Optical Camera Characterization for Feature-Based Navigation in Lunar Orbit

Abstract

Accurate localization is a key requirement for deep-space exploration, enabling spacecraft operations with limited ground support. Upcoming commercial and scientific missions to the Moon are designed to extensively use optical measurements during low-altitude orbital phases, descent and landing, and high-risk operations, due to the versatility and suitability of these data for onboard processing. Navigation frameworks based on optical data analysis have been developed to support semi- or fully-autonomous onboard systems, enabling precise relative localization. To achieve high-accuracy navigation, optical data have been combined with complementary measurements using sensor fusion techniques. Absolute localization is further supported by integrating onboard maps of cataloged surface features, enabling position estimation in an inertial reference frame. This study presents a navigation framework for optical image processing aimed at supporting the autonomous operations of lunar orbiters. The primary objective is a comprehensive characterization of the navigation camera’s properties and performance to ensure orbit determination uncertainties remain below 1% of the spacecraft altitude. In addition to an analysis of measurement noise, which accounts for both hardware and software contributions and is evaluated across multiple levels consistent with prior literature, this study emphasizes the impact of process noise on orbit determination accuracy. The mismodeling of orbital dynamics significantly degrades orbit estimation performance, even in scenarios involving high-performing navigation cameras. To evaluate the trade-off between measurement and process noise, representing the relative accuracy of the navigation camera and the onboard orbit propagator, numerical simulations were carried out in a synthetic lunar environment using a near-polar, low-altitude orbital configuration. Under nominal conditions, the optical measurement noise was set to 2.5 px, corresponding to a ground resolution of approximately 160 m based on the focal length, pixel pitch, and altitude of the modeled camera. With a conservative process noise model, position errors of about 200 m are observed in both transverse and normal directions. The results demonstrate the estimation framework’s robustness to modeling uncertainties, adaptability to varying measurement conditions, and potential to support increased onboard autonomy for small spacecraft in deep-space missions.

1. Introduction

Autonomous navigation is a fundamental functionality to enable next-generation robotic probes devoted to deep-space exploration. Accurate onboard estimation of the spacecraft state is essential to support the execution of orbit correction maneuvers during transit and proximity operations. Despite advances in onboard systems, spacecraft orbit determination remains largely dependent on ground-based radio tracking data, acquired through communication with Earth-based tracking stations [1]. The inherent communication delays and limited availability of ground assets hinder real-time interaction, underscoring the need for a paradigm shift toward onboard state estimation.

To overcome these limitations, novel techniques are being developed to enable autonomous navigation for future lunar and planetary missions [2,3]. These approaches rely on the onboard processing of data from multiple sensors [4,5], including rangefinders [6,7], inter-satellite communication systems [8,9], and optical cameras [10,11], thereby facilitating real-time estimation of spacecraft state independently of ground-based infrastructure.

Optical navigation (OpNav) techniques have emerged as a promising approach for real-time onboard state estimation, extracting measurements such as the central body’s centroid or limb from images acquired by spacecraft-mounted cameras. A key constraint of OpNav performance is the relative distance between the spacecraft and the observed body. In the unresolved regime, the target appears as a point source, offering limited navigation information. As the spacecraft approaches, the body enters the resolved regime, where its shape, limb, and shading variations become distinguishable. At a sufficiently close distance, in the surface features regime, individual landmarks on the surface can be resolved and tracked within the image frame [12,13]. This study focuses on navigation in the surface feature regime, where high-resolution imaging enables precise feature tracking to support accurate onboard state estimation.

Landmark-based OpNav estimates the spacecraft state relative to the observed body by detecting and tracking identifiable surface features, including craters, ridges, or other terrain landmarks, in sequential images acquired by onboard cameras. By comparing the observed locations of these features in the image plane with their known coordinates in a planetary reference frame, the spacecraft state is retrieved using geometric constraints and filtering algorithms, such as, for example, the extended Kalman filter (EKF). This approach is effective during close flybys or orbital operations, where high-resolution images of the surface are available.

A key technique supporting this capability is stereophotoclinometry (SPC), which facilitates the generation of high-resolution digital terrain models (DTMs) by combining stereo imaging with two-dimensional photoclinometry [14]. Stereo constraints from multiple viewpoints enhance the spatial localization of features, while photoclinometry exploits brightness variations under changing illumination conditions to infer surface topography. SPC enables the development of comprehensive landmark catalogs with associated DTMs, providing robust reference surface datasets for navigation.

