Gnss-Based Lunar Orbit Estimation Achieves High-Precision with Stochastic Cloning UD Filter

Scientists are tackling the complex challenge of precise lunar navigation, crucial for future missions and sustained lunar presence. Keidai Iiyama and Grace Gao, from their respective institutions, alongside et al., present a novel framework for estimating lunar orbit and clock behaviour using signals from terrestrial Global Navigation Satellite Systems (GNSS). Their research introduces a stochastic-cloning UD-factorized filter, significantly improving the stability of calculations when dealing with the weak signals received from lunar distances. By meticulously modelling relativistic effects, lunar time-scale transformations and signal delays, this work achieves unprecedented accuracy , demonstrating meter-level orbit and sub-millimeter-per-second velocity precision in simulations, paving the way for reliable Lunar Augmented Navigation Systems (LANS) .

Scientists are tackling the complex challenge of precise lunar navigation, crucial for future missions and sustained lunar presence.

Lunar orbit and clock estimation via GNSS

This breakthrough is particularly significant given the increasing international interest in establishing a sustainable lunar presence, with NASA, ESA, and JAXA all planning lunar communication and navigation systems. By leveraging terrestrial GNSS signals, the research circumvents these limitations, enabling autonomous orbit and clock estimation for lunar satellites. Initial studies utilized snapshot least-squares and Extended Kalman Filtering (EKF) with pseudorange measurements, but achieved kilometer-level positioning accuracy, insufficient for demanding LANS requirements. Recent flight experiments, such as the Lunar GNSS Receiver Experiment (LuGRE) mission, successfully tracked GNSS signals from lunar orbit using pseudorange measurements, but were limited by short experiment durations and the inherent noise in pseudorange data.

However, TDCP measurements introduce complexities, violating standard Kalman filter assumptions due to their dependence on both current and past satellite states. To resolve this, the scientists employed stochastic cloning, coupled with a UD-factorized filter, to maintain numerical stability and prevent filter divergence, a common issue when processing highly precise TDCP measurements. Furthermore, the research introduces a novel smoothing algorithm to refine past state estimates using all available data, overcoming the limitations of traditional recursive equations in the presence of delayed-state measurements.

Lunar GNSS orbit and time synchronisation framework

Scientists developed a novel terrestrial GNSS-based orbit determination and time synchronisation (ODTS) framework specifically for lunar navigation satellites. These simulations also included detailed lunar satellite dynamics and ionospheric/plasmaspheric delay computations derived from empirical electron density models, ensuring a comprehensive assessment of performance under realistic conditions. Researchers harnessed stochastic cloning to address the non-independence of measurements dependent on both current and past states, while a delayed-state Extended Kalman Filter (EKF) further refined the estimation process. Previous studies indicated that incorporating TDCP measurements improved orbit and clock estimation accuracy by 10, 30% compared to pseudorange-only solutions, achieving sub-10 meter accuracy in simulation; however, these methods were prone to divergence due to low system observability.

To mitigate this, the team implemented a UD-filter, avoiding computationally expensive square-root operations while maintaining covariance matrix symmetry and positive semi-definiteness. The approach rigorously accounts for both general and special relativistic effects on GNSS signals and lunar satellite clocks, crucial for accurate modelling and minimising systematic biases. Results demonstrate that the proposed method surpasses previous limitations, achieving a position signal-in-space error (SISE) of less than 40 meters (95th percentile) and a velocity SISE of 10 millimeters per second (95th percentile), aligning with LunaNet Interoperability Specification (LNIS) requirements0.43 meters (3σ) and a velocity SISE of 1.2 millimeters per second (3σ). Measurements confirm that utilizing TDCP measurements alongside pseudorange data significantly improves accuracy, building upon initial research that demonstrated the feasibility of terrestrial GNSS-based lunar orbit and clock estimation. The team addressed the limitations of earlier methods by implementing a UD-factorized filter, which maintains numerical stability and prevents filter divergence, a common issue when processing highly precise TDCP measurements0.68m RMS position SISE and 0.39mm/s RMS velocity SISE. The delayed-state smoother further refined these results, achieving sub-4m position accuracy and sub-mm/s velocity accuracy, a significant step towards reliable lunar navigation. The authors acknowledge limitations related to the tuning of masking angles and measurement noise inflation to mitigate plasmaspheric-delay biases. Future research will concentrate on refining these parameters and reducing reliance on masking through explicit estimation of residual ionospheric and plasmaspheric terms, as well as investigating raw pseudorange and carrier phase processing for improved robustness in various receiver configurations. This work, supported by the Nakajima Foundation, represents a substantial advancement in the field of lunar navigation and lays the groundwork for more precise and dependable positioning services beyond Earth.