Backpropagation Neural Network-Assisted Helmert Variance Model for Weighted Global Navigation Satellite System Localization in High Orbit

Abstract

In high-orbit space missions, the significant attenuation of Global Navigation Satellite System (GNSS) signals due to long transmission distances and complex environmental interferences has led to a drastic degradation in the accuracy of traditional positioning models, which has attracted great attention in recent years. Although multi-system GNSS fusion positioning can alleviate the problem of insufficient satellite visibility, the existing methods are difficult to effectively cope with the challenges of multi-source noise coupling and inter-system error differences unique to high orbit. In this paper, we propose an adaptive GNSS positioning optimization framework for a high-orbit environment, which improves the orbiting reliability under complex signal conditions through dynamic weight allocation and a multi-system cooperative strategy. Different from the traditional weighting model, this method innovatively constructs a two-layer optimization mechanism: (1) Based on BP neural network, it evaluates the noise characteristics of pseudo-range observations in real time and realizes the adaptive suppression of receiver thermal noise, ionospheric delay, etc.; (2) it introduces Helmert variance component estimation to optimize the weighting ratio of GPS, GLONASS, BeiDou, and Galileo and reduces the impact of signal heterogeneity on the positioning solution of the multi-system. Simulation results show that the new method reduces the root-mean-square error of positioning by 32.8% compared with the traditional algorithm to 97.72 m in typical high-orbit scenarios and significantly improves the accuracy loss caused by the defective satellite geometrical configurations under multi-system synergy.

1. Introduction

With the rapid development of space technology, high-orbit spacecraft are of great significance in the fields of satellite communications, weather detection, unknown objects, and environmental exploration [1]. In contrast to ground-based and low- to medium-orbit users, High Earth Orbit (HEO) spacecraft operate at altitudes exceeding those of GNSS constellations. Additionally, GNSS satellite transmission antennas are typically oriented toward the Earth’s center; therefore, the GNSS signal propagation in high-orbit space is not only affected by the earth’s blockage, but also the propagation distance and propagation loss of the signal will be significantly increased, which will result in a serious signal fading and the phenomenon of extremely uneven intensity distribution [2]. Therefore, in the high-altitude orbital environment, the quality of Global Navigation Satellite System signals will become very poor, and the composition of the measurement value error is more complicated, which places a higher demand on the quality control of GNSS signals.

In the high-orbit environment, several cases of flight tests have been carried out at home and abroad to verify the feasibility of GNSS positioning. These include the American AMSAT OSCAR-40 satellite, the Chinese Chang’e 5-T lunar exploration, and the Practice 17 satellite [3,4,5]. The results show that above the orbital altitude of 6 × 10⁴ km, the specially designed high-sensitivity GNSS receiver can track its main and side flap signals, laying the foundation for GNSS high-orbit applications [6]. However, the positioning solution of GNSS receivers for high-orbit spacecraft is mainly realized by the least-squares method, and all available GNSS signals come from a relatively narrow cone, which can be seen by the poor satellite geometrical configuration and the large DOP value. In this case, the large number of available satellites does not help, and the positioning error may still reach hundreds of meters [3,4,5]. There are methods to enhance the positional precision of high-altitude orbital spacecraft by adding auxiliary information. Ref. [7] utilized both GNSS and inertial sensors for combined navigation to reduce the inaccuracy in the positioning of spacecraft. Ref. [8] described how GNSS and an Orbit Propagator (OP) can be combined to enhance the accuracy of the positioning of the high-orbit spacecraft; for the problems of few available satellites and poor positioning accuracy of high-orbit satellite navigation receivers, ref. [9] proposed a clock-assisted positioning algorithm that utilizes the frequency stability of thermostatic crystals to inhibit the observation noise and further improve the positioning accuracy; for the roughness that may occur in the observations, ref. [10] proposed an Orbit Propagator-assisted anti-differential positioning algorithm of high-orbit GNSS to down-weight or exclude the observations with roughness to enhance the resilience of the positioning outcomes.

At present, the quality control of GNSS signals in high-orbit space is mainly studied from the perspectives of anti-differential and the introduction of auxiliary information. And the a priori weighting model commonly used on the ground, i.e., assigning appropriate weights to GNSS observations before they are involved in the localization solution, can enhance the quality of GNSS observation signals from high-orbit spacecraft in highly dynamic, weak-signal, and strong-interference environments without the introduction of additional external equipment [11]. Ref. [12] has systematically elaborated and discussed a variety of mainstream weighting methods based on the two main influencing factors of satellite elevation angle and carrier-to-noise ratio (CNR) or signal-to-noise ratio (SNR). Due to the differences in observation noise and orbit accuracy among GNSSs, the use of empirical weighting ratios for combined positioning may not be able to achieve the best results. Ref. [13] introduced the Helmert post-test variance model in Global Positioning System (GPS), Global Navigation Satellite System in Russian (GLONASS), and Beidou Navigation Satellite System (BDS) combined positioning related research to reasonably allocate the weights of the observations of each system in the combined single-point positioning and baseline solving of GPS/GLONASS/BDS. Aiming at the problems that the raw pseudo-range observation values of smartphones have large measurement noise and are susceptible to errors such as multipath, which makes the conventional data processing methods unable to satisfy the requirements for higher accuracy positioning, ref. [14] proposed a method based on Doppler smoothing pseudo-range. Aiming at the rapid attenuation of the accuracy of the navigation satellite ranging signal with the increase in the distance, ref. [15] proposed an adaptive Kalman filtering algorithm based on the quantitative new information, which effectively suppresses the attenuation of the accuracy of the navigation signal.

