Autonomous Navigation and Station-Keeping on Near-Rectilinear Halo Orbits

Technical Summary: Autonomous Navigation and Station-Keeping on Near-Rectilinear Halo Orbits

Problem Statement
Operating spacecraft in cislunar space, particularly within the 9:2 resonant Near-Rectilinear Halo Orbit (NRHO) planned for the Lunar Gateway, presents significant challenges due to the vast distance from Earth. This distance limits the availability of ground-based position, navigation, and timing (PNT) updates, necessitating autonomous guidance, navigation, and control (GN&C). The NRHO is inherently unstable, requiring regular station-keeping (SK) maneuvers to counteract unstable modes caused by modeling errors, state estimation inaccuracies, and control execution errors. While horizon-based optical navigation (OPNAV) has been demonstrated in previous missions (e.g., EQUULEUS, Orion, CAPSTONE), there is a need to evaluate the feasibility of a fully autonomous GN&C pipeline that integrates horizon-based OPNAV with SK control schemes under high-fidelity dynamics.

Methodology
The authors developed an end-to-end simulation pipeline to evaluate autonomous GN&C on the NRHO. The framework consists of three primary components:

1. High-Fidelity Dynamics and Simulation Environment:

   • Spacecraft motion is modeled in a Moon-centered inertial frame using a full ephemeris model. Perturbations include gravitational forces from the Earth, Moon, and Sun, lunar $J_2$ oblateness, and solar radiation pressure (SRP).
   • A numerical integrator (Runge-Kutta Prince-Dormand 9(8)) handles the quasi-periodic nature of the NRHO and the high sensitivity near perilune.
   • Synthetic images of the Moon are generated using Blender to simulate realistic horizon-based measurements.

2. Autonomous Navigation (OPNAV and Filtering):

   • Measurement Generation: The Christian-Robinson algorithm, a non-iterative, horizon-based method, processes synthetic images to estimate the spacecraft's position relative to the Moon.
   • Covariance Modeling: The authors augment the standard measurement covariance model to account for camera attitude uncertainty in addition to limb detection errors. This is achieved by deriving an analytical expression for the covariance of the position estimate in the Moon's principal axes frame, incorporating both pixel detection noise and attitude perturbation.
   • Filtering: An Extended Kalman Filter (EKF) is employed for state estimation. The prediction step accounts for process noise due to SRP uncertainties. The measurement update utilizes the derived covariance model.
   • Sensor Configuration: The study investigates the trade-offs of camera Field of View (FOV) and measurement timing. Due to the high eccentricity of the NRHO (ranging from ~3,000 km to >70,000 km), a single camera cannot optimally image the Moon at all points. The study focuses on selecting an FOV that provides reliable measurements near apolune (where SK maneuvers typically occur) while accepting degraded performance or no measurements near perilune.

3. Station-Keeping Control:

   • Control Scheme: The paper evaluates the $x$-axis crossing control scheme, which targets the $x$-component of velocity ($v_{EM}^x$) in the Earth-Moon rotating frame at a specific future perilune (typically 7 revolutions ahead).
   • Implementation Variants: Two approaches to solving the targeting problem are compared:
     • Differential Correction (DC): A root-finding approach using a shooting function and minimum-norm updates.
     • Sequentially Linearized Minimization Problem (SLMP): A constrained minimization approach that linearizes the dynamics and solves for the minimum $\Delta V$ to satisfy the targeting constraint.
   • Enhancements:
     • Unscented Transform (UT) Targeting: Instead of targeting the mean state derived from a single propagated estimate, the authors propose using the UT to propagate sigma points from the filter's covariance. The control law then targets the expected mean state $E[\psi(t_f)]$, aiming to improve prediction accuracy over long horizons.
     • Hysteresis: A mechanism is introduced where the trigger threshold for executing a maneuver ($v_{x,trig}$) is set higher than the targeting tolerance ($v_{x,tol}$). This prevents "chattering" and reduces cumulative $\Delta V$ by avoiding maneuvers for small deviations that would otherwise be corrected by the controller's inherent convergence properties.

