GNSS-based Lunar Orbit and Clock Estimation With Stochastic Cloning UD Filter

1. Problem Statement

The realization of the Lunar Augmented Navigation Service (LANS) requires lunar navigation satellites (e.g., NASA's LCRNS, ESA's LCNS, JAXA's LNSS) to achieve high-precision orbit and clock determination. The LunaNet Interoperability Specification (LNIS) mandates a Signal-In-Space Error (SISE) of <40 meters for position and <10 mm/s for velocity (95th percentile), with stricter requirements for specific missions (e.g., <13.43 m and 1.2 mm/s).

Current challenges in achieving these goals using terrestrial GNSS signals include:

• Low Observability: At lunar distances, GNSS satellites are viewed primarily through sidelobes or spillover mainlobes, resulting in poor geometric diversity and slow filter convergence.
• Measurement Limitations: Previous attempts using only pseudorange measurements (e.g., the LuGRE mission) achieved only kilometer-level accuracy due to high noise and short observation windows.
• Delayed-State Measurements: Utilizing Time-Differenced Carrier Phase (TDCP) measurements improves precision but violates standard Kalman Filter assumptions because TDCP depends on both current and past states.
• Numerical Instability: Processing high-precision TDCP data in low-observability scenarios often causes standard Kalman filters to diverge due to the loss of positive semi-definiteness in the covariance matrix.
• Unmodeled Physics: Prior works often neglect relativistic coupling between orbit and clock, lunar time-scale transformations, and significant plasmaspheric delays (which can reach >10 meters).

2. Methodology

The authors propose a comprehensive framework integrating a Stochastic-Cloning UD-Factorized Extended Kalman Filter (UD-EKF) with a Fixed-Interval Smoother.

A. Dynamics and Measurement Modeling

• State Vector: Includes position, velocity, clock bias/drift/acceleration (scaled by $c$), and a Solar Radiation Pressure (SRP) coefficient, all defined in the Lunar Celestial Reference Frame (LCRF).
• Relativistic Effects: The model explicitly accounts for:
  • Time Dilation: Coupling orbital dynamics with clock drift via gravitational and kinematic time dilation.
  • Time Scales: Rigorous transformations between Terrestrial Time (TT), Geocentric Coordinate Time (TCG), Barycentric Coordinate Time (TCB), Lunar Coordinate Time (TCL), and Lunar Time (LT).
  • Shapiro Delay: Calculated for signal propagation through the Sun's gravitational field.
• Measurement Models:
  • Dual-Frequency: Uses Ionosphere-Free (IF) combinations of L1/L5 (or E1/E5a) pseudoranges to mitigate first-order ionospheric/plasmaspheric delays.
  • TDCP: Time-differenced carrier phase measurements are used to eliminate integer ambiguities and provide high-precision relative range changes.
  • Error Sources: The model includes realistic broadcast ephemeris errors, receiver thermal noise, and ray-tracing-based ionospheric/plasmaspheric delays using the IRI and GCPM models.

B. The Stochastic-Cloning UD Filter

To handle TDCP measurements (which depend on $x_k$ and $x_{k-1}$) without violating Kalman assumptions:

• Stochastic Cloning: The state vector is augmented at each step to include a "cloned" copy of the previous state ($x_{k-1}$). This allows delayed-state measurements to be processed within a standard filtering framework.
• UD Factorization: Instead of propagating the full covariance matrix $P$, the filter propagates its factorization $P = UDU^T$ (where $U$ is unit upper-triangular and $D$ is diagonal).
  • Benefit: This ensures numerical stability and guarantees the covariance remains symmetric and positive semi-definite, preventing filter divergence even with high-precision, low-noise TDCP data.
• Computational Optimization: The authors derive a method to process measurements dependent only on the current state (e.g., pseudorange) after processing delayed-state measurements (TDCP), allowing for the skipping of unnecessary matrix updates to reduce computational cost.

C. Fixed-Interval Smoothing

• Standard Rauch-Tung-Striebel (RTS) smoothers fail with delayed-state measurements because the Markov assumption is violated.
• The authors derive new smoothing equations based on the joint posterior distribution of the augmented state (current + cloned past).
• This allows for retrospective refinement of past states using all available future measurements, significantly improving accuracy.

3. Key Contributions

1. Numerically Robust Filter: Development of a UD-factorized filter capable of handling stochastic cloning for TDCP measurements, solving the divergence issues common in low-observability lunar navigation.
2. Novel Smoother: Derivation of fixed-interval smoothing equations specifically tailored for systems with delayed-state measurements, enabling post-processing accuracy improvements.
3. High-Fidelity Physics Model: A comprehensive dynamics and measurement model that explicitly includes relativistic orbit-clock coupling, lunar time-scale transformations, and ray-traced plasmaspheric delays (often neglected in prior work).
4. Comprehensive Simulation: Evaluation using high-fidelity Monte Carlo simulations incorporating realistic multi-constellation (GPS, Galileo, QZSS) geometry, broadcast ephemeris errors, and empirical electron density models.

4. Simulation Results

The framework was evaluated over a 180-hour simulation (approx. 6 lunar orbits) using three measurement configurations:

1. L1 Pseudorange Only: Achieved ~18.4 m RMS position error (baseline).
2. Ionosphere-Free (IF) Pseudorange Only: Improved to ~6.05 m RMS position error.
3. IF Pseudorange + TDCP:
   • Filtering: Achieved 4.53 m RMS position error (25% improvement over IF-only) and 0.45 mm/s velocity RMS.
   • Smoothing: Further reduced errors to 2.79 m RMS position and 0.30 mm/s velocity.
   • Percentiles: The 95th percentile position error was reduced to 4.47 m, and the 99.7th percentile to 7.31 m.

Key Findings:

• Combining IF pseudorange with TDCP is critical for meeting LANS requirements.
• The smoother effectively suppresses transient spikes in velocity and clock drift caused by occasional TDCP outliers.
• Tangential altitude masking (e.g., >5000 km for TDCP) is necessary to ensure ionospheric residuals remain smaller than receiver noise.

5. Significance

This paper provides a critical pathway toward autonomous, high-precision navigation for lunar satellites. By demonstrating that meter-level orbit accuracy and sub-millimeter-per-second velocity accuracy are achievable using terrestrial GNSS signals alone, the work validates the feasibility of the LANS architecture. The proposed Stochastic-Cloning UD Filter and Delayed-State Smoother offer a robust mathematical solution to the specific numerical and observability challenges of deep-space GNSS navigation, addressing gaps left by previous EKF-based approaches and paving the way for future lunar exploration missions.