High-orbit spacecraft orbiting faces many problems, such as weak signals, low ranging accuracy, and poor visibility, which makes the traditional weighting method unable to give accurate and effective observation value weights. Therefore, in this research paper, after conducting an analysis of the composition of signal errors and typical tracking loops of high-orbit spacecraft, we comprehensively consider the effects of satellite elevation angle and CNR on the quality of GNSS observation signals, give accurate a priori weights within a sole satellite-based navigation system with the assistance of a BP neural network model, and employ Helmert variance component estimation method to calculate the weights among different satellite navigation systems, thus designing a GNSS observation weighting scheme suitable for high-orbit space.

2. GNSS Pseudo-Range Error Analysis in High-Orbit Environment

In order to realize the quality control of GNSS signals from high-orbit spacecraft, the error modeling and error analysis of GNSS signals are indispensable. The observation equation for the high-orbit GNSS pseudo-range observation $L_r^g$ is given in Equation (1):

$$L_r^g={\rho }_r^g+c{t}_r-c{t}^g+I_r^g+T_r^g+D_r+N_r+{\delta }_r^g$$

(1)

Here, ${\rho }_r^g$ represents the geometric distance between the high-orbit receiver and the GNSS satellite, with the subscript $r$ and superscript $g$ indicating the high-orbit receiver and the GNSS satellite, respectively, measured in meters. The terms $t_r$ and $t^g$ denote the clock offsets of the receiver and satellite, respectively, expressed in seconds, while $c$ stands for the speed of light. The variables $I_r^g$ and $T_r^g$ correspond to the ionospheric and tropospheric delays, respectively, measured in meters. Additionally, $D_r$, $N_r$, and ${\delta }_r^g$ represent the multipath error, the receiver’s thermal noise, and unmodeled errors in pseudocode measurements, all presented in units of meters.

In the high-orbit GNSS receiver, the clock offset error and orbit error of navigation satellites belong to the error at the transmitter side of the satellite signal. In the Geostationary Earth Orbit (GEO) spacecraft orbit solving, the GNSS orbit error and clock offset error are directly extracted from the ephemeris file, and the error of the file itself will directly affect the geometric distance calculation results. A large number of measured data show that the pseudo-range error caused by the satellite clock offset error and orbit error is about 1.0 m [9]. Moreover, due to the periodic changes of the orbit and the influence of the space environment, the satellite clock deviation and orbital error contain significant periodic terms, and both of them show quasi-periodic sinusoidal characteristics [16,17].

Errors in the propagation path of the high-orbit GNSS signal include tropospheric delay, ionospheric delay, and so on. Compared with thermal noise and multipath interference, ionospheric and tropospheric errors vary slowly, with small fluctuations in ionospheric and tropospheric errors for the same navigation satellite over a period of several minutes. When the vertical distance from the Earth’s center to the GNSS signal path is smaller than the sum of the Earth’s radius and the ionosphere’s thickness, the GNSS signal will penetrate the ionosphere; at this time, the pseudo-range observations by the ionosphere are more affected. When the vertical distance from the Earth’s center to the GNSS signal path is smaller than the sum of the Earth’s radius and the thickness of the troposphere (about 60 km), the GNSS signal passes through the troposphere and crosses the ionosphere for the second time, and at this time, the signal error is larger. Even the GNSS signal is gradually close to the surface of the earth is obscured by the earth, the signal will be refracted by the atmosphere, so that part of the signal even from the center of the earth is very close to still be received; that is, phenomenon of occultation occurs, which may interfere with the navigation and positioning results. However, relevant experiments in Refs [18,19] show that the time of GNSS signal crossing the ionosphere and troposphere merely constitutes a minor fraction of the whole visible time, and these satellites can be deduced from the ephemeris and then filtered out or coarsely processed in the design of actual receiver algorithms.

Errors at the high-orbit receiver end mainly consist of receiver thermal noise and multipath interference, both of which have relatively fast-changing values. When the vertical distance from the center of the earth to the transmission path of GNSS signal is larger than the sum of the Earth’s radius and the ionosphere’s thickness, the multipath effect dominates, and its variation value is mainly within ±100 m [18], which is noisier than the ground data and varies more rapidly than the ionospheric error. The factors affecting the pseudo-range multipath error of the on-board GNSS signals mainly cover several aspects, including mutual interference between different signals (e.g., occultation signals), antenna design, satellite attitude, and solar sail attitude [20], and the effect of multipath interference can be significantly mitigated by means such as proper antenna layout. The satellite signals captured by high-orbit receiver antennas are extremely faint, making receiver thermal noise a significant concern. Research data indicate that when the CNR threshold is set at 24 dB-Hz, the thermal noise error for the main lobe signal is approximately 7 m; while for the side lobe signal, it can escalate to around 20 m [21]. Consequently, thermal noise error emerges as the predominant error source in high-orbit receivers. Within the code tracking loop, specifically the Delay Lock Loop (DLL), thermal noise stands out as the primary error contributor. This noise is intrinsically linked to the signal’s CNR, meaning that variations in the CNR of GNSS signals lead to differing levels of thermal noise error introduced through the code loop. The standard deviation of the code loop’s thermal noise, as it relates to the input signal’s CNR, is mathematically expressed in Equation (2) [22]:

$${\sigma }_{t\mathrm{DLL}}={\lambda }_L\sqrt{{\displaystyle \frac{B_n}{2CNR}}D\left[1+{\displaystyle \frac{2}{T\cdot CNR(2-D)}}\right]}$$

(2)

where ${\lambda }_L$ represents the width of the code slice of the ranging code in m; $B_n$ is the equivalent noise bandwidth of the loop in Hz; $T$ is the coherent integration time in s; $CNR$ is the carrier-to-noise ratio in dB-Hz; and $D$ (number of code slices) represents the spacing of the correlator.

