Toward GNSS real-time relative orbit determination for satellite formations using adaptively robust factor graph optimization

Abstract

Satellite formations have been widely applied in scientific missions such as Earth gravity field measurements, where real-time relative orbit determination (RTROD) plays a critical role in ensuring the success of the mission. Factor graph optimization (FGO), which addresses nonlinear problems through multiple iterations and re-linearization, has recently gained popularity due to its flexibility and superior robustness in challenging environments such as urban canyons, compared to the Extended Kalman Filter (EKF). In this paper, we propose an FGO-based RTROD, which is realized by a priori factor constructed by a sliding window in combination with Schur complement. We introduce the receiver-differenced time-differenced (RDTD) carrier phase in FGO to simultaneously eliminate ambiguity effects and GNSS satellite orbit and clock errors, which is challenging to achieve for EKF-based methods under real-time constraints. In this paper, we newly propose an adaptively robust FGO (ARFGO) scheme, which uses equivalent weights to mitigate the impact of a maximum outlier in the current iteration, introduces an adaptive factor based on kinematic RTROD solutions and predictions to address relative dynamics anomalies, such as unknown satellite maneuvers, accidental collisions, and gas leaks. Experimental results based on GRACE-FO data show that the proposed FGO-based RTROD achieves higher accuracy compared to the basic EKF implementation under the same measurement conditions. The runtime of the RDTD carrier phase-based FGO is reduced by 73.4 % compared to the FGO based on the receiver-differenced carrier phase, demonstrating significant computational efficiency improvements. Furthermore, the proposed equivalent weight effectively reduces the FGO error from 4.125 cm to 3.795 cm. During satellite maneuvers, schemes without the adaptive factor exhibit meter-level deviations, while the adaptive factor can stabilize the maximum positioning error within 2 dm and the overall error to approximately 5 cm. Validation from the K-band ranging (KBR) system, a high-precision inter-satellite ranging system, demonstrates that the along-track direction accuracy of the proposed ARFGO maintains a precision of approximately 3 cm even in the presence of satellite maneuvers.