LEO Ephemeris Error Modeling Enabling Long Baseline Correction for Improved PNT

Joe Saroufim and Zaher Kassas

Peer Reviewed

Abstract:	The vulnerability of global navigation satellite systems (GNSS) signals to unintentional interference [1] or intentional cyberattacks [2] led researchers over the past decade to study the exploitation of signals of opportunity (SOPs) for positioning, navigation, and timing (PNT) [3]. In recent years, there has been a rapid emergence of space vehicles (SVs) in the low Earth orbit (LEO), the majority of which are launched for broadband communications, Internet-of-Things (IoT) and autonomous vehicle connectivity applications. Aside from these applications, LEO satellites have received significant interest from the research community, government agencies, and private industry, as key enablers for PNT [4]. This is due to their inherently desirable characteristics for PNT. First, the number of LEO satellites is growing dramatically rather quickly, with SpaceX currently leading the race, having around 6,400 active satellites, with a plausible extension to 42,000 [5]. This abundance offers favorable geometric dilution of precision (GDOP), which leads to improved positioning accuracy [6]. Second, LEO SVs have significantly smaller orbiting periods as compared to medium Earth orbit (MEO) satellites, which yields informative LEO Doppler measurements. Third, transmitting from low altitudes results in a higher received signal power than GNSS satellites that reside in MEO. Fourth, the diversity of frequencies at which different LEO constellations transmit their signals reinforces their resilience against unintentional and deliberate interference. On the other hand, current research focuses on addressing the challenges of opportunistically exploiting LEO SVs for PNT. First, LEO signals’ properties are not necessarily disclosed to the public. This challenge was addressed by some recently introduced receiver design architectures [7, 8]. Second, the LEO SV’s clock states are unknown, revealing some clocks synchronization methods and adaptive clocks estimation techniques [9]. Third, some LEO satellites’ signals are subject to ionospheric and tropospheric attenuation, based on their carrier frequency and geophysical conditions. Finally, unlike GNSS, LEO satellites’ precise ephemerides are not publicly communicated in their downlink signals. However, an estimate of the LEO SVs’ states can be calculated from the two-line element (TLE) sets published and updated by the North American Aerospace Defense Command (NORAD). Every TLE set consists of designated and temporal data on the first row, while the second row lists the SV’s standard orbital elements (inclination angle, right ascension of ascending node, eccentricity, argument of perigee, mean anomaly, and mean motion) defined at a single epoch. Every TLE set can be used to initialize orbit propagation algorithms, such as simplified general perturbation 4 (SGP4), used to estimate the corresponding satellite’s position and velocity at any time epoch. Nevertheless, the estimated Keplerian elements suffer from certain errors, which would accumulate and project into the propagated ephemeris, and the orbit propagation algorithms are unable to accurately model the complex perturbations in the LEO, leading to a bias in the satellites’ states, ranging from couple hundreds of meters to few kilometers, mostly concentrated in the SV’s along-track direction. The ephemeris error challenge has been approached in four different aspects: (i) closed-loop tracking [10, 11]; (ii) machinelearning orbit determination [12]; (iii) LEO-augmented GNSS [13]; and (iv) differential navigation [14]. Nevertheless, each of these aspects holds some limitations. First, for closed-loop tracking, the cross-track and radial position errors are less observable than the along-track error. Second, machine-learning techniques require large training data, in addition to the knowledge of ground truth ephemeris. Third, LEO-augmented GNSS requires the installation of GNSS receivers onboard LEO SVs. Finally, differential navigation requires accurate time synchronization and short baseline constraints among receivers. A recent study presented a characterization of the 2-D ephemeris error and its impact on non-differential and differential ranging measurements [15]. The study only tackled the orbit error in the orbital plane of each SV, i.e., the propagation of the in-track and radial errors onto the ranging measurement, demonstrating the benefit of differential measurements in compensating for the ephemeris error effect. The study was then generalized to tackle the 3-D ephemeris error by introducing a method to disambiguate spatiotemporal errors from the LEO ranging measurements [16]. This study builds upon the ephemeris error characterization introduced in [16] and introduce a method to estimate the 3-D ephemeris error and provide long range ephemeris corrections by making the following contributions. First, an expression for the time-varying ranging error due to a 3-D ephemeris error is derived and parameterized in terms of two unknown parameters. Second, a method is introduced to estimate the two unknown parameters of each SV at a reference receiver and communicate ephemeris corrections to any unknown navigating vehicle. Third, the spatiotemporal validity