3. Traditional Weighting Model for GNSS Observations

In high-altitude orbits, the Earth’s presence restricts the signal reception area of high-altitude satellites to the ring-shaped cone defined by the main beam edge. When some main lobe signals pass through the ionospheric layer, it leads to distinct signal qualities among various satellites. Meanwhile, the loss of signals in free space causes a sharp decline in the visibility and quality of GNSS navigation satellite signals. The error makeup in the measurement values also gets more complex, which poses challenges to accurately utilize these signals for navigation and positioning purposes. This situation requires more sophisticated signal processing and error correction techniques to ensure the reliability and precision of high-altitude satellite applications. As a key link in satellite positioning, the random error weighting model of measurement value is essential to reasonably allocate the size of the role played by each satellite in the solution. Currently, common weighting models are usually based on CNR weighting and satellite elevation angle weighting [12]. The basic idea is that satellite signals with lower satellite elevation angles or smaller carrier-to-noise ratios typically exhibit more typical multipath effects and more significant atmospheric residual delay errors. However, in high-orbit environments, it is often the case that satellites with smaller absolute values of elevation angles or smaller carrier noise ratios have less transmit power and are more susceptible to random noise interference, with a consequent degradation in the level of accuracy of the observed data. Developing an accurate stochastic model to weight GNSS observation data in high-orbit environments is crucial for enhancing the positioning performance of high-orbit spacecraft.

3.1. Carrier-to-Noise Ratio Weighted Model

Errors caused by receiver thermal noise, ionospheric and tropospheric delays, and multipath interference are unavoidable during GNSS signal propagation from high-orbiting spacecraft. These errors can affect the accuracy of the observations in various ways. Therefore, the quality of observations cannot be improved by simply adopting an equal-weighting model. When considering the difference between the terrestrial and space scenarios, a spacecraft’s departure from Earth leads to a marked rise in the measurement noise of GNSS signals. The free-space environment causes signal propagation losses, which in turn cause an inescapable degradation of the positioning geometry. Given the variable nature of the measurement noise that is close to the actual situation, applying a CNR weighting model is a prudent approach. This model allows for the proper weighting of pseudo-range observations in the high-orbit context, enhancing the accuracy and reliability of the positioning data obtained from GNSS signals. This is crucial for various space-based applications, such as satellite navigation and orbital control.

The CNR represents the quotient of the carrier signal intensity received by the receiver and the noise intensity. This ratio can efficiently mirror the quality of satellite signals captured by the antenna of a high-altitude orbital receiver [22]. To attain a superior weighting outcome for Global Navigation Satellite System (GNSS) observations, this research paper employs an empirical stochastic model that relies on the weighting of the CNR.

$${\sigma }_i=\sqrt{a+b\times 10^{-(\mathrm{CNR}{)}_i/10}}$$

(3)

where $a$, $b$ are the model parameters, the reference values in this paper are 395 and 23,500, respectively, and the parameters need to be adjusted according to the weighting effect when used in practice.

When weighting the GNSS observations, we usually use the inverse of the signal error variance to weight the signal, i.e.,

$${\omega }_i={\displaystyle \frac{1}{{\sigma }_i^2}}$$

(4)

where ${\omega }_i$ denotes the weight of the $i$th channel signal.

Nonetheless, the utilization of the traditional CNR weighting model encounters limitations when addressing complex error sources within the high-altitude orbital environment. Firstly, by analyzing the CNR modeling Equations (5) and (6), it becomes evident that the CNR of GNSS signals for high-altitude spacecraft depends on several influencing factors. These factors encompass the signal’s transmission power, the transmitting antenna’s gain, the signal’s free-space propagation loss, and the receiving antenna’s gain. In other words, when gauging signal quality solely from the perspective of the received signal power at the receiver, the information reflecting the signal quality characteristics is inevitably incomplete.

Conversely, considering receiver thermal noise as the primary error source in the high-altitude orbital environment, Equation (3) fails to precisely depict the non-linear correlation between the magnitude of the receiver thermal noise and the CNR. Moreover, this model is unable to address the inaccuracies stemming from error sources like ionospheric delay, tropospheric delay, and multipath interference that are present in the high-altitude orbital setting. Consequently, in real-world scenarios, the conventional CNR weighting model often falls short in delivering the required levels of accuracy and reliability.

As per the findings in [10], the received power of Global Navigation Satellite System (GNSS) satellite signals can be formulated as follows:

$$P_r=P_t+G_t+20\mathit{log} \left({\displaystyle \frac{\lambda }{4\pi d}}\right)+G_r$$

(5)

Here, $P_r$ represents the power received by the receiver, while $P_t$ denotes the transmission power of the GNSS satellite signal. The gain of the navigation satellite’s transmitting antenna is given by $G_t$. The term $20\mathit{log}\left({\displaystyle \frac{\lambda }{4\pi d}}\right)$ accounts for the free-space propagation loss of the GNSS signal, where $\lambda$ is the signal’s wavelength and $d$ is the distance over which the signal propagates. Additionally, $G_r$ signifies the gain of the receiver’s antenna.

