Andrea López, University of Colorado Boulder; Jackson Kulik, Utah State University; Jay McMahon, University of Colorado Boulder
https://doi.org/10.64861/THAZ5447
Keywords: cislunar navigation, angles-only navigation, relative orbit estimation, nonlinear observability, nonlinear estimation, Gaussian mixture filtering
Abstract:
Angles-only relative navigation using passive optical sensors is attractive for cislunar proximity operations due to its low size, weight, and power requirements, but it has a fundamental limitation: bearing measurements alone cannot instantaneously recover range. In the Earth-Moon Circular Restricted Three-Body Problem (CR3BP), the nonlinear multi-body dynamics can provide weak range observability over time through state coupling and geometric evolution, but this observability is local, directional, and strongly dependent on both the orbital geometry and the relative configuration between spacecraft. Understanding when and where observability is strong or weak is important for designing estimation strategies that remain robust as the information content of the measurements changes over time._x000D_
This paper develops and applies several nonlinear observability tools to characterize the directional and time-varying observability of cislunar angles-only relative navigation. Two established methods are employed alongside two proposed extensions. The Hermann-Krener (H-K) observability matrix, constructed from Lie derivatives of the measurement function along the system dynamics and computed via automatic differentiation, provides local observability information at a given state: its singular value decomposition yields directional scores that quantify the instantaneous sensitivity of the measurements to perturbations along each state-space direction. The empirical observability Gramian (EOG), computed over a sliding window by simulating perturbed trajectories and accumulating measurement deviations, captures the observability information accumulated over a finite time horizon along an actual trajectory segment._x000D_
Building on these tools, we propose a sigma-point observability Gramian that replaces the EOG’s axis-aligned perturbations with unscented transform sigma points. This produces an observability measure conditioned on the estimator’s current uncertainty rather than on an arbitrary perturbation scale. We also develop higher-order observability measures based on the second-order partial derivative tensor of the composed dynamics-measurement map, isolating curvature information that the first-order tools cannot separate out on their own. These measures identify which state-space directions carry the most second-order sensitivity and provide a second-order-corrected observability Gramian that can be compared directly against the uncertainty-conditioned sigma-point Gramian, distinguishing curvature effects captured analytically from those absorbed empirically into the sigma-point ensemble._x000D_
The observability framework is applied to angles-only relative navigation scenarios in cislunar space. The analysis computes observability metrics over the orbital period, producing time histories that reveal how observability evolves with orbital phase and relative configuration. The results identify regions of strong and weak observability along the orbit and characterize which state-space directions are most and least informed by the bearing measurements as the geometry evolves. The comparison between the different tools highlights their relative strengths: the H-K analysis captures local structure, the EOG and sigma-point Gramian capture accumulated information over different assumptions, and the higher-order measures reveal where nonlinear effects dominate._x000D_
The observability characterization is placed in the context of Gaussian mixture filtering for angles-only navigation. Under weak observability, mixture component weights can become unreliable: two components differing primarily along a poorly observable direction produce nearly identical measurement predictions, so the highest-weight component does not necessarily correspond to the most accurate state estimate. This paper connects the observability metrics developed here to this behavior, providing initial insight into when weight-based filter decisions such as pruning can be trusted and when they cannot. The broader motivation is to inform the design of observability-aware adaptive mixture management strategies for cislunar relative navigation._x000D_
Date of Conference: September 15-18, 2026
Track: Cislunar SDA