Orbit Prediction Methods for ONEWEB Constellation

Abstract

This study aims to enhance Low Earth Orbit (LEO) satellite orbit prediction accuracy. We propose the Precise Orbit Determination with Optimized Perturbations (PODOP) method, considering Earth’s non-spherical gravity, atmospheric drag, etc., and a Long Short-Term Memory (LSTM)-based approach for orbital element time series. Validation shows that PODOP’s 10-day median error is 8.1 km (19% larger than Simplified General Perturbations (SGP4)’s 10.1 km) and LSTM’s 10-day median error is 5.3 km, outperforming SGP4 (48.5 km) and PODOP and improving constellation management and collision prevention.

1. Introduction

In the era of rapid aerospace development, Low Earth Orbit (LEO) satellite constellations, like the ONEWEB constellation, play a crucial role in various fields [1]. As of 2025, the ONEWEB constellation, with over 600 satellites in orbit, ranks second globally. Its satellites, orbiting at an about 1200 km altitude with an 87.9-degree inclination and evenly distributed across 12 planes (49 satellites per plane), offer wide coverage [2].

Accurate orbit prediction is essential for the ONEWEB constellation’s operation [3,4]. In the crowded LEO space, with over 23,000 debris pieces larger than 10 cm, precise prediction helps avoid satellite collisions [5]. For communication tasks, it ensures stable links, which are crucial for services like global internet access [6].

Traditional orbit prediction using Two-Line Elements (TLEs) has limitations [7]. TLE data, which employs the Simplified General Perturbations 4 (SGP4) model for satellite orbit description, provides an approximate account of satellite orbits. However, it overlooks complex physical factors, as the SGP4 model simplifies the real-world orbital dynamics by not fully considering all the intricate physical processes at play [8]. Over time, this leads to increasing prediction errors [9].

Artificial intelligence, especially the Long Short-Term Memory (LSTM) network, offers new solutions [10]. LSTM can better handle satellite orbit time-series data, overcoming the problems of traditional Recurrent Neural Networks [11].

Factors such as observation quality, dynamic model accuracy, and data errors affect orbit prediction performance [12,13]. Observation-related issues, like weather and equipment problems, impact TLE data quality [14]. The choice of dynamic models for perturbation factors and dealing with data outliers also matters.

Research on ONEWEB constellation orbit prediction has practical and academic value. Practically, it improves constellation operation and service quality. Academically, it enriches satellite orbit theory, providing a basis for future aerospace missions. This research aims to advance the field of satellite orbit prediction and aerospace technology.

This paper begins by introducing the orbital and positional distributions of the ONEWEB constellation. It then details orbit prediction methods: LSTM-driven orbit prediction, Precise Orbit Determination, and the Orbit Prediction (PODOP) method. Following this, the results of a comparison of the orbit prediction errors from the three methods are presented. Next, the factors affecting the performance of both PODOP and LSTM are analyzed. Finally, the conclusion summarizes the findings.

2. Methods

2.1. LSTM-Driven Orbit Prediction Using TLEs

The input to the LSTM network is the historical TLE orbital elements, denoted as $X={\left[x_1,x_2,\dots ,x_n\right]}^T$. Here, $x_i$ represents the TLE orbital element vector at the i-th time step, and $x_i={\left[n_i,e_i,i_i,{\mathsf{\Omega }}_i,{\omega }_i,M_{0i},B_i^{*}\right]}^T$ includes $n$ (mean motion, a value related to orbital period), $e$ (eccentricity defining orbit shape), $i$ (inclination indicating the angle between the orbital and reference planes), $\mathsf{\Omega }$ (Right Ascension of the Ascending Node locating the orbital plane), $\omega$ (argument of perigee determining the ellipse orientation in the orbital plane), $M_0$ (mean anomaly at epoch representing the satellite’s position at a reference time), and $B^{*}$ (drag coefficient factor related to atmospheric drag). Despite being mean values from TLE that do not represent the true position of the space object, these elements can be employed in the training of the LSTM model. The elements derived after training, which bear resemblance to those in TLEs, still necessitate the utilization of the SGP4 model for the calculation of the true position.

An LSTM unit consists of an input gate $i_t$, a forget gate $f_t$, an output gate $o_t$, and a memory cell $C_t$. The input gate $i_t=\mathsf{\sigma }(W_{ii}{x}_t+W_{hi}{h}_{t-1}+b_i$), where $W_{ii}$ and $W_{hi}$ are weight matrices for $x_t$ and $h_{t-1}$ respectively, $b_i$ is a bias vector, and $\mathsf{\sigma }$ is the sigmoid function that maps values to the range.