The formula for the CNR is as follows:

$$CNR=P_r-10\lg \left(T\right)+228.6+L_{\mathrm{AD}}$$

(6)

where $T$ = 290 K, for the ambient noise temperature, 228.6 is the Boltzmann constant in dB; $L_{\mathrm{AD}}$ is the signal quantization loss during A/D conversion, here 3 dB is taken.

3.2. Model Based on Elevation Angle

The high-orbit environment is different from the ground, and the weighting effect is limited by directly adopting the sine function model or cosine function model, so we adopt the empirical function model as in Equations (7) and (8), and adjust the empirical parameters $a$, $b$ according to the weighting algorithm localization effect in the high-orbit environment; $E_i$ is the satellite elevation angle of the $i$th signal channel.

1.

Empirical sine function model

$${\sigma }_i=\sqrt{a^2+{\displaystyle \frac{b^2}{{\mathrm{s}\mathrm{i}\mathrm{n}}^2\left(E_i\right)}}}$$

(7)

2.

Empirical cosine function model

$${\sigma }_i=\sqrt{a^2+{\displaystyle \frac{b^2}{{\mathrm{c}\mathrm{o}\mathrm{s}}^2\left(E_i\right)}}}$$

(8)

On the ground, the satellite elevation angle-based weighting is due to the fact that the satellite signal passes through the atmosphere, which results in low elevation angle satellites that tend to have more typical multipath characteristics and larger residual delay errors, so the elevation angle-based weighting method generally determines the weights of each satellite by constructing a function that monotonically increases with elevation angle and decreasing the weights of those satellites with low elevation angles. In the high-orbit environment, most of the satellite signals do not pass through the atmosphere, and based on the reason why the satellite elevation angle weighting is effective, we can analyze it from the perspective of signal power.

From Figure 1, it is easy to see that the sum of the absolute value of the satellite elevation angle and the absolute value of the receiver antenna reception angle is 90°. However, with the movement of the GEO spacecraft, the receiver antenna receiving angle changes, and the transmitting angle of the GNSS satellite signal also changes, which causes the navigation satellite transmitting antenna gain to change. Combined with Equations (5) and (9), the free propagation loss of the signal changes as the reception angle changes, while the transmit power and receive antenna gain are generally fixed values. The weighting based on the satellite elevation angle in the high-orbit environment is essentially weighted from the perspective of the satellite signal power.

The relationship between the distance from the satellite to the receiver and the receiving angle of the receiver antenna [2] is as follows:

$$d=\left\{\begin{array}{cc}{R}_{\mathrm{GEO}}+R_{\mathrm{GNSS}}& \theta =0°\\ {\displaystyle \frac{R_{\mathrm{GNSS}}}{\sin \theta }}\mathit{sin}\left[\mathit{arcsin}\left({\displaystyle \frac{R_{\mathrm{GEO}}\sin \theta }{R_{\mathrm{GNSS}}}}+\theta \right)\right]& 0°<\theta \le 39.0°\end{array}\right.$$

(9)

where $R_{\mathrm{GEO}}$ is the radius of the GEO spacecraft orbit; $R_{\mathrm{GNSS}}$ is the radius of the GNSS navigation satellite orbit.

4. Design of BP Neural Network-Assisted Weighting Method for Helmert’s Post-Test Variance Model in High-Track Environment

In view of the complexity of the satellite signal error composition of high-orbit spacecraft and the existence of some errors that are difficult to model accurately, it is difficult for the traditional weighting model to give effective weights for the observations. Based on the analysis in the first subsection, this paper treats the multipath interference at the top of the ionosphere and the signal that crosses the ionosphere only once as roughness and adopts Helmert’s post-test variance model to regulate the weights between the satellite navigation systems to suppress the roughness; for the occultation signals that are less than 60 km away from the ground and the signals that cross the ionosphere for the second time, the rejection operation is taken directly; for the main error sources, thermal noise error as well as orbital error and satellite clock deviation, a neural network is used to fit their magnitudes, which in turn weights the pseudo-range observations.

4.1. Helmert’s Post-Test Variance Model

In the orbit determination process of high-orbit spacecraft, different types of pseudo-range observations have significant differences in observation accuracy [21]. Even within a single satellite navigation system, there is a significant difference between the noise magnitude of the observations of the main flap signals and the side flap signals, and it is difficult to determine the reasonable weights of the observations by a priori weighting models such as CNR weighting and satellite elevation angle-based weighting in the orbit determination processing, and Helmert’s a posteriori variance component estimation method for updating the weighting arrays makes up for this point. In this paper, the GNSS observations are categorized into four major satellite systems, GPS, BDS, GLONASS, and Galileo (Galileo Satellite Navigation System), and the error equations of each system are shown in Equation (10):

$$v_j=B_j\widehat{x}-l_j$$

(10)

where $v_j$ is the number of corrections to the residuals of the pseudo-range observations for each system; $B_j$ is the linearized geometric matrix for each system; $\widehat{x}$ is the parameter to be estimated; and $l_j$ is the residual of the observations. The initial weighting ratio of the four systems, GPS, BDS, GLONASS, and Galileo, is set to 1:1:1:1, and the a priori weighting matrix of observations within the same satellite navigation system is determined according to a weighting model based on the satellite elevation angle or the CNR and is set to $P_j$; $j=1, 2, 3, 4$ represent the four major satellite navigation systems, respectively.

