update module is set to once every 6 hours, and the orbit parameter change thresholds are set to a position deviation greater than 1 km (adjustable range 500m~5km) or a velocity deviation greater than 1 m / s (adjustable range 0.5~5m / s).

[0086] Specific implementation of the result output layer:

[0087] The result formatting module encapsulates the output data into JSON or NMEA-0183 format. The output interfaces include RS-232 serial port (115200 baud rate), Ethernet interface (TCP / IP protocol), and 4G / 5G wireless interface.

[0088] like Figure 2 As shown, the specific workflow of this invention is as follows:

[0089] Step S1 (System Initialization):

[0090] After the system powers on, the FPGA loads the configuration file, sets the receiver center frequency to 1.575GHz (compatible with GPS L1 band, facilitating the use of existing antenna facilities), reads antenna array calibration parameters from EEPROM, obtains TLE files from local storage or network updates, loads orbital dynamics model parameters (Earth's gravitational constant, atmospheric drag coefficient, etc.) into DSP memory, and writes positioning calculation thresholds to registers: satellite number threshold = 4 (adjustable range 3~5), Doppler iteration convergence threshold = 0.1m (adjustable range 0.05~1m), orbital parameter change recalculation threshold = position deviation 1km / velocity deviation 1m / s (adjustable range position 500m~5km, velocity 0.5~5m / s), positioning error closed-loop trigger threshold = 10m (adjustable range 5~50m), EM iteration convergence threshold = log-likelihood relative change 10 -4 (Adjustable range 10) -3 ~10 -5 The data association tolerance threshold is set to frequency ±500Hz / angle of arrival ±5° (adjustable range: frequency ±200~1000Hz, angle ±2~10°). The initial noise covariance value is set to an empirical value (Doppler noise variance = 1Hz). 2 The variance of the noise at the angle of arrival is 0.01 rad. 2 ).

[0091] Step S2 (Signal Reception and Conditioning):

[0092] The antenna array receives spatial signals at a sampling rate of 10Hz. The receiver acquires and tracks the signal through a carrier tracking loop and a code tracking loop. The coherent integration time is set to 1ms, and the number of incoherent integrations is set to 20 to improve the signal-to-noise ratio. The signal conditioning module amplifies the intermediate frequency signal by 60dB gain, filters it with a 10MHz bandwidth, and samples it using a 14-bit ADC, outputting a digital intermediate frequency signal to the data processing layer.

[0093] Step S3 (Data Preprocessing and Feature Extraction):

[0094] The preprocessing module performs outlier detection on 10,000 accumulated sampling points every second (outliers exceeding 3σ are discarded) using a 5-point moving average filter. The feature extraction module performs 4096-point FFT frequency estimation on the data every second, combined with phase-locked loop fine tracking to obtain Doppler frequency shift (accuracy better than 0.5Hz). It calculates the phase difference matrix (4×4) for the received signal from the 4-element antenna and uses the MUSIC algorithm to estimate the signal angle of arrival (azimuth accuracy better than 2°, elevation accuracy better than 3°), calculated as θ=arctan2(e,n) and φ=arcsin(u / ||d||), where (e,n) are the east and north components, and d is the direction vector. Noise variance is calculated statistically from the observation residuals over 10 consecutive seconds. The data association module uses the signal carrier frequency (tolerance ±500Hz) and angle of arrival (tolerance ±5°) as association criteria, grouping continuous observation data by individual satellites, achieving a data association accuracy higher than 99%.