This method has shown high-quality performance in missions to small bodies, such as Dawn [15] and OSIRIS-REx [16], where the repeated imaging of specific surface regions allowed for the simultaneous refinement of topographic models and spacecraft trajectory. Onboard image processing was carried out by OSIRIS-REx to support its high-precision, touch-and-go sampling maneuver. However, the SPC applicability is limited in orbit-based operations around slowly rotating bodies like the Moon, where constrained mission timelines and limited viewing geometries reduce opportunities for repeated surface observations.

Among surface features, impact craters represent a suitable class of landmarks for OpNav. The extensive mapping of the lunar surface by orbital missions has resulted in the development of high-resolution crater catalogs [17,18,19], which can be downsampled and stored onboard to support image-to-catalog matching for real-time feature identification. Image-to-database crater matches are integrated into a filtering framework, which incorporates observed feature locations to update the spacecraft state estimate and support autonomous navigation relative to the observed body.

A critical component of landmark-based optical navigation is the reliable detection of craters in the image data. This task has been addressed using computer vision methods, such as edge detection [20] and k-means clustering for image segmentation [21], and deep-learning approaches, including object detection and instance segmentation networks [22,23]. The achievable accuracy of crater detection directly influences the quality of state estimation and must therefore be quantitatively assessed to ensure consistency with the navigation system’s precision requirements [24].

The accuracy of crater detection directly affects the quality of the optical measurements used for navigation and, consequently, the overall performance of the state estimation process. In an optical navigation framework, measurement precision is governed not only by the resolution and intrinsic characteristics of the imaging system but also on the performance of the algorithms used to extract and identify landmark features. In addition, accurate localization requires reliable prediction of the spacecraft state between image acquisitions, which is contingent on the precise modeling of the orbital dynamics. Errors in trajectory propagation can compromise orbit reconstruction, even with high-quality optical data acquisition.

To mitigate these effects, an accurate formulation of the dynamical equations of motion is required. However, achieving this level of fidelity often imposes significant computational demands on the onboard navigation system. As a result, simplified or truncated dynamical models are adopted to enable real-time computation of the spacecraft trajectory. This introduces a fundamental trade-off between orbit determination accuracy and computational efficiency. The uncertainties arising from these modeling approximations are accounted for by incorporating process noise into the estimation filter. To assess the combined impact of process and measurement noise parameters on the state estimation performance, numerical simulations were carried out, providing insights into the achievable navigation accuracy.

This paper presents the development and performance assessment of a crater-based optical navigation system for spacecraft operating in low lunar orbit, with a focus on characterizing the role of the navigation camera in orbit determination accuracy. The system architecture and filtering framework are detailed in Section 2, followed by a discussion of the optical measurement noise characterization in Section 3. Section 4 presents the results of numerical simulations conducted in a synthetic lunar environment to evaluate the joint impact of measurement and process noise on navigation performance. The analysis quantifies the attainable localization accuracy under realistic conditions, accounting for crater detection performance, camera resolution, and onboard computational limitations. Implications for the design of robust autonomous navigation systems for future lunar missions are addressed in Section 5.

2. Methods and Modeling

The feature-based OpNav system leverages the sequential processing of optical images acquired by the navigation camera onboard the spacecraft. A key advantage of optical measurements over ground-based observables (e.g., radio tracking data) is their suitability for real-time processing directly onboard following image acquisition. This enables continuous updates of the spacecraft trajectory, limiting reliance on ground-based tracking and supporting autonomous navigation capabilities [11].

The navigation filter integrated into the OpNav toolchain is the EKF that sequentially processes optical data extracted from the images to refine the spacecraft state (e.g., inertial position and velocity) and the associated covariance matrix.