The application process of Helmert’s post-test variance model in the high-orbit environment is as follows.

1.

First, the weighting model based on satellite elevation angle or CNR to determine the a priori weights within a single system is used: According to the pseudo-range observations of each system, the initial empirical weight ratio between systems is set to $P_1=P_2=P_3=P_4=1$, and the relationship between different observations, i.e., unit-weighted variance is constructed based on the a priori information, and the same weight matrix construction method is used for the observations within a single satellite system. See Equations (3)–(8) in Section 3.1 and Section 3.2;

2.

Least squares parity is sought for $v_j^T{P}_j{v}_j; j=1, 2, 3, 4$ represent the four major satellite navigation systems, respectively. Since the observations are independent of each other among the systems, the normal equation of Equation (10) is as follows:

$$\sum _{j=1}^4{B}_j^T{P}_j{B}_j\widehat{X}-\sum _{j=1}^4{B}_j^T{P}_j{L}_j=0$$

(11)

Among them, $N=\sum _{j=1}^4 N_j=\sum _{j=1}^4 B_j^T{P}_j{B}_j,W=\sum _{j=1}^4 B_j^T{P}_j{L}_j,$ solve for $\widehat{X}=N^{-1}W,$ $v_j^T{P}_j{v}_j$ is solved by substituting back into Equation (10);

3.

The unit weight variance of the observations is calculated by type of satellite system through Equation (12).

$$\left[\begin{array}{c}{\widehat{\sigma }}_1^2\\ {\widehat{\sigma }}_2^2\\ {\widehat{\sigma }}_3^2\\ {\widehat{\sigma }}_4^2\end{array}\right]=S^{-1}\left[\begin{array}{c}{v}_1^T{P}_1{v}_1\\ v_2^T{P}_2{v}_2\\ v_3^T{P}_3{v}_3\\ v_4^T{P}_4{v}_4\end{array}\right]$$

(12)

$$S=\left[\begin{array}{ccc}{S}_{11}& \cdots & S_{14}\\ ⋮& \ddots & ⋮\\ S_{41}& \cdots & S_{44}\end{array}\right]$$

(13)

$$\left\{\begin{array}{c}{S}_{ii}=n_i-2\mathrm{tr}\left(N^{-1}{N}_i\right)+\mathrm{tr}(N^{-1}{N}_i{)}^2\\ S_{ij}=\mathrm{tr}\left(N^{-1}{N}_i{N}^{-1}{N}_j\right)\end{array}\right.$$

(14)

where ${\widehat{\sigma }}_1^2,{\widehat{ \sigma }}_2^2,{\widehat{ \sigma }}_3^2,{\widehat{ \sigma }}_4^2$ is the estimated new unit-weight variance, respectively, and $n_j$ is the number of observations $L_j$ [13];

4.

The weight matrix for each system is updated:

$$p_i^{\left(k+1\right)}={\displaystyle \frac{c}{{\sigma }_i^2}}{p}_i^{\left(k\right)},i=1, 2, 3, 4$$

(15)

where $c$ is a nonzero constant that tends to take $c={\sigma }_1^2$ (In the case of fusion positioning of the four major systems, GPS, BDS, GLONASS, and Galileo, the GPS satellite observations are used as a baseline to adjust the weights of the other systems.);

5.

Finally, the above steps are repeated until the following condition ${\sigma }_1^2\approx {\sigma }_2^2\approx {\sigma }_3^2\approx {\sigma }_4^2$ is satisfied, then the iteration is stopped and the final obtained weight matrix is applied to weight the observations. In this paper, the iteration termination condition executed at the time of programming is $\mid {\displaystyle \frac{1-{\widehat{\sigma }}_{min}^2}{{\widehat{\sigma }}_{max}^2}}\mid \le 0.01$.

4.2. Neural Network Modeling

Combined with the analysis in the second subsection, in order to provide more comprehensive a priori feature information for the neural network to fit the nonlinear factors in the conversion process of CNR, satellite elevation angle, and observation error, eliminate the influence due to the modeling error, and improve the accuracy of GNSS navigation, the network model is constructed as follows:

The mapping relationship was constructed by taking the carrier-to-noise ratio (CNR_i, i = 1, …, n, with n being the number of GNSS satellites) and the satellite elevation angle (E_i) as inputs and the pseudo-range error (σ_i, i = 1, …, n) as the output.

$$\left({\sigma }_i\right)=f\left(CN{R}_i,E_i\right)$$

(16)

If a suitable neural network model is used to fit the above mapping relationship, it can replace the traditional a priori weighting model, thus eliminating the nonlinear error between the a priori feature information of the GNSS signals and the weights of the observed signals. In this paper, a BP neural network (BPNN) model with an optimal number of hidden layer nodes is used to fit the magnitude of random noise as well as other nonlinearities in the conversion relationship during the computation of observation weights.

A BP (Backpropagation) neural network is a feedforward multilayer network trained using the error backpropagation algorithm. Its typical structure comprises an input layer, at least one hidden layer, and an output layer [23], as depicted in Figure 2.