Forget gate:

$$f_t=\sigma \left(W_{if}{x}_t+W_{hf}{h}_{t-1}+b_f\right)$$

(1)

Output gate:

$$o_t=\sigma \left(W_{io}{x}_t+W_{ho}{h}_{t-1}+b_o\right)$$

(2)

Candidate memory cell:

$$\widetilde{C_t}=\mathit{tanh}\left(W_{ic}{x}_t+W_{hc}{h}_{t-1}+b_c\right)$$

(3)

Memory cell update:

$$C_t=f_t\odot C_{t-1}+i_t\odot \widetilde{C_t}$$

(4)

Output:

$$h_t=o_t\odot \mathit{tanh}\left(C_t\right)$$

(5)

where $\odot$ denotes element-wise multiplication.

The historical TLE orbital elements are fed into the LSTM network for training. The model parameters are adjusted by minimizing the loss function $L={\sum }_{t=1}^n|\widehat{x_t}-x_t{|}^2$, where $\widehat{x_t}$ is the predicted value and $x_t$ is the true value. After training, by inputting the latest historical TLE orbital elements, the model can predict the TLE orbital elements at future time steps.

To ensure the reproducibility and clarity of the LSTM-driven orbit prediction approach, specific parameters of the LSTM model are detailed as follows. The model architecture consists of five LSTM layers followed by three dense layers. The LSTM layers are designed to capture temporal dependencies in the TLE data, with each layer sequentially processing and extracting features from the input time series. The dense layers then perform nonlinear mapping on the outputs of the LSTM layers to generate the final predicted orbital elements.

The loss function employed for model training is the Mean Absolute Error (MAE), defined as $MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_i-\widetilde{y_i}\right|$, where $y_i$ is the true value and $\widetilde{y_i}$ is the predicted value. This loss function effectively minimizes the average absolute difference between predicted and true orbital elements, enhancing the model’s accuracy in capturing orbital trends.

For training stability and reproducibility, the deep learning model is initialized with a random seed of 3407. The dataset is split into training and testing sets using an 80%/20% ratio, ensuring that the model is evaluated on unseen data to assess its generalization capability. Additionally, the input to the LSTM model includes not only the orbital elements but also their changes relative to the initial time step, which are formatted to maintain consistency with TLE data structure, facilitating direct integration with the SGP4 model for subsequent position calculation.

2.2. PODOP

The positions $r_k$ of the satellite are calculated based on the TLE parameters using the SGP4 model. The observed positions $r_k^{obs}$ are obtained, where $k = 1, 2,\dots ,N$ ( $N$ is the number of TLE data points).

A dynamic equation of satellite motion is established taking into account the main forces such as the Earth’s gravity. In the Cartesian coordinate system, the motion equation is

$$\ddot{r}=-{\displaystyle \frac{GMr}{|r{|}^3}}+F_{pert}$$

(6)

where $G$ is the gravitational constant, $M$ is the mass of the Earth, $r$ is the satellite position vector, and $F_{pert}$ represents the perturbation forces (such as atmospheric drag, solar radiation pressure, etc., whose specific forms are not elaborated here). Given the initial state $r_0$ and $\dot{r_0}$, the positions $r_k^{pred}$ at different time steps can be predicted through numerical integration methods (such as the Runge–Kutta or Cowell method).

The error function $J$ is defined as the sum of the squared differences between the observed positions and the positions predicted by the dynamic model:

$$J={{\displaystyle \sum }}_{k=1}^N{\left(r_k^{obs}-r_k^{pred}\right)}^T\left(r_k^{obs}-r_k^{pred}\right)$$

(7)