The orbital state equations are propagated by using a custom orbit propagator designed for real-time operations in planetary or close-proximity scenarios [25]. Given the best estimated spacecraft state at time $t_{k-1}$ (${\widehat{\mathbf{x}}}_{k-1}$), the integrator propagates the state vector at time $t_k=t_{k-1}+\Delta t$ (${\overline{\mathbf{x}}}_k$) according to a user-defined spacecraft dynamical model. For the lunar mission scenario considered in this work, in addition to the gravity field of the Moon, modeled through a spherical harmonic expansion truncated at degree and order $l_{max}$, we also accounted for the point-mass contributions of the Sun and the Earth, and the solar radiation pressure (SRP). A fourth-order Runge–Kutta integrator is used for the state propagation step, enabling fixed-step integration of the dynamical equations, a key requirement for real-time onboard systems. Variable-step integrators, while more efficient and accurate in ground-based applications, introduce non-deterministic computational loads and variable timing, which are incompatible with the strict timing and predictability constraints of onboard processing environments.

The system covariance matrix is propagated forward in time by using the discrete-time form, as

$${\overline{\mathbf{P}}}_k=\mathsf{\Phi }(t_k,t_{k-1}){\widehat{\mathbf{P}}}_{k-1}\mathsf{\Phi }{(t_k,t_{k-1})}^T+{\mathbf{Q}}_k,$$

(1)

where ${\widehat{\mathbf{P}}}_{k-1}$ is the estimated covariance matrix at time $t_{k-1}$, $\mathsf{\Phi }(t_k,t_{k-1})$ is the state transition matrix (STM), and ${\mathbf{Q}}_k$ is the discrete-time system noise covariance matrix (or process noise matrix), defined as:

$${\mathbf{Q}}_k={\int }_{t_{k-1}}^{t_k}\mathsf{\Phi }(t_k,\tau ){\mathbf{G}}_k\left(\tau \right){\mathbf{Q}}_c{\mathbf{G}}_k{\left(\tau \right)}^T\mathsf{\Phi }{(t_k,\tau )}^{T}d\tau .$$

(2)

The process noise power spectral density matrix ${\mathbf{Q}}_c$ must be sufficiently large to account for uncertainties arising from unmodeled accelerations and linearization errors in the orbital dynamics. The matrix ${\mathbf{G}}_k$ defines the mapping of process noise into the state space [26]. In this formulation, the process noise increments are assumed to be temporally uncorrelated [27].

At each new image acquisition, craters are extracted using a crater detection algorithm (CDA) and matched with known craters from an onboard catalog. For each matched pair, the discrepancy between the centroid of the observed crater (${\mathbf{z}}_{k,i}$) and the predicted centroid of the corresponding catalog crater (${\mathit{h}}_{k,i}$), computed according to the optical measurement model, provides information used to update the spacecraft state and covariance through the Kalman filter equations.

By stacking the optical residuals and the corresponding measurement partials for each of the N crater pairs into aggregated vectors and matrices, the Kalman filter update is performed as follows,

$$\begin{array}{cc}\hfill {\widehat{\mathbf{P}}}_k& =\left[{\mathbf{I}}_{6\times 6}-{\mathbf{K}}_k{\mathbf{H}}_k\right]{\overline{\mathbf{P}}}_k\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\widehat{\mathbf{x}}}_k& ={\overline{\mathbf{x}}}_k+{\mathbf{K}}_k({\mathit{z}}_k-{\mathit{h}}_k),\hfill \end{array}$$

(4)

where ${\mathbf{H}}_k$ is the measurement Jacobian matrix and ${\mathbf{K}}_k$ is the Kalman gain matrix, which is computed as

$${\mathbf{K}}_k={\overline{\mathbf{P}}}_k{\mathbf{H}}_k^T{\left({\mathbf{H}}_k{\overline{\mathbf{P}}}_k{\mathbf{H}}_k^T+{\mathbf{R}}_k\right)}^{-1}.$$

(5)

The block-diagonal matrix ${\mathbf{R}}_k$ represents the measurement noise covariance (see Section 3). A detailed analysis of the influence of process and measurement noise covariances on filter convergence is presented in Section 4.