[0095] Step S4 (Determining the initial orbit of the satellite):

[0096] Taking a low-Earth orbit satellite as an example, assuming its orbital altitude is approximately 550 km (a typical Starlink satellite orbit). The first step involves using continuous 3-second single-epoch Doppler observations (3 Doppler frequency shifts) and employing a multi-stage improved search method combining hierarchical dimensionality reduction, adaptive step size, and prior constraints. This method performs a constrained spatial hierarchical fine search in the orbital elements space (semi-major axis, eccentricity, inclination, right ascension of the ascending node, argument of perigee, and true anomaly) to obtain a coarse orbit estimate. The computational cost is reduced by more than two orders of magnitude compared to the traditional full-space blind search. The second step involves recursively extending the coarse orbit to the next 10 epochs and calculating the residuals with the measured Doppler values. A match where the residual is less than a threshold (1 Hz) is verified. The third step uses the verified coarse orbit as the initial value and employs the least squares method to fit the Doppler observations of all available epochs. After 5-8 iterations, the fit converges, and the initial orbital parameters are output.

[0097] Step S5 (Precise satellite orbit determination):

[0098] Based on the initial orbital parameters, an orbital dynamics model was constructed, incorporating Earth's non-spherical gravity (J2 term), atmospheric drag (using the NRLMSIE-00 atmospheric model), and solar radiation pressure. The SGP4 perturbation model was introduced for preliminary correction. The state equations and observation equations (Doppler observation equations + angle of arrival observation equations) were constructed into a nonlinear state-space model, and a CKS+EM joint estimation scheme was adopted.

[0099] The Doppler observation equation is expressed as follows:

[0100]

[0101] Where f cWhere n is the signal carrier frequency, c is the speed of light, and n is the signal carrier frequency. d The Doppler observation noise follows a zero-mean Gaussian distribution with variance . .

[0102] The equation for the angle of arrival observation is expressed as follows:

[0103]

[0104] Where R E R N n is the Earth's radius of curvature correction term. θ n φ The angular observation noise has variances of [missing information]. , .

[0105] The joint observation equation is z=h(r) u )+n, the Jacobian matrix of the observation equation with respect to the terminal position The matrix is ​​3M×3 (M is the number of satellites). A weighted least squares (WLS) framework is used to fuse the two types of observations, with the weight matrix W=R. -1 That is, the weight of each observation is the reciprocal of its noise variance—higher Doppler accuracy results in a higher weight, while higher angle-of-arrival noise automatically reduces the weight, ensuring that high-quality observations dominate the solution. The latest estimate is then iteratively derived from the observation residuals in real time using a subsequent CKS+EM joint estimation scheme. , , The weight matrix is ​​dynamically updated. When the electromagnetic environment deteriorates or the noise of a certain type of observation increases, its weight automatically decreases to avoid lowering the overall positioning accuracy. Before calculation, the two types of observations are normalized to eliminate the influence of dimensional differences on weight allocation.

[0106] In the EM algorithm, the observed variable is the Doppler shift observation f of all epochs. d The observed values ​​are: azimuth (k), elevation (k), and φ (k). The latent variables are the satellite state sequence X (k), process noise covariance matrix Q, and observation noise covariance matrix R. The iterative process is as follows:

[0107] (1) Initialization: Given the initial orbital state, initial noise covariance empirical values ​​Q0 and R0, the number of iterations t=0, and the convergence threshold ε=10. -4 ;

[0108] (2) E-step: Substitute the current noise parameters, perform CKS fixed interval smoothing, and obtain the full epoch state estimate. ;

[0109] (3) M-step: Update the noise covariance Q based on the smoothing result (t+1) R(t+1) ;

[0110] (4) Convergence criterion: Calculate the relative change of the log-likelihood function. If ΔL < ε, the iteration terminates and the smooth trajectory and final noise parameters are output; otherwise, t = t + 1, and the process returns to step (2). Auxiliary convergence criterion: The Frobenius norm change of the noise parameters in three consecutive iterations is less than the threshold to avoid misjudgment caused by log-likelihood fluctuations.

[0111] After 5-10 alternating iterations, the noise covariance converges to a stable value, and the satellite's precise orbital parameters are output (position accuracy better than 100 meters, velocity accuracy better than 0.1 m / s).