The Jacobian matrix H is derived by taking partial derivatives of the error function J concerning the parameters $\mathit{x}$. These parameters $\mathit{x}$ include the initial position components $p_i,p_j,p_k$ and the initial velocity components $v_i,v_j,v_k$. Each element $H_{ij}$ of the matrix is the partial derivative of J concerning the jth parameter $x_j$ evaluated at time $t_j$. The normal equation ${\mathit{H}}^{\mathit{T}}\mathit{H}\mathbf{\Delta }\mathit{x}=-{\mathit{H}}^{\mathit{T}}b$ is constructed, where ${\mathit{b}}_{\mathit{i}}={\mathit{r}}_{\mathit{i}}^{\mathit{o}\mathit{b}\mathit{s}}-{\mathit{r}}_{\mathit{i}}^{\mathit{p}\mathit{r}\mathit{e}\mathit{d}} \mathrm{a}\mathrm{n}\mathrm{d} \mathbf{\Delta }\mathit{x}$ is the parameter correction. By iteratively solving the normal equation $\mathbf{\Delta }\mathit{x}=-{\left({\mathit{H}}^{\mathit{T}}\mathit{H}\right)}^{-1}{\mathit{H}}^{\mathit{T}}b$, the parameters are continuously updated to minimize the error function $J$. Subsequently, following the acquisition of the corrected orbital parameters, numerical integration can be executed for orbit prediction. The accuracy of the predicted orbit attained by this means is highly likely to outperform that of the TLE. The enhanced dynamic performance, which may be superior to that of the SGP4 model, stems from the utilization of a more comprehensive mechanical model.

To ensure the accuracy and robustness of the PODOP method, the force model incorporates a comprehensive set of perturbations tailored to the dynamics of LEO satellites. The Earth’s gravitational field is modeled using the EGM2008 model up to degree and order 60 × 60, accounting for non-spherical gravitational effects that induce periodic orbital perturbations. Atmospheric drag, a dominant perturbation for LEO objects, is computed using the NRLMSIS model, which accounts for solar and geomagnetic activity to better capture variations in atmospheric density—critical for satellites orbiting at an ~1200 km altitude like those in the ONEWEB constellation. Additionally, third-body gravitational forces from the Sun and Moon are included using the DE406 ephemeris to account for their gravitational influences on orbital motion. Solar radiation pressure is modeled with a cannonball model, considering the satellite’s area-to-mass ratio to estimate the pressure-induced acceleration. Minor perturbations such as solid Earth tides are also integrated to further refine the dynamic model.

For numerical integration, the Cowell method is employed with an 11th-order algorithm and a fixed step size of 60 s, ensuring the efficient and precise propagation of orbital elements through the computed force field. The iterative estimation process adopts a convergence criterion where iterations terminate when the position difference between two consecutive iterations is less than 1 m. In terms of data quality control, outliers are identified using a 3σ criterion, where residuals exceeding three times the standard deviation of orbit determination residuals are classified as gross errors and excluded from the estimation process. This multi-faceted force model, combined with rigorous numerical integration and data screening, ensures that the PODOP method accurately reflects the complex orbital mechanics, enabling more precise orbit determination and prediction compared to simplified models like SGP4.

3. Orbital and Positional Distributions of the ONEWEB Constellation

The ONEWEB constellation aims to deploy hundreds of LEO satellites. As of 2025, it holds the second-largest satellite constellation globally, with over 600 satellites in orbit [15]. These satellites orbit at around a 1200 km altitude, with an 87.9-degree inclination, and are spread across 12 orbital planes, with 49 satellites in each plane. This configuration enables global coverage, covering an area of about 134% of the Earth’s surface. Each satellite has an orbit period of approximately 109.3 min and can stare at a specific area for around 2.2 min, ensuring continuous service to that area within a certain time frame.

ONEWEB satellites possess unique technical features. Each weighs 147.5 kg, has a 5-year designed lifespan, and measures approximately 1 m × 1 m × 1.3 m. Powered by two solar panels, they use a Busek BHT-350 Hall effect thruster for orbit maintenance and attitude control [1].

Figure 1 depicts the distribution of the Right Ascension of the Ascending Node (RAAN) of 655 ONEWEB satellites in October 2024. Presented as scatter plots, the data points are ordered by RAAN values, clearly showing the satellite distribution across different RAAN planes. The satellites are relatively evenly distributed among the 12 RAAN planes, demonstrating the regularity of the constellation’s orbital layout design. This layout is conducive to achieving broader and more stable coverage. The few satellites not on these 12 RAAN planes might be due to not yet reaching their intended orbits.

Taking ONEWEB satellite 44058 as an example, Figure 2 presents the trends of its orbital elements in 2024. This set of graphs presents the trends of multiple key orbital elements of ONEWEB satellite 44058 in 2024, which is of great significance for studying its orbital characteristics and operational patterns. The abscissa of all sub-graphs is the day of the year in 2024, representing the time progression, while the ordinate corresponds to the measurement units of different orbital elements.

The upper-left sub-graph shows the variation in the RAAN. The degree-change data exhibit two distinct horizontal segments, indicating that the RAAN of this satellite underwent significant changes during 2024. This phenomenon occurs because the angular value range of RAAN is from 0 to 360 degrees. When the angle crosses 360 degrees, it appears as a “discontinuity” on the graph. In subsequent calculations, this “discontinuity” needs to be processed to restore it to a continuous curve.