2.1. Definition of the Process Noise Covariance

A critical aspect of sequential navigation filters is the treatment of unmodeled or mismodeled accelerations during the covariance propagation step. This is conventionally addressed through covariance inflation, an approach known as state-noise compensation (SNC) [26]. A rigorous approach to modeling process noise requires computing an analytical solution to Equation (2), which formally propagates the continuous-time process noise through the system dynamics via the state transition matrix (STM). However, deriving this solution is generally challenging, particularly when the spacecraft state is expressed in Cartesian coordinates, due to the complexity introduced by the STM [28]. To circumvent this, a commonly adopted approximation assumes a kinematic model with zero nominal acceleration, leading to a simplified STM structure. This allows a closed-form solution to Equation (2), which remains valid for sufficiently small integration time steps $\Delta t$ [26], as

$${\mathbf{Q}}_k=\left[\begin{array}{cc}{\displaystyle \frac{1}{3}}\Delta t^3{\mathbf{Q}}_c& {\displaystyle \frac{1}{2}}\Delta t^2{\mathbf{Q}}_c\\ {\displaystyle \frac{1}{2}}\Delta t^2{\mathbf{Q}}_c& \Delta t{\mathbf{Q}}_c\end{array}\right].$$

(6)

When the state vector includes only the spacecraft position and velocity, the process noise spectral density matrix ${\mathbf{Q}}_c$ is commonly approximated as ${\mathbf{Q}}_c={\sigma }_u^2{\mathbf{I}}_{3\times 3}$, where ${\sigma }_u$ represents the diffusion coefficient [29]. This parameter characterizes the intensity of unmodeled accelerations and is typically fine-tuned to promote filter convergence. While this tuning is often empirical, prior knowledge of the dynamical environment can be leveraged to inform the selection of an appropriate diffusion coefficient [27].

To enable real-time computation, onboard navigation filters typically employ a truncated gravity field model of the central body. This simplification is necessary to reduce the computational cost associated with evaluating high-degree and high-order spherical harmonics, which would otherwise exceed the capabilities of typical onboard processors. However, such truncation introduces modeling errors in the force model, leading to discrepancies in trajectory propagation that can degrade state estimation performance. These errors are typically addressed through covariance inflation in the estimation filter.

In this study, the diffusion coefficient used to model process noise was estimated by isolating the impact of the central body’s gravity field truncation. The analysis compared acceleration profiles generated by two dynamical models: a simplified model used for onboard propagation, incorporating a lunar gravity field truncated at degree and order 5, and a higher-fidelity reference model representing the ground truth, which includes a 10 × 10 lunar gravity field [25]. All other perturbing forces, such as third-body gravitational influences from the Earth and the Sun, modeled as point masses, and solar radiation pressure, were treated consistently in both models. This approach allowed for a controlled and realistic estimation of the process noise level associated with the onboard gravity model limitations.

2.2. Image Processing Pipeline

The image processing pipeline is designed to extract and associate crater landmarks from monocular images for use in optical navigation [24]. The spacecraft is assumed to be equipped with a framing camera that operates in the visible light spectrum, allowing crater detection only on Sun-illuminated regions of the lunar surface. The camera features a field of view of 67° × 67° and a 1024 × 1024 px optical sensor, resulting in a ground sampling resolution of ∼65 m at an altitude of 50 km (

Table 1. Camera intrinsic parameters.

Parameter	Value	Unit
Focal length, f	10	[mm]
Pixel pitch, $\sigma$	13	[μm]
FoV	67 × 67	[deg × deg]
iFoV	1.3 × 1.3	[mrad × mrad]
Pixel scale, s (@ 50 km)	∼65	[m]
Image size	1024 × 1024	[px]
Principal point, $\left(p_x,p_y\right)$	$\left(512,512\right)$	[px]

).

The pipeline frontend consists of a crater detection stage, implemented using fine-tuned convolutional neural networks (CNNs) trained for object detection. The network processes the input image and outputs a set of crater candidates, each characterized by its pixel coordinates, bounding box dimensions, and confidence score.

Concurrently, known crater landmarks from the onboard database are geometrically projected onto the image plane using a perspective pinhole camera model. This projection accounts for the spacecraft’s estimated position and attitude at the time of image acquisition, as well as the intrinsic parameters of the imaging system. The result is a set of predicted crater centroids, corresponding to cataloged landmarks visible under the current viewing geometry.

To establish reliable associations between detected and projected craters, a geometric matching strategy based on crater triads is employed. This approach compares the invariant properties of crater triplets, such as relative distances and internal angles, between the two sets, improving robustness to detection noise and partial occlusions. Once correspondences are established, the optical residuals are computed as the vector differences between the centroids of matched crater pairs, forming the measurement input to the state estimation filter (see Figure 1).

This integrated approach enables the robust identification and association of surface features under varying illumination and viewing conditions, supporting real-time optical navigation in planetary orbit.