[0112] Step S6 (Terminal Positioning Calculation):

[0113] Assuming that 5 low-Earth orbit satellites (≥4) are observed at the current moment, the positioning solution module switches to multi-satellite mode: First, based on the signal angle of arrival information and precise orbit information of the 5 satellites, the least squares positioning algorithm is used to calculate the coarse positioning result of the terminal (accuracy of about 500 meters); This coarse positioning result is used as the initial value for the Doppler positioning iteration, and substituted into the Doppler observation equation system (each of the 5 satellites provides a Doppler equation), and the Gauss-Newton method is used for iterative optimization. After 3-5 iterations, the system converges and outputs the final positioning result of the terminal (horizontal accuracy better than 8 meters, elevation accuracy better than 15 meters, 95% confidence level; the horizontal accuracy in the optimal scenario can be controlled within 5 meters).

[0114] Assuming that only 2 low-Earth orbit satellites (<4) are observed at another time, the positioning solution module switches to the least-satellite mode: it fuses the signal arrival angles of the 2 satellites (each providing azimuth and elevation angles, a total of 4 equations) and Doppler shift (2 equations) and the satellite's precise orbit information (known), and constructs an overdetermined system of equations (6 equations, 3 unknowns) with the terminal's three-dimensional position (x, y, z) as unknowns. It then uses the least squares method to directly solve the equations and outputs the terminal positioning result (accuracy of approximately 50~150 meters, depending on the number of satellites: ≤40 meters with 3 satellites, ≤150 meters with 2 satellites).

[0115] A weighted fusion transition scheme is used between the two modes to achieve smooth switching and avoid abrupt changes in positioning results; when the number of satellites fluctuates around the threshold, a dual mechanism of hysteresis threshold + time window decision is used to achieve anti-jitter.

[0116] Step S7 (Satellite Orbit Prediction and Update):

[0117] Based on the precise satellite orbit parameters (epoch time t0) obtained in step S5, a variable-step fourth-order Runge-Kutta (RK4) integrator is used, with a dual-step error prediction mechanism to dynamically adjust the integration step size to predict the orbit parameters for the next 7 days. During integration, an interval recursive hot-start strategy is adopted, utilizing the strong correlation between orbits in adjacent time intervals (each interval is 10 minutes). The integration result of the previous interval is used as the initial value for the next interval, avoiding repeated integration and redundant calculations, significantly reducing the computational load. The orbit update module receives new observation data every 6 hours, compares the predicted orbit with the observed values, calculates the orbit drift error, and corrects it. If the deviation between the predicted position and the actual observed value of a satellite exceeds the position threshold of 1 km or the velocity threshold of 1 m / s, the recalculation process in steps S4-S5 is triggered. The precise orbit position error obtained from 1 hour of observation is ≤100 meters, and the predicted orbit position error for 7 days is ≤2 km, both of which are better than the accuracy performance of traditional TLE and SGP4 models.

[0118] Step S8 (Result Output and Closed-Loop Correction):

[0119] The results formatting module encapsulates precise satellite orbital parameters (position, velocity, epoch time), terminal positioning results (longitude, latitude, altitude, positioning accuracy), and orbit prediction data (orbit ephemeris for the next 7 days) into JSON format. This data is then output to the user terminal in real-time via an Ethernet interface at a frequency of 1Hz. Simultaneously, the system continuously monitors positioning residuals and orbit prediction residuals.

[0120] Definition of location observation residual:

[0121]

[0122] Definition of trajectory prediction residual:

[0123]

[0124] If the positioning residual exceeds the preset threshold (10 meters) for 10 consecutive seconds, the recalculation process will be automatically triggered.

[0125] (1) The error monitoring module triggers a recalculation instruction, and the system temporarily stores the positioning result of the previous epoch for output retention;

[0126] (2) Data rollback: Roll back the original observation data from the past minute and re-execute preprocessing, feature extraction and data association;

[0127] (3) Orbit recalculation: The satellite orbit is updated throughout the entire process, from the initial orbit determination using the three-step method to the precise orbit determination using CKS+EM.