The upper-right sub-graph depicts the argument of perigee. Its degree curve fluctuates periodically. Starting at approximately 100 degrees at the beginning of the year, it drops to around 50 degrees and then rises, reaching a peak in the middle of the year and then decreasing again, reflecting the dynamic changes of this element throughout the year.

The middle-left sub-graph records the mean anomaly. Its degree curve has a distinct periodicity, with values fluctuating between 220 and 300 degrees, reflecting the periodic variation in the satellite’s relative position in its orbit.

The middle-right sub-graph shows the semi-major axis. Its length (in kilometers) fluctuates slightly linearly around 7576 km for most of the time, with some deviant points, indicating that the semi-major axis is generally stable but is subject to minor changes due to certain factors.

The lower-left sub-graph presents the eccentricity. The curve has a period of approximately 100–150 days, alternating between peaks and troughs, indicating that the eccentricity of the satellite’s orbit changes dynamically and regularly throughout the year, reflecting the periodic adjustment of the orbit shape.

The lower-right sub-graph records the inclination. Its degree fluctuates around 87.9 degrees, being relatively stable at the beginning and end of the year but with significant changes in the middle of the year, reflecting the dynamic adjustment of the angle between the satellite’s orbital plane and the reference plane throughout the year.

In addition, except for the RAAN, there are deviant points in the other orbital elements. These deviant points may be gross errors and need to be removed by pre-processing algorithms in subsequent calculations.

Figure 3 depicts the 2024 variation in the $B^{*}$ (drag coefficient factor related to atmospheric drag) value for ONEWEB satellite NORAD ID 44058. The x-axis represents the day of the year in 2024, and the y-axis shows the $B^{*}$ value. Mostly, the $B^{*}$ value fluctuates slightly around 0, indicating a relatively stable state. Notably, in the six orbital element graphs and this $B^{*}$ graph, data points deviating from regular curves likely correspond to abnormal TLEs. Such abnormal TLE data negatively impacts orbit prediction performance. Both the PODOP and the LSTM model may experience reduced accuracy and reliability in orbit prediction because of this.

Figure 4 illustrates the orbital coordinate distribution of ONEWEB satellite NORAD ID 44058 from 1 October 2024 to the next seven days. The x-axis, labeled “Duration/day”, shows the time difference in days from the start on 1 October 2024, ranging from 0 to 7 days, depicting the satellite’s orbit change during this period.

The y-axis is split into two parts. The left y-axis, in kilometers (km), measures the satellite’s spatial position. Different-colored lines represent coordinates in three axis directions: the gray line for the X coordinate, the red line for the Y coordinate, and the blue line for the Z coordinate. The X coordinate fluctuates widely between −9000 km and 9000 km. The Y coordinate has relatively smaller fluctuations, mainly between −3000 km and 3000 km. The Z coordinate’s fluctuation range is similar to the X coordinate’s, varying between −9000 km and 9000 km. This reveals dynamic changes in the satellite’s three-dimensional position and significant periodicity within seven days.

The right y-axis is also labeled “Duration/day”, indicating the time difference between the moment of the TLE data used for corresponding coordinates and the initial moment (1 October 2024). The scale on the right y-axis corresponding to the short horizontal line “T” of black scatter points shows the TLE data time points for each coordinate data. These time points are unevenly distributed, suggesting that TLE data were selected at different moments to determine the satellite’s orbital coordinates.

4. Results

The TLE data for 655 ONEWEB constellations were acquired in 2024. A particular date (e.g., 8 October 2024) was designated as a time node. All satellites are used for LSTM model training, but future data is used for validation, and such future data is not involved in model computation at all. As per the research design and following the standard practice of separating data for evaluation purposes, TLE data after this node were solely used to generate the true orbit. This true orbit serves as a reference for evaluating the performance of orbit prediction methods.

The true orbit, computed by integrating TLE data with the SGP4 model, represented the satellite’s position and velocity as a time series with a 5 min interval. This served as the reference for evaluating method performance.

For error calculation, to objectively assess different methods, a set of predicted orbits was derived using the TLE data closest to the pre-node time with the SGP4 model. The LSTM method utilized pre-node historical TLE data for calculation, output future-time TLE data, and then combined with the SGP4 model to obtain the predicted orbit. The PODOP method operated on pre-node historical TLE data and directly obtained the predicted orbit. These predicted orbits were compared with the true orbit to calculate prediction errors.