3. Characterization of Optical Sensors Noise

The performance of the optical navigation framework described in previous sections relies on the accurate detection and matching of surface landmarks, specifically lunar craters, extracted from images acquired by the onboard camera. The precision of these measurements directly impacts the quality of the optical residuals used in the state estimation process. However, the observed crater centroids are affected by measurement uncertainty, which arises from different noise sources that must be properly characterized to ensure robust filter performance.

In this work, optical measurement noise is modeled as the combination of independent error sources, including uncertainties from camera calibration, landmark detection, catalog accuracy, and attitude determination.

Calibration noise results from inaccuracies in estimating the intrinsic parameters of the optical camera, which are required to correct lens distortions in the raw images [30]. These parameters are obtained through dedicated calibration procedures conducted prior to the mission and, when feasible within mission constraints, refined using in-flight calibration campaigns. A thorough optical characterization of the imaging system enables sub-pixel reprojection accuracy. To conservatively account for residual calibration uncertainty, a standard deviation of ${\sigma }_{\mathrm{CAL}}=0.5$ px is assumed in this study.

Measurement uncertainty is also influenced by errors in the automatic detection of surface landmarks, particularly in the localization of crater centroids. Methods based on conventional computer vision (CV) techniques and more recent artificial intelligence (AI)-based approaches have been developed to address this task. CV-based methods typically exploit the morphological features of craters, such as elliptical rims and characteristic illumination–shadow patterns [31]. Techniques like the fuzzy Hough transform have been used to identify circular and elliptical shapes corresponding to crater boundaries [32]. Template matching has also been applied, where predefined crater templates are correlated with image content to detect similar patterns [33,34]. While these methods can be effective under controlled conditions, their performance often degrades in the presence of variable illumination, complex terrain morphology, and overlapping craters.

Recent advances in AI-based methods, particularly those employing deep learning, have significantly improved crater detection accuracy and robustness. Convolutional neural networks (CNNs), embedded within object detection frameworks, have demonstrated strong performance across diverse lunar terrains and imaging conditions [35,36]. These models generalize effectively across different lighting and geometric configurations, making them well suited for onboard applications.

The crater detection network adopted in this study is based on a fine-tuned AI model integrated within the proposed image processing pipeline. This approach has demonstrated improved accuracy in estimating crater centroids compared to computer vision techniques, particularly under varying illumination and terrain complexity. As reported in recent benchmarking studies, AI-based methods consistently achieve sub-3 pixel centroid localization errors [37,38]. Based on these results and the observed performance of the adopted detection framework [24], a standard deviation of ${\sigma }_{\mathrm{DET}}=2$ px is assumed in this study. This value reflects a conservative yet realistic characterization of detection uncertainty within the context of the employed navigation architecture.

Accurate knowledge of the coordinates of pre-mapped crater centroids is essential for ensuring unbiased projection of these features onto the image plane. The crater database used in this study is derived from the Robbins catalog, which represents the current state-of-the-art for lunar impact craters larger than 1 km in diameter [39]. In this dataset, crater locations are defined as the centers of circles or ellipses fitted to manually sampled points along the crater rims. In addition to the estimated coordinates of the crater center, the Robbins’ catalog provides associated one-sigma formal uncertainties in latitude and longitude (expressed in degrees), enabling an estimate of the positional uncertainty due to catalog limitations. These angular uncertainties correspond to a mean position error of approximately 50 m on the lunar surface, consistent with previous studies [40]. By scaling this value with the camera ground sampling distance (Table 1), the catalog-induced error is estimated as ${\sigma }_{\mathrm{CAT}}=0.75$ px. This value is used to account for the contribution of catalog uncertainty to the overall optical measurement error budget.

Errors in spacecraft attitude estimation directly impact the accuracy of projecting catalog crater positions onto the image plane. Even small angular uncertainties in the estimated orientation can lead to significant misalignments between predicted and observed crater locations, particularly when using wide field-of-view optical systems. This study assumes the use of a compact star tracker for onboard attitude determination. The system provides a conservative one-sigma accuracy of ${\sigma }_{\mathrm{ATT},\mathrm{rad}}=0.5$ milliradians [41,42]. This angular error is propagated to the image plane by scaling it with the camera’s instantaneous field of view (iFoV), resulting in an equivalent positional error of approximately ${\sigma }_{\mathrm{ATT}}=0.4$ px. This component is incorporated into the overall measurement noise model to account for attitude-induced uncertainty in crater projection.