Key Contributions

• Augmented OPNAV Covariance: The development of an analytical measurement covariance expression that explicitly incorporates camera attitude uncertainty, validated via Monte-Carlo experiments.
• Control Scheme Comparison: A rigorous comparison of DC and SLMP approaches for $x$-axis crossing control, highlighting that SLMP's short-sighted per-maneuver optimization can lead to higher cumulative costs compared to DC's quadratic convergence properties.
• UT-Based Prediction: The proposal and validation of a targeting scheme that utilizes the filter's state covariance via the Unscented Transform to improve long-horizon state predictions, thereby reducing cumulative station-keeping costs.
• Hysteresis Mechanism: The introduction of a tunable hysteresis mechanism within the control loop to desensitize the system to noise and reduce unnecessary maneuvers.
• End-to-End Pipeline Validation: The first integrated assessment of an autonomous GN&C pipeline for NRHO using only horizon-based OPNAV, demonstrating the interplay between sensor limitations (FOV, measurement location) and control performance.

Results
Monte-Carlo experiments were conducted to assess the pipeline's performance over 60 revolutions (approx. 393 days):

• Navigation Performance: The EKF successfully tracks the spacecraft state using horizon-based OPNAV. The filter's estimation accuracy exhibits a periodic structure correlated with the spacecraft's position; standard deviations are minimized near apolune (around true anomaly $170^\circ$) and degrade as the spacecraft moves away due to the reduced apparent diameter of the Moon.
• Control Performance:
  • DC vs. SLMP: Differential Correction generally outperformed SLMP in terms of cumulative $\Delta V$ cost, particularly when hysteresis was not applied. SLMP's tendency to satisfy constraints exactly at the tolerance limit resulted in higher costs compared to DC's natural convergence to values well within the tolerance.
  • UT Impact: Incorporating the Unscented Transform for targeting significantly reduced the 95th percentile and maximum yearly $\Delta V$ costs for both DC and SLMP. This confirms that accounting for state covariance in the prediction step mitigates the impact of initial estimation errors over long propagation times.
  • Hysteresis: The hysteresis mechanism effectively reduced cumulative costs by preventing frequent, small maneuvers, particularly for the SLMP approach.
• Full Pipeline: In the full GN&C simulation (using $f=360$ mm camera, 3 measurements/rev, $v_{x,trig}=20$ m/s, $v_{x,tol}=1$ m/s), the UT-DC controller achieved a mean yearly $\Delta V$ of ~48 cm/s and a 95th percentile cost of ~62.5 cm/s. This is lower than controller-only experiments due to the realistic, lower-than-assumed navigation errors provided by the EKF.
• Maneuver Location Sensitivity: Unlike scenarios assuming constant navigation covariance, the cumulative $\Delta V$ cost in this OPNAV-based pipeline varied with the maneuver location. The best performance was observed at a true anomaly of $170^\circ$, coinciding with the minimum in the filter's velocity standard deviation.

Significance and Claims
The paper claims to demonstrate the feasibility of autonomous GN&C for the NRHO using only horizon-based optical navigation. It establishes that:

1. A limited number of horizon-based measurements (three per revolution) are sufficient to maintain state estimation accuracy for station-keeping.
2. The choice of camera FOV and measurement timing is critical; a "best" camera exists that balances measurement quality near apolune against the inability to image near perilune.
3. The periodic structure of the navigation filter's covariance, driven by the varying geometry of the Moon, directly influences station-keeping costs, making the maneuver location a tunable parameter for cost reduction.
4. The integration of Unscented Transform-based targeting and hysteresis mechanisms offers a robust strategy for reducing cumulative $\Delta V$ costs, with the UT-DC approach achieving a 95th percentile yearly cost of approximately 60 cm/s.

The authors conclude that while horizon-based OPNAV has limitations near perilune, it is a viable primary navigation source for autonomous operations on the NRHO, provided that control strategies are adapted to account for the periodic nature of the estimation errors.