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

Keidai Iiyama, Grace Gao

TL;DR

The paper tackles precise GNSS-based lunar navigation by addressing low observability and numerical stability when processing time-differenced carrier phase measurements. It introduces a stochastic-cloning UD-filter with a fixed-interval smoother, enabling stable handling of delayed-state TDCP observations and robustly accounts for relativistic effects, time-scale transformations, and ionospheric/plasmaspheric delays. Through high-fidelity Monte Carlo simulations with multi-constellation GNSS geometry and ray-traced delays, the approach achieves meter-level orbit accuracy and sub-millimeter-per-second velocity accuracy, meeting upcoming Lunar Augmented Navigation Service targets. The work demonstrates that combining ionosphere-free pseudorange with TDCP, together with smoothing, yields substantial improvements over pseudorange-only methods and offers a viable path toward autonomous lunar satellite navigation.

Abstract

This paper presents a terrestrial GNSS-based orbit and clock estimation framework for lunar navigation satellites. To enable high-precision estimation under the low-observability conditions encountered at lunar distances, we develop a stochastic-cloning UD-factorized filter and delayed-state smoother that provide enhanced numerical stability when processing precise time-differenced carrier phase (TDCP) measurements. A comprehensive dynamics and measurement model is formulated, explicitly accounting for relativistic coupling between orbital and clock states, lunar time-scale transformations, and signal propagation delays including ionospheric, plasmaspheric, and Shapiro effects. The proposed approach is evaluated using high-fidelity Monte-Carlo simulations incorporating realistic multi-constellation GNSS geometry, broadcast ephemeris errors, lunar satellite dynamics, and ionospheric and plasmaspheric delay computed from empirical electron density models. Simulation results demonstrate that combining ionosphere-free pseudorange and TDCP measurements achieves meter-level orbit accuracy and sub-millimeter-per-second velocity accuracy, satisfying the stringent signal-in-space error requirements of future Lunar Augmented Navigation Services (LANS).

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

TL;DR

The paper tackles precise GNSS-based lunar navigation by addressing low observability and numerical stability when processing time-differenced carrier phase measurements. It introduces a stochastic-cloning UD-filter with a fixed-interval smoother, enabling stable handling of delayed-state TDCP observations and robustly accounts for relativistic effects, time-scale transformations, and ionospheric/plasmaspheric delays. Through high-fidelity Monte Carlo simulations with multi-constellation GNSS geometry and ray-traced delays, the approach achieves meter-level orbit accuracy and sub-millimeter-per-second velocity accuracy, meeting upcoming Lunar Augmented Navigation Service targets. The work demonstrates that combining ionosphere-free pseudorange with TDCP, together with smoothing, yields substantial improvements over pseudorange-only methods and offers a viable path toward autonomous lunar satellite navigation.

Abstract

This paper presents a terrestrial GNSS-based orbit and clock estimation framework for lunar navigation satellites. To enable high-precision estimation under the low-observability conditions encountered at lunar distances, we develop a stochastic-cloning UD-factorized filter and delayed-state smoother that provide enhanced numerical stability when processing precise time-differenced carrier phase (TDCP) measurements. A comprehensive dynamics and measurement model is formulated, explicitly accounting for relativistic coupling between orbital and clock states, lunar time-scale transformations, and signal propagation delays including ionospheric, plasmaspheric, and Shapiro effects. The proposed approach is evaluated using high-fidelity Monte-Carlo simulations incorporating realistic multi-constellation GNSS geometry, broadcast ephemeris errors, lunar satellite dynamics, and ionospheric and plasmaspheric delay computed from empirical electron density models. Simulation results demonstrate that combining ionosphere-free pseudorange and TDCP measurements achieves meter-level orbit accuracy and sub-millimeter-per-second velocity accuracy, satisfying the stringent signal-in-space error requirements of future Lunar Augmented Navigation Services (LANS).
Paper Structure (53 sections, 60 equations, 16 figures, 9 tables, 2 algorithms)

This paper contains 53 sections, 60 equations, 16 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Lunar Orbit and Clock Estimation with Terrestrial GNSS
  3. State Definition
  4. Orbit Dynamics
  5. Relativistic Effects and Clock Dynamics
  6. Lunar Coordinate Time (TCL) and Lunar Time (LT)
  7. Relativistic Clock Drift with respect to TCL
  8. Shapiro Time Delay
  9. Stochastic Clock Drift Model
  10. GNSS Observables
  11. Pseudorange and Carrier-Phase Measurements
  12. Multi-Constellation Inter-System Biases (ISB)
  13. Light-Time Equation
  14. Ionospheric and Plasmaspheric Delays and Mitigation
  15. Time-Differenced Carrier Phase Measurements
  16. ...and 38 more sections

Figures (16)

  • Figure 1: GNSS sidelobe and portion of mainlobe signals spilling into cislunar space (not to scale)
  • Figure 2: Numerically integrated difference between TCL and TCG for 2027/01/01–2027/12/31. The secular drift component is removed.
  • Figure 3: The simulated LCRNS (LDN-1) satellite orbit in Moon-centered inertial frame over 6 orbits (180 hours). The direction to Earth and Sun at the initial epoch is shown as blue and red arrows, respectively.
  • Figure 4: GNSS transmitter orbits in the J2000 frame generated from IGS final products (SP3). Blue, orange, and green markers denote GPS, Galileo, and QZSS satellites, respectively. Dots indicate satellite positions at the initial simulation epoch (March 1, 2025, 12:00:00 UTC).
  • Figure 5: Antenna patterns of the selected GNSS satellites and the GNSS receiver onboard the satellite (From left: GPS SVN74 (Block III, L1), GALILEO E1, QZS-2 (L1), Lunar GNSS receiver)
  • ...and 11 more figures