The cumulative effect of the individual error sources is modeled as a combination of independent, zero-mean Gaussian random variables. Under this assumption, the total optical measurement noise is approximated by the root-sum-square (RSS) of the one-standard-deviation associated with each contribution, yielding a nominal measurement uncertainty of ${\sigma }_R\approx 2.5$ px. This value is used as the reference input for the observation noise covariance matrix ${\mathbf{R}}_k$ (see Equation (5)).

Accurate modeling of the optical measurement noise is critical for defining ${\mathbf{R}}_k$, which, together with the process noise covariance, governs the behavior and convergence properties of the navigation filter. A precise characterization of both matrices is therefore essential to ensure reliable state estimation during the sequential processing of optical data.

4. Validation and Testing Campaigns

The lunar environment provides a well-characterized and data-rich setting for the development, validation, and testing of vision-based navigation systems. Extensive mapping efforts conducted by past and ongoing missions have made the Moon one of the most thoroughly documented celestial bodies. High-resolution topographic models and detailed morphological datasets have enabled the construction of comprehensive crater catalogs [17,18,19], which are essential for implementing and evaluating landmark-based navigation techniques.

In this study, numerical simulations were carried out using a spacecraft in a low-altitude, near-polar lunar orbit. The reference trajectory is quasi-circular, with an eccentricity of $e=0.0014$, and is characterized by a semi-major axis of ∼1790 km, resulting in an orbital period of ∼113 min. This configuration closely resembles the 50 km average-altitude science orbit of the Lunar Reconnaissance Orbiter (LRO) mission [43,44]. Adopting this geometry enables the evaluation of the navigation system under representative operational conditions, including realistic surface visibility, camera resolution, and crater density across varying lunar terrains.

To generate model-based optical data within the navigation algorithm, the lunar crater database compiled by Robbins serves as the onboard catalog [18]. This comprehensive database includes over 1.2 million craters, offering a dense and georeferenced set of surface landmarks suitable for optical navigation. A key characteristic of the lunar surface is the non-uniform crater distribution, which reflects the geological dichotomy between the near and far sides of the Moon, as illustrated in Figure 2.

In the simulated orbital scenario, the spacecraft passes over Sun-illuminated regions of the lunar maria, which are known to exhibit significantly lower crater densities compared to the adjacent highlands due to their younger geological age [45].

The navigation sensor integrated into the proposed system is a visible-light optical camera with an acquisition frequency of 0.01 Hz (i.e., one image every 100 s), featuring specifications representative of sensors commonly deployed on small satellites (Table 1). The camera operates in nadir-pointing mode and captures images only when the lunar surface is illuminated by the Sun. The navigation filter runs at 10 Hz and propagates the spacecraft trajectory using a fixed-step fourth-order Runge–Kutta integrator. Upon each image acquisition, crater landmarks are detected and matched against entries in the onboard catalog. When more than three valid correspondences are established, the filter performs a measurement update, correcting both the estimated spacecraft state and its associated covariance.

To assess the contribution of optical measurements to navigation performance, the initial spacecraft state is perturbed by introducing position and velocity errors drawn from a zero-mean multivariate Gaussian distribution with covariance ${\widehat{\mathbf{P}}}_0$. This matrix represents the a priori state covariance and characterizes the uncertainty associated with the initial knowledge of the spacecraft state. It is modeled as a diagonal matrix of the form

$${\widehat{\mathbf{P}}}_0=\left[\begin{array}{cc}{\widehat{\sigma }}_{r_0}^2{\mathbf{I}}_{3\times 3}& {\mathbf{0}}_{3\times 3}\\ {\mathbf{0}}_{3\times 3}& {\widehat{\sigma }}_{v_0}^2{\mathbf{I}}_{3\times 3}\end{array}\right]$$

(7)

where ${\widehat{\sigma }}_{r_0}$ and ${\widehat{\sigma }}_{v_0}$ denote the 1–$\sigma$ a priori uncertainties in spacecraft position and velocity, respectively. In this study, conservative values of 300 m for position and 1 m s⁻¹ for velocity are assumed.