of the ephemeris corrections is analyzed for optimal receiver distribution to obtain full correction coverage. Fourth, simulation results are presented showing accurate navigation of ground and aerial vehicles with ephemeris error corrections from multiple base stations. Finally, experimental results are presented, demonstrating the effectiveness of long range ephemeris corrections on accurate receiver localization. The proposed method derives first an expression for the time-varying range error due to a 3-D ephemeris error in terms of only two constant unknown parameters. The first parameter represents the magnitude of the error, while the second represents the angle of the error vector in each satellite’s NTW frame (Frenet coordinate system), characterizing the cross-track and radial error components. Although, these parameters vary along the SV’s orbit and the ephemeris error increases with the age and propagation time of each TLE file, they can be assumed to remain invariant over a short period of time. A new approach is introduced to estimate these two parameters along with the relative clock bias and drift between the receiver and each SV at a reference receiver. The receiver will then communicate the two parameters to any unknown receiver to correct its erroneous measurements extracted from TLE+SGP4. A simulation study was conducted to demonstrate the performance of the ephemeris corrections using 4 LEO constellations, namely Starlink, OneWeb, Orbcomm, and Iridium NEXT. Preliminary results have shown that a 3-D position RMSE of 17.9 m can be achieved on an unmanned aerial vehicle (UAV) traveling a distance of 30.5 km with ephemeris corrections communicated over a 557 km baseline distance. Finally, experimental results are presented to validate the efficacy of the proposed method. A reference receiver with known position, located in St. Louis, Missouri, USA, estimated the ephemeris parameters of 7 overhead Starlink LEO SVs over a period of 10 minutes. The reference receiver communicated these parameters, over a baseline distance of 635 km, to an unknown receiver in Columbus, Ohio, reducing its 2-D positioning error from 2.41 km (with TLE+SGP4) to 8.78 m (with ephemeris corrections). The paper will build upon the aforementioned results and extend the study to address the long-term accuracy of the estimated ephemeris corrections. As the propagated ephemeris accumulate error over time, the accuracy of these corrections degrades with the time and distance traveled by every SV. The paper will analyze the statiotemporal variation of the ephemeris correction accuracy and its impact on the PNT performance. The study aims to find an optimal distribution and number of reference receivers to secure full correction coverage over a certain defined area (e.g., United States). The study considers ground or aerial vehicles, navigating with pseudorange measurements from visible LEO SVs with poorly known ephemerides, where a certain number of reference receivers provide the necessary ephemeris corrections for a specific level of navigation accuracy. References [1] S. Ji, W. Chen, X. Ding, Y. Chen, C. Zhao, and C. Hu, “Potential benefits of GPS/GLONASS/GALILEO integration in an urban canyon – Hong Kong,” Journal of Navigation, vol. 63, pp. 681–693, October 2010. [2] R. Ioannides, T. Pany, and G. Gibbons, “Known vulnerabilities of global navigation satellite systems, status, and potential mitigation techniques,” Proceedings of the IEEE, vol. 104, pp. 1174–1194, February 2016. [3] Z. Kassas, “Position, navigation, and timing technologies in the 21st century,” vol. 2, ch. 38: Navigation with Cellular Signals of Opportunity, pp. 1171–1223, Wiley-IEEE, 2021. [4] F. Prol, R. Ferre, Z. Saleem, P. Valisuo, C. Pinell, E. Lohan, M. Elsanhoury, M. Elmusrati, S. Islam, K. 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Soares, “Precise and efficient orbit prediction in LEO with machine learning using exogenous variables,” in Proceedings of IEEE Congress on Evolutionary Computation, pp. 1–8, 2024. [13] X. Li, Y. Yuan, X. Han, X. Li, and Y. Fu, “Toward wide-area and high-precision positioning with LEO constellation augmented PPP-RTK,” IEEE Transactions on Instrumentation and Measurement, vol. 73, pp. 1–13, 2024. [14] J. Saroufim, S. Hayek, and Z. Kassas, “Simultaneous LEO satellite tracking and differential LEO-aided IMU navigation,” in Proceedings of IEEE/ION Position Location and Navigation Symposium, pp. 179–188, April 2023. [15] J. Saroufim, S. Hayek, and Z. Kassas, “Analysis of satellite ephemeris error in differential and non-differential navigation with LEO satellites,” in Proceedings of IEEE Aerospace Conference, pp. 1–9, 2024. [16] J. Saroufim, S. Hayek, S. Kozhaya, and Z. Kassas, “Improved LEO PNT accuracy enabled by long baseline ephemeris corrections,” in Proceedings of ION GNSS+ Conference, pp. 1219–1229, 2024.
Published in:	2025 IEEE/ION Position, Location and Navigation Symposium (PLANS) April 28 - 1, 2025 Salt Lake Marriott Downtown at City Creek Salt Lake City, UT
Pages:	625 - 630
Cite this article:	Saroufim, Joe, Kassas, Zaher, "LEO Ephemeris Error Modeling Enabling Long Baseline Correction for Improved PNT," 2025 IEEE/ION Position, Location and Navigation Symposium (PLANS), Salt Lake City, UT, April 2025, pp. 625-630.
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