The reference orbit is derived from TLE data. Since TLE data is generally considered to have errors at the kilometer level, we assess the performance of the method by analyzing long-term orbital propagation errors. This approach takes into account two aspects: on the one hand, not all satellites can provide orbits with meter-level or even millimeter-level accuracy to all users; on the other hand, it is commonly believed that a method with small long-term orbital prediction errors, such as those for a week, typically does not yield excessively large errors in the short term, say within 1 days.

Errors were presented in multiple dimensions, including along-track, normal, and radial errors and position distance differences. The error for a specific prediction day was calculated as the root mean square error of the differences between the true and predicted orbit positions for all time series on that day, enabling a comprehensive and accurate evaluation of each method’s orbit prediction performance at different time scales.

Figure 5 displays the orbit prediction error graphs of four ONEWEB satellites (numbered 44058, 44059, 44060, and 45132) using three methods: SGP4, PODOP, and LSTM. The x-axis, labeled “Prediction days,” ranges from 0 to 11 days, while the y-axis, “Error/km,” indicates the error in kilometers. In the figure, blue squares represent SGP4 method errors, red circles denote PODOP method errors, and green triangles signify LSTM method errors.

As shown, in the early prediction stage (first 2–3 days), all three methods have small and similar orbit prediction errors. As the number of prediction days increases, the SGP4 method’s error grows more notably, especially in the later prediction stage (after 8 days), with a rapid increase. The PODOP method’s error also rises throughout the prediction period, but more gradually. The LSTM method’s error grows relatively steadily for most of the prediction time. Compared to the SGP4 method, after a certain number of prediction days (e.g., around the 6th day for satellite 44058), the LSTM method’s error advantage becomes evident and stays at a relatively low level.

In Figure 6, at each prediction time point, there is a box plot for each prediction method. These box plots are based on calculations of 655 ONEWEB satellite data. The box in a box plot represents the interquartile range (from the 25th to 75th percentile), showing the middle 50% data concentration. The black line inside is the median, indicating the data’s intermediate level. The whiskers above and below the box show the data’s approximate distribution range, though there may be outliers outside, generally reflecting data dispersion.

From

Table 1. Statistics of 10-day orbit prediction errors for the three methods.

10	Method	Mean Errors/m	Improved Rate	Median Errors/m	Improved Rate
1	SGP4	1309		527
	PODOP	2487	−90%	1464	−178%
	LSTM	2664	−104%	1859	−253%
2	SGP4	2097		1036
	PODOP	3540	−69%	2182	−111%
	LSTM	3033	−45%	2544	−146%
3	SGP4	3133		1285
	PODOP	4770	−52%	2943	−129%
	LSTM	3073	2%	2366	−84%
4	SGP4	4515		1407
	PODOP	6744	−49%	3811	−171%
	LSTM	3331	26%	2574	−83%
5	SGP4	7089		1790
	PODOP	9268	−31%	4334	−142%
	LSTM	3458	51%	2616	−46%
6	SGP4	10,234		2866
	PODOP	12,055	−18%	4970	−73%
	LSTM	3956	61%	3043	−6%
7	SGP4	13,736		4187
	PODOP	15,129	−10%	5918	−41%
	LSTM	5484	60%	3387	19%
8	SGP4	18,444		5865
	PODOP	17,169	7%	6522	−11%
	LSTM	6795	63%	3963	32%
9	SGP4	27,249		7705
	PODOP	23,981	12%	7424	4%
	LSTM	8832	68%	5248	32%
10	SGP4	36,319		10,097
	PODOP	31,045	15%	8129	19%
	LSTM	10,930	70%	5290	48%

, we can obtain a wealth of information about the 10-day orbit prediction errors of the three methods.

First, looking at the SGP4 method, its mean errors show significant variation across the 10 days. On day 1, the mean error is 1309 m, and it increases to 36319 on day 10, indicating a tendency for prediction errors to grow as the prediction period extends. The median errors of SGP4 also follow a similar upward trend, from 527 m on day 1 to 10097 m on day 10.

For the PODOP method, the mean errors present a more erratic pattern. On day 1, the mean error is 2487 m, which is much higher than that of SGP4. Then, it fluctuates, with values like 3540 m on day 2, 4770 m on day 3, and 31045 m on day 10. The median errors of PODOP are smaller, such as 1464 m on day 1, 2182 m on day 2, and 8129 m on day 10.