The model equations are as follows:

The output of the hidden layer node is as follows:

$$y_j=f_1\left(\sum _{i=0}^n {\nu }_{ij}{x}_i\right), j=\mathrm{1,2},\dots ,m$$

(17)

The output of the output layer node is as follows:

$$z_k=f_2\left(\sum _{j=0}^m{w}_{jk}{y}_j\right),k=\mathrm{1,2},\dots ,l$$

(18)

The output error of the BPNN network is as follows:

$$E={\displaystyle \frac{1}{2}}\sum _{k=1}^l\{d_k-f_2\left[\sum _{j=0}^l{w}_{jk}{f}_1\left(\sum _0^n{\nu }_{ij}{x}_i\right)\right]{\}}^2$$

(19)

where $x_i$ is the input variable of the neural network; $v_{ij}$, $w_{jk}$ are the connection weights of each layer; $z_k$ is the output value of the neural network; $d_k$ is the output expectation value; $f_1$, $f_2$ are the hidden layer transfer function and the output layer transfer function, respectively.

The neural network’s basic architecture includes 2 input nodes, 1 output node, and a hidden layer with N nodes, where N = 2 + a and a ranges from 1 to 10. Prior to training, the optimal number of hidden layer nodes is determined by evaluating 10 different configurations. For each configuration, the mean square error (MSE) of the training set is computed, and the configuration with the lowest MSE is chosen as the optimal setup. The hidden layer uses the tangent sigmoid (tan-sigmoid) activation function, while the output layer employs a linear function (purelin). The BP neural network is configured with a learning rate of 0.01, a maximum of 1000 training iterations, and a target training error of 0.01. The trainlm algorithm, which combines Newton’s method and gradient descent, is utilized for training to enhance both speed and accuracy.

4.3. Weighting Algorithm Design

The flowchart of the weighting algorithm in this paper is shown in Figure 3; ${\delta }_i^j$ ($j$ = 1, 2, 3, 4, representing the four major satellite navigation systems, respectively, $i$ = 1, 2, 3, …, n, n represents the number of satellites available for the navigation system) is the weight of each system observation in the output of Helmert’s post-test variance model.

Within a single satellite navigation system, the weights of observations are determined, and this paper uses the BP neural network model with the best hidden layer instead of the traditional weighted model of carrier-to-noise ratio/satellite elevation angle; for the inter-system weights determination, this paper adopts the Helmert’s post-test variance model, and the a priori weights for the inputs to the model are also adopted as the a priori weights fitted by the BP neural network.

To optimize computational efficiency and reduce algorithmic complexity while pre-serving the limited on-board processing capabilities, the training of the neural network model begins with the CNR and pseudo-range error values of a single visible satellite. This approach allows for the training and testing of a straight-forward BP neural network architecture. After the training accuracy meets the requirements or the number of training times reaches the upper limit, the training is completed, and then the model is applied to the pseudo-range error prediction of a four-system visible star. Following this, the estimated pseudo-range error is applied to Equation (4) to derive the GNSS observation weights. These weights are utilized in a weighted least squares algorithm to generate the positioning outcomes.

5. Simulation Verification

5.1. Design of Simulation Conditions

In this paper, the simulation software STK 11.6 (Satellite Tool Kit) and MATLAB R2022b are used to build the simulation environment for high-orbit spacecraft. In the navigation satellite position simulation, the GNSS navigation constellation simulation integrates the four major global navigation satellite systems: GPS, BDS, GLONASS, and Galileo. To ensure accuracy, the simulation employs high-precision satellite ephemeris data obtained from Wuhan University’s IGS Data Center, corresponding to the precise moment of 00:00:00 on 14 May 2024. These ephemeris data are utilized to generate the precise orbital positions of satellites across all four systems. The simulation is configured with a total of 116 satellites, distributed as follows: 31 for GPS, 41 for BeiDou, 22 for GLONASS, and 22 for Galileo, representing the operational constellations of each system at the specified time.

In the high-orbit spacecraft orbit design, this paper takes the geosynchronous orbit high-resolution optical remote sensing satellite Gaofen 4 (GF-4) launched by China Aerospace Science and Technology Group (CASTG) as the object, simulates the operation orbit of the high-orbit GEO satellites, and generates the root of the high-orbit orbits by using the Satellite Toolkit STK. The orbital plane of the GF-4 satellite is overlapped with the equatorial plane, and the specific orbital parameter settings are shown in

Table 1. GEO satellite orbital parameters.

Orbital Parameters	GF-4
Radius/km	42,164.100
Orbital eccentricity	0
Track inclination/(°)	0
Ascending node of equinoxes/(°)	0
Perigee angle/(°)	21.5
True near point angle/(°)	0

, and the generated orbital data including the position and velocity information of the satellite in this orbit use MATLAB to write the IGS precision ephemeris interpolation program to get the position and velocity information matching the orbit sampling rate of the high-orbit GEO satellite. In order to completely show the changes of the GNSS satellite zenith angle in one cycle and minimize the data volume, the simulation time is set to 12 hours (h), and 8640 orbital data are generated in 5s steps; the simulated orbit is exported in STK, and the coordinate system is the geocentric geoid coordinate system.

