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### Session F3c: Beyond GNSS: Emerging Trends in LEO-Based and Terrestrial Signals of Opportunity for PNT

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**LSTM-Driven Prediction of Orbital Parameters for Accurate LEO Opportunistic Navigation**

*Md Sahat Mahmud, Zihong Zhou and Bing Xu, Department of Aeronautical and Aviation Engineering, The Hong Kong Polytechnic University*

**Date/Time:** Thursday, Sep. 19, 11:26 a.m.

Peer Reviewed

Background / Objective:

The use of Low Earth Orbit (LEO) satellite constellations (Iridium, Starlink, Orbcomm, etc.) for Position, Navigation and Timing (PNT) application has become popular among Global Navigation Satellite System (GNSS) researchers in recent years due to advantages such as higher signal strength, enhanced security and potential to solve multipath issue with their rapid change in geometry (Lawrence et al., 2017). Due to weak signal strength and high vulnerability to interference, GNSS satellites fail to provide accurate positioning results in places such as urban and indoor environments. For such cases, LEO satellites are being considered for opportunistic navigation (Reid et al. 2020). Unlike GNSS constellations, most LEO satellite constellation operators do not share their ephemeris data, which leaves publicly available Two-Line Elements (TLE) data from North American Aerospace Defense Command (NORAD) as the only information on satellite orbital parameters to propagate LEO satellites’ orbit for opportunistic positioning. However, the TLE data itself can be inaccurate sometimes, and the error in orbit propagation with dynamic models like Simplified General Perturbations 4 (SGP4) increases over time if the TLE data is not updated frequently (Kassas et al., 2019). A study by Jiang et al. (2022) shows that, if the estimated satellite position from TLE is deviated from the true line-of-sight, the positioning accuracy can have an error of at least 1.5 km. This error is large in terms of PNT as the current state-of-the-art GNSS positioning algorithms can achieve an accuracy within 10 m of root mean square errors (Zhong & Groves, 2022; Ng et al. 2023). To solve this problem, there have been numerous studies for error correction of the TLE elements or error modeling of the orbital parameters using dynamic models, deep learning, Satellite Laser Ranging (SLR) approaches, etc. (Bizalion et al., 2022; Bennett et al., 2012). However, there has been very little research to utilize the capabilities of Recurrent Neural Network (RNN) models like Long Short-Term Memory (LSTM) to predict future orbital parameters for a satellite and use them to estimate satellite position and velocity. The LSTM network uses additional gates to control the vanishing gradient problem. This allows the network to learn long-term dependencies of the input data, which is particularly useful for orbital parameter prediction due to their time-dependent characteristics (Osama et al., 2022; Ren et al. 2019).

This research proposes an RNN-based orbital element prediction algorithm that uses the LSTM network. The algorithm learns from the historical TLE data of a LEO satellite and predicts the future values of orbital elements of the specific epoch for which the satellite position and velocity are to be estimated. The predicted values of orbital parameters can then be used by the SGP4 propagator to estimate the values of satellite position and velocity. This process will run until the next TLE becomes available. This approach is expected to solve the problem of error propagation resulting in a better estimation of satellite position and velocity, which will eventually provide more accurate positioning results for LEO opportunistic navigation.

Methodology / Key innovative steps:

After acquiring the satellite TLE data, the proposed model pre-processes the data for training the LSTM model. Pre-processing includes the following steps - Conversion of the epoch data to ISO 8601 format, taking sin values of RAAN (?) as sudden jumps may cause the problem of discontinuity and normalization of the data since the orbital elements are measured on different scales. For the normalization purpose min-max normalization is used to map all values to z in a range between 0 and 1, where z is computed as (x – min(x))/(max(x)-min(x)).

As the final format for training data, the model uses the following orbital elements and time features decoded from the TLE dataset:

1) Date-time decoded from the epoch data.

2) Time difference between the epochs of two consecutive TLE data.

3) Right Ascension of Ascending Node (?).

4) Mean motion (n).

5) Mean anomaly (M).

6) Eccentricity (e).

7) Argument of perigee (?).

8) Inclination (i).

As the ordering of data points can be crucial for orbital mechanics, the time difference between the epochs of two consecutive TLE data is included in the input data as it helps the neural network to understand the sequence of observations. LSTM networks, in particular, are designed to capture sequential dependencies in data and using time difference helps in improving model accuracy, which has been observed during the experiments on the model.

The training dataset for the proposed model is dynamic and keeps changing depending on the target value of the RNN to achieve higher accuracy and optimization, which is an innovative approach in this model. For example, if RAAN is the target value, the features selected as input candidates for the RNN would be date-time (yyyy-mm-dd hh:mm:ss), time difference between the epochs of two consecutive TLE data and inclination angle (i). These combinations of input features are not arbitrary, rather obtained through numerous experiments on the proposed model.

Instead of implementing conventional hyper-parameter tuning, the model uses Bayesian Search technique with Tree Structured Parzen Estimation (TPE) algorithm, which is also a novel idea in time series forecasting of orbital parameters. This algorithm dynamically models the objective function x?=argmaxx ? x f(x) by making a series of evaluations i.e. x1, …, xT of f such that the optimum of f is found in the quickest time (Nguyen, 2019). The RNN also uses early stopping to avoid over-fitting of the training data and decreases the learning rate incrementally as it was found to be a useful technique by Haidar-Ahmad et al. (2022).

Currently, the proposed model is configured to predict future values of orbital parameters based on test features which can be used by SGP4 to predict LEO’s position and velocity. Further work will be done to predict the values of TLE elements for the specific epoch for which the satellite position and velocity is to be estimated. The authors plan to structure the model in such a way that the proposed model will predict those orbital parameters first that are not significantly affected by any other parameters than time and carry on predicting other parameters based on predicted parameters.