A comprehensive set of numerical simulations was carried out to assess the robustness and reliability of the crater-based optical navigation framework. The analysis focuses on the influence of crater detection accuracy and orbital dynamics mismodeling on overall navigation performance, highlighting the system’s sensitivity to both measurement and process uncertainties.

4.1. Benchmark Case with High-Fidelity Propagation and Accurate Optical Data

As a preliminary test case, state estimation is performed under idealized conditions to establish a reference baseline. A high-performance onboard computer is assumed, allowing for the implementation of the same high-fidelity dynamical model used to generate the ground-truth trajectory. Additionally, high-quality optical data are considered, with a measurement noise level of ${\sigma }_R=2.5$ px. The total simulation duration is approximately two hours, corresponding to a complete orbital period of the spacecraft.

Figure 3 presents the position and velocity estimation errors resolved along the radial, transverse, and normal directions, along with the associated 1–$\sigma$ and 3–$\sigma$ formal uncertainties, shown as shaded areas. The plot provides a detailed view of the filter behavior and the navigation system performance across the orbital arc. At the initial epoch, the spacecraft enters the Sun-illuminated maria region (see Figure 2), enabling the use of crater-based optical measurements for state estimation. A convergence phase is observed during the first 30 min of filter operation, after which the estimation errors stabilize. The sawtooth pattern visible in the uncertainty bands reflects the sequential measurement updates performed at each image acquisition step. As expected from the nadir-pointing monocular camera configuration, estimation errors and formal uncertainties are larger along the radial direction, aligned with the camera boresight, compared to the along-track and cross-track directions. At steady state, 3–$\sigma$ position uncertainties reach approximately 100 m in the radial direction and around 75 m in the transverse and normal directions. The corresponding velocity uncertainties remain below 20 cm/s across all components.

After approximately 40 min, the spacecraft traverses into the shadowed highlands, where optical measurements are no longer available. During this acquisition gap, the navigation filter relies solely on the dynamical model for state propagation. As a result, formal uncertainties grow noticeably, particularly in the radial and transverse components. Nevertheless, estimation errors remain bounded and relatively low, owing to the accurate trajectory estimates achieved prior to the loss of optical updates and the consistency between the onboard model and the ground truth. When the spacecraft re-enters a Sun-illuminated region, optical measurements are reacquired, allowing the filter to perform sequential corrections and reduce uncertainties to levels comparable to those reached before the shadowed interval.

Figure 4 shows the 2D distribution of the optical innovations (see Equation (4)) associated with image updates prior to entering the shadowed region. Lighter shades indicate measurements acquired at later time steps. The residuals are normally distributed around the origin, confirming that the state estimates are statistically unbiased. The red ellipse denotes the 1–$\sigma$ confidence region, which is consistent with the assumed measurement noise model.

4.2. Constrained Onboard Computing with Simplified Dynamics and Accurate Optical Data

Onboard computing constraints can limit the complexity of the dynamical models used for real-time orbit propagation. While the inclusion of high-order gravitational and non-conservative force models enhances the accuracy of trajectory predictions between image acquisitions, it also increases computational load and memory usage, potentially compromising real-time operations. This introduces a fundamental trade-off between orbit prediction accuracy and computational efficiency.

To investigate the impact of dynamical mismodeling on the performance of the optical navigation filter, a test scenario was considered in which the onboard orbit propagator employs a truncated lunar gravity field model limited to degree and order 5. In contrast, the ground-truth trajectory is generated using a more accurate 10 × 10 gravity field model. As in the previous assessment, high-quality optical data are assumed, with a measurement noise level of ${\sigma }_R=2.5$ px.

Figure 5 presents the time evolution of position and velocity estimation errors, along with the corresponding 1–$\sigma$ and 3–$\sigma$ formal uncertainties. Compared to the idealized scenario where the onboard model matched the reference trajectory dynamics, a higher diffusion coefficient is adopted in the estimation filter to accommodate the increased process noise resulting from gravity field truncation. A transient phase is observed during the initial segment of the trajectory while the filter refines the state estimate. At convergence, the 3–$\sigma$ formal uncertainties reach ∼150 m in the radial direction and ∼100 m in the transverse and normal directions. The velocity uncertainties remain below ∼30 cm/s across all components. As the spacecraft enters the shadowed highlands, optical measurements are no longer available and state updates are suspended. During this interval, dynamical errors accumulate due to the simplified onboard model. However, position and velocity errors remain bounded within the 3–$\sigma$ formal uncertainty confidence level, confirming consistency between the estimated uncertainty and the actual filter performance. Upon re-entry into Sun-illuminated regions, optical measurements resume and allow the filter to recover, progressively reducing estimation errors and formal uncertainties to levels comparable to those achieved prior to the shadowed segment.

