Pseudorange and Doppler-Based State Estimation from MEO to LEO: A Comprehensive Analysis of Maximum Likelihood Estimators

Recent research into state estimation algorithms for Low Earth Orbit (LEO) satellites, driven in part by the emergence of broadband LEO constellations, has stimulated interest in their application to Position, Navigation and Timing (PNT). This study investigates the effectiveness of Doppler measurements as a complementary solution for positioning and compares them with conventional pseudorange-based methods in Global Navigation Satellite Systems (GNSSs). The study highlights a limitation in Maximum Likelihood (ML) estimators, especially Least Squares (LS), which affect the convergence of positioning algorithms, especially when dealing with satellites in the LEO region. To analyse this problem, we developed a theoretical framework called "satellite scale-down". We then used Monte Carlo simulations to investigate the relationship between the initialisation point of the algorithm and orbital altitude. We also examined the algorithm's performance with satellites positioned at different orbital altitudes. The results indicate that decreasing the orbital radius enhances the performance of the Doppler-based positioning algorithm; however, it makes the LS positioning algorithm more sensitive to the initialisation point. In fact, without any a priori data on the receiver, the algorithm fails to converge. Therefore, it is necessary to make a trade-off between the achievable performance and the available information about the receiver.