Preliminary Findings:

Preliminary results indicate the promising capabilities of the proposed Neural Network model. Initial experiments have been conducted using the TLE data set of IRIDIUM NEXT 160, covering the period from 01/01/2020 to 01/02/2024. The proposed model exhibits strong performance in capturing patterns associated with the target parameters. The results show that the RNN model is capable of predicting precise future values of TLE elements. This indicates that the algorithm can be used to predict the orbital parameters for future epochs which can be used to estimate satellite position and velocity with higher precision, needed for achieving better results in LEO opportunistic navigation.

Significance:

The SGP4 propagator estimates satellite position and velocity with an error of 3 km for LEO satellites if the TLE is not updated for a long time (Kassas et al., 2019). This error becomes significant as the TLE gets outdated (Lazzaro & Bettanini., 2022), causing significant errors in LEO opportunistic positioning. A lot of research has been conducted to correct the errors in TLE data, but the problem of error propagation over time has been overlooked in the GNSS community for a long time. This research provides not only an algorithm to predict the orbital parameters for specific future epochs but will also motivate future work to solve the problem of error propagation of satellite position and velocity for LEO opportunistic positioning. The LSTM networks are very useful for time series predictions like orbital parameter prediction due to their ability to capture temporal and long-range dependencies in sequential data unlike conventional neural networks, and this advantage is utilized by the proposed approach. Moreover, the proposed algorithm is written in Python which is very easy to integrate with existing orbit propagation algorithms like SGP4, making it convenient for researchers who plan to further investigate the problem of error propagation.

Innovation:

Unlike most research, where Deep Learning (DL) techniques have been used for error correction or error modeling (Osama et al., 2022; Ren et al. 2019), this study adopts an approach that predicts orbital elements for a specific epoch in the future. These predicted values will later be used directly to estimate satellite position and velocity using an orbit propagation model such as SGP4. Moreover, rather than training the proposed model on the same set of input parameters every time, the training dataset has been made dynamic. This approach helps the model to learn dynamically and provide better accuracy and optimization, as deep learning techniques usually take a long time to execute. The usage of Bayesian search for optimized hyper-parameter using TPE is also a novel idea in orbital element prediction algorithms. This approach improves the accuracy of the algorithm by using the most optimized hyperparameters.

References:

Bennett, J., Sang, J., Smith, C., & Zhang, K. (2012). Improving Low-Earth Orbit Predictions Using Two-line Element Data with Bias Correction.

Bizalion, H., Guillot, A., Petit, A., & Lucken, R. (2022). Systematic TLE data improvement by neural network for most cataloged resident space objects. Advances in Space Research.

Haidar-Ahmad, J., Khairallah, N., & Kassas, Z. M. (2022, July). A hybrid analytical-machine learning approach for LEO satellite orbit prediction. In 2022 25th International Conference on Information Fusion (FUSION) (pp. 1-7). IEEE.

Jiang, M., Qin, H., Zhao, C., & Sun, G. (2022). LEO Doppler-aided GNSS position estimation. GPS Solutions, 26. https://doi.org/10.1007/s10291-021-01210-2

Kassas, Z., Morales, J. J., & Khalife, J. J. (2019). New-age satellite-based navigation STAN: simultaneous tracking and navigation with LEO satellite signals. Inside GNSS, 14(4), 56–65.

Lawrence, D., Cobb, H.S., Gutt, G., O’Connor, M., Reid, T.G., Walter, T. Navigation from LEO, GPS World, July 2017.

Lazzaro, R., Bettanini, C. Evaluation of Satellite’s Point-Ahead Angle Derived from TLE for Laser Communication. Aerotec. Missili Spaz. 101, 7–15 (2022).

Nguyen, V. (2019). Bayesian optimization for accelerating Hyper-Parameter tuning. 2019 IEEE Second International Conference on Artificial Intelligence and Knowledge Engineering (AIKE). https://doi.org/10.1109/aike.2019.00060.

Ng, H.-F., Hsu, L.-T., & Zhang, G. (2023). Multi-Epoch Kriging-Based 3D Mapping-Aided GNSS and Doppler Measurement Fusion using Factor Graph Optimization. NAVIGATION: Journal of the Institute of Navigation, 70(4), navi.617. https://doi.org/10.33012/navi.617.

Osama, A., Raafat, M., Darwish, A., Abdelghafar, S., & Hassanien, A. E. (2022). Satellite Orbit Prediction Based on Recurrent Neural Network using Two Line Elements. In 2022 5th International Conference on Computing and Informatics (ICCI) (pp. 298–302). IEEE.

Reid, T.G.R., Walter, T., Enge, P.K., Lawrence, D., Cobb, H.S., Gutt, G., O'Connor, M. and Whelan, D. (2020). Navigation from Low Earth Orbit. In Position, Navigation, and Timing Technologies in the 21st Century. https://doi.org/10.1002/9781119458555.ch43a.

Ren, H., Chen, X., Guan, B., Wang, Y., Liu, T., & Peng, K. (2019). Research on Satellite Orbit Prediction Based on Neural Network Algorithm. In Proceedings of the 2019 3rd High Performance Computing and Cluster Technologies Conference (pp. 267–273).

Zhong, Q., & Groves, P. D. (2022). Multi-Epoch 3D-Mapping-Aided Positioning using Bayesian Filtering Techniques. NAVIGATION: Journal of the Institute of Navigation, 69(2), navi.515. https://doi.org/10.33012/navi.515.

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