The LSTM method demonstrates more favorable performance in terms of mean errors. On day 1, the mean error is 2664 m, comparable to PODOP but higher than SGP4. However, as the days progress, LSTM’s mean errors increase at a slower rate compared to SGP4 and PODOP. By day 10, the mean error of LSTM is 10930 m, which is lower than both SGP4 (36319 m) and PODOP (31045 m) on that day. The median errors of LSTM also show a relatively better trend. On day 1, the median error is 1859 m, higher than SGP4 but lower than PODOP. As the prediction days increase, LSTM’s median errors remain more controlled, with 5290 m on day 10, which is significantly lower than SGP4′s 10097 m and PODOP’s 8127 m.

Additionally, the improved rate column provides insights into how each method performs relative to others. For example, when comparing LSTM with SGP4 or PODOP on different days, there are positive improved rates, suggesting that LSTM outperforms the other two methods in many cases. The improved rates for LSTM are often positive and relatively large, especially as the prediction days increase, indicating that LSTM becomes increasingly advantageous compared to SGP4 and PODOP over longer prediction periods.

Notably, the true orbit here is from TLE data combined with the SGP4 model. In a short time (e.g., 5 days), due to TLE data accuracy and SGP4 model simplification of orbital perturbations, the true orbit has non-negligible errors. So, in the short term, the SGP4 algorithm seems best, as it has a small error compared to the true orbit, but this can be misleading. Theoretically, for longer-term orbit predictions (e.g., 7 days and above), analyzing prediction errors can reduce the impact of true-orbit errors on method evaluation, more accurately revealing each method’s capabilities in orbit prediction.

5. Discussion

5.1. Factors Affecting the Performance of PODOP

The performance of the PODOP method is closely related to several key aspects.

5.1.1. Data Distribution

The PODOP method takes as input the position information generated from multiple TLEs. It has been determined that using 10-day TLE data provides optimal orbit determination results [16]. A 10-day time window effectively balances data volume and quality, thereby preventing the loss of crucial information that could occur with scant data and avoiding the introduction of excess noise associated with overly abundant data. Shorter intervals, such as 3 days, are likely to overlook long-term trends, while longer ones, like 15 days, may introduce noise and obsolete information. It has been demonstrated that the time distribution of TLE data significantly affects orbit determination accuracy and stability [17]. Given that satellites experience distinct perturbations during ascending and descending orbits, resulting in orbital characteristic disparities, during ascending orbits, short-span and uniformly-distributed TLE data are more effective in capturing orbit changes. In contrast, during descending orbits, extending data duration and ensuring data continuity are beneficial for accurate orbit parameter determination.

5.1.2. Dynamic Models

The precision of the dynamic model is crucial for PODOP performance. Satellites in orbit are affected by complex perturbations like the Earth’s non-spherical gravitational field, atmospheric drag, solar radiation pressure, and the gravitational forces of the Sun and Moon [18].

Earth’s gravitational field: The non-spherical part causes periodic orbit perturbations. Higher-order models like EGM-2008 improve orbit determination accuracy, especially in LEO [19].

Atmospheric drag: A major perturbation for LEO satellites is that its accurate modeling is difficult due to atmospheric density variations [20]. Models like NRLMSIS 2.0 that account for solar and geomagnetic activity enhance orbit prediction accuracy during high solar activity years [21].

5.1.3. Gross Error Detection

TLE data may contain gross errors due to measurement errors, sensor failures, etc. These errors, if not removed, affect PODOP performance [22]. As PODOP is based on least squares, gross errors in observations lead to solution deviations. The key to gross error detection is setting a reasonable threshold. A dynamic adaptive threshold method is used here:

(1)

Determine an appropriate time window based on satellite orbit element change laws.

(2)

Calculate position differences between consecutive TLEs within the window.

(3)

Compute the root mean square error (RMS) of these differences.

(4)

Set two or three times the RMS as the threshold for gross error judgment.

The detection process includes initial detection, sliding window detection, and iterative detection until no TLE is removed.

5.1.4. Maneuver Detection and Processing

During gross error detection, satellite maneuvers, which can cause sudden orbit element changes and be misjudged as gross errors, need to be considered [23]. Satellite operating status and mission plans should be combined to determine if a maneuver has occurred [10]. If so, TLE data around the maneuver should be specially processed, such as introducing a maneuver model during orbit determination or removing maneuver-period data during pre-processing.