[0128] (4) Positioning recalculation: Recalculate the terminal position using the new track and verify that the residual falls back to within the threshold;

[0129] (5) Output update: Replace the retained positioning results and restore normal real-time output.

[0130] On an 8-core DSP platform, the full-process recalculation time for a single satellite is about 0.5 to 2 seconds, while the time for multiple satellites to be processed in parallel is about 3 to 5 seconds, and the positioning output interruption time is less than 1 second.

[0131] Step S9 (Continuous System Operation and Maintenance):

[0132] The system continuously repeats steps S2-S8 at a frequency of 1Hz to achieve continuous tracking of non-cooperative low-Earth orbit satellites and real-time positioning of the terminal. The system self-test module performs a status check on each module every 24 hours, including signal-to-noise ratio monitoring at the signal receiving layer, data integrity statistics at the data processing layer, and convergence status checks at the positioning solution layer. It automatically alarms and logs any anomalies detected. The orbital dynamics model parameters and noise covariance parameters are updated monthly based on long-term observation data.

[0133] Typical application scenarios

[0134] Scenario 1: UAV positioning in GNSS denied environments.

[0135] In complex electromagnetic environments or urban canyons, when the GNSS signal of a UAV is interfered with or blocked, the system of this invention is activated to achieve continuous positioning of the UAV by receiving opportunistic signals from low-orbit satellites such as Starlink. When four or more satellites are observed, the positioning accuracy is better than 8 meters (95% confidence), serving as an effective backup for GNSS; when only two to three satellites are observed, it can still output positioning results of 50 to 150 meters, achieving a flexible PNT capability of "positioning whenever satellites are available".

[0136] Scenario 2: Non-cooperative satellite orbit monitoring.

[0137] Ground monitoring stations track a low-Earth orbit satellite with unknown orbital parameters and continuously observe its opportunistic signals using the system of this invention. After one hour of observation and joint estimation with CKS+EM, its precise orbital parameters can be determined with a position error ≤100 meters, which is better than the accuracy of publicly available TLE data (TLE typically has a position error of hundreds of meters to kilometers); the 7-day orbit prediction position error is ≤2km, which is also better than the accuracy performance of traditional TLE and SGP4 models.

[0138] Scenario 3: Emergency location and rescue.

[0139] At disaster sites, GNSS signal infrastructure may be damaged. Rescue personnel carrying the ground terminal of this invention (requiring only a single ground station + a 4-element antenna array + a wideband receiver, without the need for additional base stations or sensors) can achieve self-positioning by receiving opportunistic signals from low-Earth orbit satellites, without relying on any satellite navigation signals, thus achieving autonomous and flexible positioning. The hardware deployment is lightweight and cost-controllable, making it suitable for large-scale deployment.

[0140] Summary of threshold parameters

[0141] The six core threshold parameters involved in the system of this invention, their typical values, and adjustable ranges are shown in the table below:

[0142] Satellite number switching threshold 4 3-5 Based on the Doppler positioning principle, balancing accuracy and usability Doppler Iteration Convergence Threshold 0.1 meters 0.05~1 meter Matching meter-level positioning targets, balancing accuracy and efficiency. Recalculation threshold for orbital parameter changes Location 1km / Speed ​​1m / s Location: 500m~5km / Speed: 0.5~5m / s Based on publicly available TLE accuracy, orbit recalculation is triggered. Positioning error closed-loop trigger threshold 10 meters 5~50 meters Corresponding to the system accuracy indicators, trigger closed-loop correction. EM Iteration Convergence Threshold <![CDATA[Log-likelihood relative variation 10 -4 > <![CDATA[10 -3 ~10 -5 ]]> Balance noise estimation accuracy and computational cost Data association tolerance threshold Frequency ±500Hz / Angle of arrival ±5° Frequency ±200~1000Hz / Angle ±2~10° Ensure data correlation accuracy >99%.