As the starting point of the signal propagation link, the satellite signal transmitting power of the GNSS signal transmitting antenna and the gain of the transmitting antenna have a significant influence on the signal receiving power of the high-orbit receiver. Since the antenna direction map parameters of BeiDou system and the key information such as transmit power are not disclosed to the outside world, for the sake of data and performance description, the corresponding performance parameters of GPS system are chosen here as an alternative description, and the antenna direction maps of GLONASS and Galileo navigation systems are also produced according to GPS. The antenna direction map released by Lockheed Martin is used for the GPS navigation satellite constellation, and in the actual simulation and verification of the high-orbit environment, when modeling the error, in order to ensure the continuity of the experiment, the data of one of the profiles should be selected as the data source, see Figure 4 for details. Substituting the set transmit antenna gain into Equations (5) and (6), we can get the corresponding signal received power of the high-orbit receiver and its CNR. When the satellite is in orbit, the receiving antenna of the high-orbit receiver points to the center of the earth.

5.2. Analysis of GNSS Availability Results

5.2.1. Number of Visible Satellites

By using the GNSS simulation constellations obtained from the simulation in Section 5.1 and the operational orbit of the high-orbit spacecraft, the satellites that simultaneously satisfy the GNSS satellite geometry visibility and carrier noise ratio threshold screening [24] are regarded as visible stars, in order to analyze the visibility of satellites from the four global navigation systems (GPS, BDS, GLONASS, and Galileo) during the operational phase of a high-orbit spacecraft; Figure 5 provides a detailed illustration. The receivers are configured with a sensitivity of −185 dBW and a CNR threshold of 20 dB-Hz. Under these parameters, the standalone GPS system fails to consistently provide the minimum of four visible satellites required for positioning. In contrast, the combined use of GPS and BDS systems ensures visibility of more than 10 satellites for most of the time. When all four systems are integrated, the number of visible satellites often exceeds 15, significantly enhancing positioning reliability. Consequently, this paper employs a four-system fusion strategy for optimal positioning performance.

5.2.2. Dilution of Precision

Positioning accuracy is one of the most critical metrics for evaluating the performance of a GNSS [25], and its magnitude is determined by the measurement error and Dilution of Precision (DOP). In the context of high-orbit spacecraft operations, measurement errors in high-orbit observations are unavoidable. To improve the positioning accuracy of such spacecraft, it is essential to reduce the DOP. The Position Dilution of Precision (PDOP), a key DOP metric, reflects how the geometric arrangement of GNSS satellites relative to the receiver affects navigation and positioning accuracy. This, in turn, determines the accuracy of GNSS single-point positioning. The PDOP of a high-orbit spacecraft is shown in Figure 6. When navigating with a single GPS system due to the number of visible stars, there will be fewer than four, resulting in too large PDOP values, which are not easy to show in the figure. When we use the GPS + BDS constellation, it reduces the PDOP significantly, with an average PDOP of 13.90 for GPS + BDS for 0.5 days, and up to an average PDOP of 11.02 when navigating with a quad system.

5.2.3. Pseudo-Range Error

In the simulation of pseudo-range error, this paper adds 50–100 m roughness to the signal propagation path from the ground 60–1200 km satellite signals to replace the multipath interference at the top of the ionosphere and the signals that cross the ionosphere only one time; the mask signals below 60 km distance from the ground and the signals that cross the ionosphere two times are regarded as invisible signals.

In the simulation of satellite clock deviation and orbit error, a set of random numbers obeying (0, 1) Gaussian distribution is generated and assigned to each navigation satellite, and during the simulation period, this random number is used as the amplitude of the sinusoidal characteristic function of the satellite clock deviation and orbit error; as for the receiver thermal noise, the mean value is set to 0 and the amplitude is calculated using the simulation of Equation (2). The orbital period of the satellite is about half a solar day [26], and in the high-orbit environment, only satellite signals from the other side of the earth can be received, so Figure 7 gives the measurement errors within 0.25 d and the corresponding sequence of transmitter angle changes in the G01, G02, G03, and G06 satellites as an example.

In order to better analyze the signal characteristics of the above four satellites, we take the signals with emission angles less than 21.5° as the main flap signals, and the signals with emission angles greater than 21.5° as the side flap signals. After removing the nulls, we calculate the root mean square error (RMSE) of the main and side flap signals of G01, G02, G03, and G06 satellites, respectively, and the calculation results are shown in

Table 2. RMSE of the main and side valve signals of G01, G02, G03, and G06 satellites.

Navigation Satellite	G01	G02	G03	G06
Main valve signal RMSE/m	—	6.38	—	6.56
Side valve signal RMSE/m	21.75	22.39	22.58	20.89

.

Combined with Figure 7 and Table 2, it can be analyzed that under the simulation conditions of this paper, the high-orbit receiver cannot receive the main flap signals of G01 and G03 satellites, and the signals of G02 and G06 satellites can only receive part of the main flap signals due to the earth’s blocking. The RMS value of the pseudo-range measurement error of the main signal is close to 7 m, and the RMS value of the pseudo-range measurement error of the side signal is about 22 m, and the pseudo-range error of the main signal is about 1/3 of that of the side signal, which is more consistent with the measured data of ref. [18]. Figure 8 presents the results derived from the RMSE analysis of the main lobe and side lobe signals across all navigation satellites. The overall average RMSE for pseudo-range measurement errors is 20.25 m. By restricting the analysis to signals with emission angles below 21.5°, the average RMSE drops to around 6.73 m. Furthermore, the pseudo-range errors associated with the main lobe signals are significantly lower than those of the side lobe signals.

5.3. Analysis of GNSS Positioning Results

According to the neural network model construction and training method proposed in Section 4.2, we utilize the a priori feature information of the G06 satellite signal with relatively comprehensive error composition and the pseudo-range observation information to train the parameters of the BP neural network model, and we can derive the image of the loss function change during the training process and its local zoom-in diagram as shown in Figure 9.