5.2. Factors Affecting the Performance of LSTM in Orbit Prediction

5.2.1. Gross Errors

Ensuring data quality and LSTM performance requires identifying and removing gross errors in orbital element time series. Orbital element data may be compromised during collection and processing by factors like instrument limitations, long-term sensor malfunctions, and space environment anomalies such as solar flares and geomagnetic storms. Unaddressed, these errors hinder the LSTM model from accurately learning orbital characteristics, reducing prediction accuracy.

A multi-faceted approach is needed for gross error identification and removal. In addition to using statistical characteristics like mean, variance, and standard deviation to set thresholds for screening abnormal points, gross error detection can be performed via a sliding window: within the window, orbital elements are fit with a linear function, and those with a fitting root mean square error greater than three times the fitting root mean square error are identified as gross errors. Due to the dynamic nature of orbital elements, simple statistical thresholds are often insufficient. An orbital dynamics model can help estimate the range of orbital element changes under different conditions, flagging data points outside this range as potential gross errors. Additionally, considering the time correlation in orbital element time series is crucial. Adjacent data points are closely related, so gross error identification should analyze not only individual points but also their relationships with neighboring ones. A data point that contradicts the trend and is outside the reasonable range is likely a gross error.

5.2.2. Irregularity of Orbital Elements

The irregularity of orbital elements, caused by complex factors such as the Earth’s inhomogeneous gravitational field, atmospheric drag, and solar radiation pressure, poses a significant challenge to the LSTM algorithm’s performance. These irregularities, manifested as data mutations, larger fluctuation amplitudes, and weak periodic changes, impede the LSTM model’s ability to extract data patterns. For example, sudden element mutations can cause the model to mis-predict, and irregular fluctuations can increase training time and computational cost and reduce the model’s generalization ability.

To counteract the negative impacts of orbital element irregularity on the LSTM algorithm, several pre-processing techniques can be employed. Data smoothing and filtering algorithms can reduce fluctuations and remove noise. Additionally, optimizing the LSTM model’s structure and parameters, such as introducing an attention mechanism [24], can enhance its adaptability to irregular data, enabling it to better focus on key features and trends and improve prediction accuracy.

5.2.3. The Impact of LSTM Main Parameter Settings on Errors

Figure 7 illustrates how different LSTM parameter settings affect prediction errors across X, Y, and Z axes and the overall position.

For layers, error declines as layers increase from 1 to 3, then stabilizes at 4 layers, suggesting that 3–4 layers balance complexity and performance. In terms of neurons per layer, reducing neurons from 256 to 64 lowers error, indicating that excessive neurons may cause overfitting, with 64–128 neurons being optimal. Regarding sliding window size, error decreases when the size rises from 10 to 15 and then plateaus, meaning a moderately sized window (15–20) captures temporal dependencies well without introducing noise.

For steps per epoch, error remains low until three steps, then spikes sharply at four steps, implying that too many steps disrupt training stability. With batch size, error is relatively stable across 32–128, showing that batch size has a minor effect within this range. As for the activation function, ReLU leads to the lowest error, while Softsign and LeakyReLU perform worse, proving ReLU’s superiority in this orbital prediction task.

For epochs, error stays low and stable across 100–250 epochs, indicating that the model converges early and additional epochs do not improve accuracy.

In summary, parameters like layers, neurons per layer, and sliding window size significantly influence error, while batch size and epochs have less impact. Selecting ReLU as the activation function and optimizing layer-related parameters can effectively reduce prediction errors.

5.3. Discussion on the Improvement of Orbital Accuracy for ONEWEB Practical Applications

The enhanced orbital accuracy, as demonstrated in Section 4, brings about substantial practical implications for ONEWEB.

In managing the ONEWEB constellation, precise orbit knowledge is fundamental. For instance, with the improved accuracy, the mean error of orbit prediction at 7 days for LSTM is 5.5 km, compared to 13.7 km for SGP4. This allows for more accurate station-keeping maneuvers. When satellites need to maintain their relative positions within the constellation for optimal coverage and communication link establishment, the reduced error ensures that fewer corrective maneuvers are required. Considering a constellation of 655 satellites like ONEWEB, each unnecessary maneuver saved can lead to significant savings in fuel, which is crucial for the long-term sustainability of the constellation.

Collision prevention is another area where improved orbital accuracy is vital. Space debris and the increasing number of satellites in Low Earth Orbit (LEO) pose a real threat. By day 10, the median error of SGP4 is 10097 km, while that of LSTM is 5290 km. With LSTM’s more accurate predictions, ONEWEB can better predict potential close approaches with other space objects. For example, if a debris object has an estimated position error of a certain range, the more precise orbital data from LSTM enables a more accurate assessment of the collision risk. This allows for timely evasive maneuvers, protecting the valuable ONEWEB satellites and reducing the potential creation of additional space debris.