[0143] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A non-cooperative low-Earth orbit satellite positioning system, characterized in that, include: A signal receiving layer is used to receive opportunistic signals radiated by non-cooperative low-Earth orbit satellites and to perform signal conditioning on the opportunistic signals; The data processing layer, which is communicatively connected to the signal receiving layer, is used to preprocess, extract features and correlate data on the conditioned signal data, extract Doppler frequency shift observations and signal angle of arrival observations, and calculate the noise variance of the two types of observations. The positioning calculation layer, which is communicatively connected to the data processing layer, is used to dynamically switch the positioning mode according to the number of observed low-Earth orbit satellites and perform orbit determination and terminal positioning calculation for non-cooperative low-Earth orbit satellites. The orbit prediction layer is communicatively connected to the positioning calculation layer and is used to predict and update the orbit based on the precise orbit parameters of the satellite. The result output layer is communicatively connected to the positioning calculation layer and the trajectory prediction layer, and is used to output the positioning results and trajectory data in a formatted manner. The corrected orbit data from the orbit prediction layer is fed back to the positioning calculation layer, forming a closed-loop collaborative mechanism.

2. The non-cooperative low-Earth orbit satellite positioning system according to claim 1, characterized in that, The signal receiving layer includes: The antenna array, employing a multi-element directional antenna, is used to receive opportunistic signals from non-cooperative low-Earth orbit satellites from different directions, enabling preliminary detection of the signal's angle of arrival. The low-orbit satellite signal receiver, employing a wideband receiver, is used to acquire, track, and synchronize the opportunistic signals, extract the Doppler frequency shift information of the signals, and cooperate with the antenna array to collect signal angle-of-arrival correlation data. The signal conditioning module is used to filter, amplify, and denoise the received satellite signals, conditioning them to a standard format that meets the requirements of data processing.

3. The non-cooperative low-Earth orbit satellite positioning system according to claim 1, characterized in that, The data processing layer includes: The preprocessing module is used to remove outliers, smooth data, correct time delay errors, and import preset satellite two-line orbit data files from the raw signal data. The feature extraction module is used to estimate Doppler frequency shift information through fast Fourier transform, calculate signal angle of arrival information through the phase difference of the antenna array, and calculate the noise variance of the two types of observations. The data association module is used to associate multi-epoch satellite signal data received by a single ground station, match the same satellite signals at different epochs, and integrate multi-epoch Doppler observations with signal angle of arrival observations.

4. The non-cooperative low-Earth orbit satellite positioning system according to claim 1, characterized in that, The positioning solution layer includes: The initial orbit determination module is used to perform a three-step algorithm to coarsely calculate the initial orbit of the satellite: First, the coarse orbit of the satellite is estimated using single-epoch Doppler observations through an improved search method; second, data matching and verification are completed by combining multi-epoch Doppler observations; third, the initial orbit of the satellite is determined by combining the coarse orbit data and multi-epoch observations using the least squares method. The precise orbit determination module is used to construct a satellite orbit dynamics model based on the initial orbit data, introduce a perturbation model to correct orbit errors, and adopt a joint estimation scheme of cubic Kalman smoother and expectation-maximization algorithm to adaptively optimize the noise covariance, and iteratively optimize the initial orbit to obtain the precise orbit parameters of the satellite. The positioning calculation module is used to dynamically switch the positioning mode according to the number of low-orbit satellites observed: when the number of satellites is greater than or equal to a preset threshold, the terminal coarse positioning information is generated based on the signal angle of arrival information and the precise satellite orbit information and used as the initial value for the Doppler positioning iterative calculation. The final positioning result of the terminal is obtained through iterative optimization; when the number of satellites is less than the preset threshold, the terminal positioning result is directly calculated by fusing the signal angle of arrival, Doppler frequency shift and satellite orbit information.