In machine learning, the loss function is a metric for evaluating the difference between a model’s predictions and actual results [27]. In the training phase, the task of the model is to optimize the parameters to reduce the loss function value so as to improve the prediction accuracy. As can be seen in Figure 9, the model has a large change in the loss value in the first 20 iterations, and the loss value is not yet able to converge stably in the first 30 iterations of the model, after which the model predicts the loss value to be basically unchanged and stable at about 0.1. The accuracy and stability of the proposed neural network model in estimating pseudo-range observation errors are validated by the consistent improvement observed with increasing iteration counts.

This study aims to evaluate the performance of a hybrid weighting approach that combines a BP neural network-assisted system internal weighting method with the Helmert variance component estimation model. The proposed approach is applied to achieve optimal inter-system weighting among GPS, GLONASS, BDS, and Galileo constellations in high-orbit scenarios. The following two weighting models are set up for the comparative experiments: the equal-weighting model and the BP neural network-assisted Helmert post-test variance model. The experimental results are shown in Figure 10, Figure 11, Figure 12 and Figure 13, the horizontal axis is the simulation time (s), and the vertical axis is the receiver position error (m).

The localization errors of the two models in each direction are taken to obtain the RMS statistical values, and the comparison of the localization errors of the two models in each direction is plotted, as shown in Figure 14. From the analysis of Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, it can be seen that compared with the equal-weight model, the positioning accuracy of the BP neural network-assisted Helmert’s post-test covariance model is significantly improved, by 30.6% in the X-direction, 33.8% in the Y-direction, 31.6% in the Z-direction, and 32.8% in the 3D-direction, which proves the validity of the weighting scheme designed in this paper in the high orbital environment.

In order to better verify the effectiveness of the designed BP neural network-assisted Helmert post-test covariance model high-orbit localization method and the enhancement effect of each component of the model, this paper designs the ablation comparison experiments and calculates the statistical value of RMSE of each model, and the specific RMSE of each model and the effect of enhancement are shown in

Table 3. RMSE and lifting effect of four models in the 3D direction.

Model	RMSE/m				Enhancement Effect/%
	X	Y	Z	3D
EWM	42.75	112.03	82.34	145.45	—
CNRWM	37.07	96.41	72.57	126.23	13.2%
BPNNWM	31.05	79.89	61.58	105.54	27.4%
BHWM	29.65	74.13	56.35	97.72	32.8%

and Figure 15.

• Equal weighting model: a model for which the weighting matrix is a unit array;
• CNR weighting model: the weighting matrix is a model that is down-weighted according to the size of the CNR of the satellite signals of each channel;
• BP neural network-based weighting model: the weighting matrix is the model of down-weighting according to the size of the pseudo-range error predicted by the BP neural network;
• BP neural network-assisted Helmert post-test variance weighting model: the weighting matrix is a model that down-weights the pseudo-range error size predicted by the BP neural network within the system and designs the Helmert post-test variance model for weighting between the systems.

As can be seen from Table 3, the root mean square error of the weighted model positioning results based on BP neural network is 105.54 m, and the root mean square error of the BP neural network-assisted Helmert’s post-test covariance model can be up to 97.72 m, which is a significant improvement in positioning accuracy compared to the unweighted positioning algorithms with the positioning accuracy increased by 27.4% and 32.8%, respectively. Among them, the weighted model based on BP neural network improves 27.4% compared with the equal-weighted model, which is much higher than the weighted effect of 13.2% of the load-to-noise weighted model, which indicates that this paper adopts the BP neural network model to weight the pseudo-range observation values, which can obtain more reasonable observation values, and the effect of the improvement of the localization accuracy is remarkable.

6. Conclusions and Future Work

In this paper, we take the latest published GPS official navigation star launch antenna direction map, use STK and MATLAB simulation tools to analyze the signal characteristics of GF-4 satellite, and conduct comparison experiments on four models, namely, equal-weighted model, carrier-to-noise-ratio-weighted model, BP neural network-based weighted model, and BP neural network-assisted Helmert post-test covariance model, and come up with the following conclusions and future work:

1.

By adopting a high-sensitivity receiver with a carrier-to-noise ratio threshold of −185 dBW, the number of visible stars can be up to more than 15 and the DOP value can be as low as about 10 in the case of multi-navigation system-compatible reception, which can realize continuous orbiting with full orbital cycle, and proves the feasibility of orbiting for high-orbiting spacecraft.

2.

By analyzing the high-orbit GNSS signal error composition and its signal characteristics, a BP neural network-assisted Helmert post-test variance model weighted positioning algorithm for high-orbit GNSS is proposed through comparative experiments involving four distinct weighted models; the proposed algorithm has been proven to effectively manage the quality of GNSS observations in high-orbit environments. Furthermore, it significantly improves the precision and reliability of high-orbit spacecraft positioning.

3.

In this study, the designed algorithm incorporates key error sources, including receiver clock errors, satellite clock errors, ionospheric and tropospheric delays, multipath effects, and thermal noise. Nevertheless, the influence of other factors, such as solar radiation pressure, relativistic effects, and plasma layers beyond the ionosphere, is not addressed in the pseudo-range error modeling [28]. And it can be considered in the future to design the weighting algorithms of high-orbiting spacecraft that take into account more error sources. The research in this paper is an important reference for further exploration of GNSS applications for high-orbiting spacecraft.