In optimizing communication systems, accurate orbits play a pivotal role. ONEWEB’s satellites provide communication services, and the quality of these services depends on the precise positioning of the satellites. The reduced errors in orbit prediction, such as the lower mean and median errors of LSTM throughout the 10-day prediction period compared to SGP4 and PODOP, ensure that the satellites are in the expected positions for maintaining stable communication links with ground stations or user terminals. This leads to improved signal strength and fewer drop-outs. In areas where the constellation aims to provide high-speed internet access, like remote regions, the enhanced orbital accuracy means more consistent service delivery. For example, in a rural area relying on ONEWEB for broadband, the better-predicted satellite positions can keep the connection stable, enabling seamless video conferencing, online learning, and other data-intensive activities.

The improvement in orbital accuracy, as shown by the data in Section 4, directly enhances the reliability and efficiency of ONEWEB’s constellation operations, collision avoidance capabilities, and communication services. It is a fundamental factor for the successful deployment and long-term operation of the ONEWEB system.

5.4. Performance Analysis of High-Precision Reference Orbits

Figure 8 presents the orbit prediction errors of the SGP4, PODOP, and LSTM methods, with the high-precision reference orbit derived from the satellite APOD-A (NORAD ID: 40903). This satellite, equipped with a Global Positioning System (GPS), enables the calculation of precision orbits with meter-level errors, serving as a reliable benchmark for evaluating the three methods.

Over the 6-day prediction period, the SGP4 method shows a steep increase in error, reaching approximately 6 km by day 6. This indicates that the simplified perturbation model of SGP4 is insufficient for long-term, accurate orbit prediction, especially when compared to the high-precision reference. The PODOP method performs better, with errors rising to around 2 km on day 6. However, it still lags behind the precision of the reference orbit. In contrast, the LSTM method demonstrates remarkable performance. Its prediction errors remain at a very low level throughout the period, closely aligning with the meter-level accuracy of the GPS-derived reference orbit.

This significant difference highlights the advantage of LSTM in capturing the complex temporal dependencies and dynamic patterns of orbital elements. While traditional methods like SGP4 and PODOP rely on physical models with inherent simplifications, LSTM can learn from historical orbital data to make more accurate predictions. The results confirm that for applications requiring high-precision orbit prediction, such as satellite collision avoidance and precise constellation management, the LSTM method, when validated against high-precision reference orbits like those from GPS-equipped satellites, offers a more reliable solution.

5.5. Time-Consuming for the Three Methods

The computational time consumption varies significantly among the SGP4, PODOP, and LSTM methods. The SGP4 method, relying on simplified orbital perturbation models, is extremely fast, with a computation time of approximately several microseconds. This makes it suitable for scenarios requiring rapid but less precise orbit predictions.

The PODOP method, which incorporates more comprehensive physical models to account for various perturbations, generally takes about several tens of seconds to several minutes. The increased time is due to the complex calculations involved in modeling multiple force effects.

In contrast, the LSTM method, as a deep learning-based approach, has a computation time ranging from several seconds to several tens of seconds. Although slower than SGP4, its time consumption is acceptable considering the significant improvement in prediction accuracy it offers, especially for long-term and high-precision orbit-prediction tasks.

6. Conclusions

This research has successfully developed and validated the PODOP and LSTM methods for LEO satellite orbit prediction. The comprehensive force model in the PODOP method effectively accounts for multiple perturbations, leading to more accurate predictions compared to simplified models like SGP4. The LSTM method, leveraging the power of deep learning, can better capture the complex temporal characteristics of orbital elements, resulting in lower errors and more stable performance.

For example, throughout the 1–10-day prediction period, LSTM consistently shows lower median errors and smaller interquartile ranges. The improved orbital accuracy has practical implications for ONEWEB and similar satellite constellations. It enables more precise satellite constellation management, reduces the risk of collisions, and optimizes communication services.

However, there are still areas for improvement. The computational cost of the PODOP method can be further reduced, and the LSTM model can be optimized to handle more complex scenarios. Future research could focus on integrating real-time data to further enhance prediction accuracy and develop more efficient algorithms to balance performance and computational resources. Overall, this study paves the way for more reliable and accurate LEO satellite orbit prediction in the future.