5. The non-cooperative low-Earth orbit satellite positioning system according to claim 1, characterized in that, The orbit prediction layer includes: The orbit prediction model employs an improved Runge-Kutta integrator, using precise satellite orbit parameters and a dynamic model to predict satellite orbit parameters over a future period, and utilizes the correlation between orbits in adjacent time intervals to reduce redundant calculations. The orbit update module is used to receive new observation data in real time and periodically correct the predicted orbit to eliminate orbit drift errors. When the changes in satellite orbit parameters exceed the preset threshold, a recalculation process is triggered.

6. A non-cooperative low-Earth orbit satellite positioning method, characterized in that, Includes the following steps: Step S1: Configure signal reception parameters, import two lines of satellite orbit data files, orbit dynamics model parameters, and positioning calculation thresholds; Step S2: Receive opportunistic signals radiated by non-cooperative low-Earth orbit satellites through the antenna array, acquire, track, and synchronize the signals, and perform filtering, amplification, and noise reduction processing. Step S3: Perform outlier removal and smoothing on the original signal data, estimate Doppler frequency shift information through fast Fourier transform, calculate signal angle of arrival information through antenna array phase difference, calculate noise variance of the two types of observations, and correlate multi-epoch signal data. Step S4: The initial orbit of the satellite is determined by a three-step algorithm, including estimating the coarse orbit of the satellite using single-epoch Doppler observations, completing data matching and verification by combining multi-epoch Doppler observations, and solving the initial orbit parameters of the satellite using the least squares method. Step S5: Construct a satellite orbit dynamics model based on the initial orbit parameters, introduce a perturbation model to correct orbit errors, and use a joint estimation scheme of cubic Kalman smoother and expectation-maximization algorithm to adaptively optimize the noise covariance. Iterate the initial orbit calculation until the convergence threshold is met, and output the accurate satellite orbit parameters. Step S6: Dynamically switch the positioning mode according to the number of observed low-orbit satellites. When the number of satellites is greater than or equal to a preset threshold, perform Doppler iterative precise positioning. When the number of satellites is less than the preset threshold, perform multi-source information fusion direct positioning. Step S7: Based on the satellite's precise orbital parameters, an improved Runge-Kutta integrator is used to predict the orbit, and new observation data is received in real time to correct the predicted orbit. Step S8: Format the output of the positioning results and track data. If the positioning accuracy or track prediction error exceeds the preset threshold, return to step S3 to reprocess and calculate the data.

7. The non-cooperative low-Earth orbit satellite positioning method according to claim 6, characterized in that, In step S6, when the number of observed satellites is greater than or equal to a preset threshold, the coarse positioning result of the terminal is calculated based on the signal angle of arrival information and the precise orbit information of the satellites. The coarse positioning result is used as the initial value for the Doppler positioning iterative solution and substituted into the Doppler observation equation to obtain the final positioning result of the terminal through iterative optimization.

8. The non-cooperative low-Earth orbit satellite positioning method according to claim 6, characterized in that, In step S6, when the number of observed satellites is less than a preset threshold, the signal angle of arrival information, Doppler frequency shift information and satellite precise orbit information are fused to construct a set of positioning equations, and the terminal positioning result is directly obtained by solving the equations.

9. The non-cooperative low-Earth orbit satellite positioning method according to claim 6, characterized in that, In step S5, the joint estimation scheme of the cubic Kalman smoother and the expectation-maximization algorithm substitutes the observed data into the nonlinear state-space model, estimates the noise covariance parameter iteratively through the expectation-maximization algorithm, and performs the optimal estimation of the state by the cubic Kalman smoother, thereby achieving adaptive optimization of the noise covariance.

10. The non-cooperative low-Earth orbit satellite positioning method according to claim 6, characterized in that, In step S7, when the changes in satellite orbit parameters exceed a preset threshold, the process of steps S4 to S5 is triggered to recalculate the precise satellite orbit.

Images