4.3. Constrained Onboard Computing with Simplified Dynamics and Low-Quality Optical Data

To assess the impact of increased measurement noise on navigation performance, simulations were conducted with a measurement noise level of ${\sigma }_R$ = 5.0 px. This configuration reflects conditions representative of resource-limited platforms such as CubeSats, where reduced attitude determination accuracy or lower crater detection performance may degrade optical measurement quality. The navigation filter continues to operate under onboard computational constraints, using a lunar gravity field model truncated at degree and order 5.

Position and velocity estimation errors and associated formal uncertainties over a single orbital period are shown in Figure 6. As expected, the increased measurement noise leads to larger state estimation uncertainties. At the end of the transient phase, the 3–$\sigma$ bounds on position and velocity are nearly twice those obtained in the previous scenario (see Section 4.2). Nevertheless, estimation errors remain bounded within the formal uncertainty confidence level, demonstrating consistency between the estimated covariance and actual filter performance.

The results obtained across all simulated scenarios demonstrate the robustness of feature-based optical navigation systems for low-altitude lunar orbiters, even under degraded measurement conditions and constrained onboard dynamics modeling. By incorporating process noise through state-noise compensation, the estimation filter maintains consistent performance across varying scenarios, effectively absorbing the effects of model mismatches and measurement degradation.

A critical challenge arises from the use of near-circular orbits combined with the Moon’s slow rotation. These conditions lead to repetitive surface observations across successive images, resulting in limited geometric diversity in the measurement updates. This redundancy can reduce the informational content of the optical measurements, causing the filter to converge too rapidly and become overconfident. As a result, the formal uncertainties may be overly optimistic, underrepresenting the true state estimation error, an effect commonly referred to as filter inconsistency.

The proposed framework mitigates this issue through two mechanisms. First, the dynamic selection of crater landmarks, based on real-time detection and matching in each image, introduces natural variability in the feature set, increasing the diversity of the measurement geometry. Second, the introduction of a non-zero process noise model compensates for the reduced information gain by maintaining a realistic level of uncertainty in the propagated state. Together, these elements prevent overconvergence, preserve filter consistency, and ensure reliable state estimation throughout the orbital arc.

5. Conclusions

This study presented the design and performance evaluation of a feature-based optical navigation system for spacecraft operating in low-altitude lunar orbits. The proposed approach leverages an onboard crater catalog in combination with an image processing pipeline that enables automated crater detection and matching. State estimation is performed through a sequential EKF, which integrates optical measurements as they become available, refining the spacecraft state in real time.

Through a series of simulation campaigns, the system was validated under varying conditions representative of real-world mission scenarios. In a benchmark configuration using a high-fidelity dynamical model and high-quality optical data, the filter achieved reliable state estimation with position uncertainties reaching approximately 75 m in the transverse and normal directions and 100 m in the radial direction. Velocity uncertainties remained within 20 cm/s at the 3–$\sigma$ confidence level. Even under degraded conditions, including increased measurement noise and simplified onboard gravity models, the navigation system demonstrated robust performance, maintaining estimation errors within the 3–$\sigma$ formal uncertainty bounds. Position uncertainties reached approximately 200 m in the transverse and normal directions and 300 m in the radial direction, while velocity uncertainties remained below 40 cm/s at the 3–$\sigma$ confidence level.

A key contribution of this work lies in the comprehensive modeling of both measurement and process noise, which are critical to achieving consistent and reliable state estimation in sequential optical navigation. The measurement noise model is derived from a detailed error budget that accounts for distinct sources of uncertainty, including camera calibration inaccuracies, crater detection errors, catalog positional uncertainties, and attitude determination noise. These contributions are treated as independent, zero-mean Gaussian processes and are combined into a unified observation covariance matrix, accurately reflecting the sensor characteristics and operational constraints of the onboard system.

In parallel, process noise is introduced through a state-noise compensation strategy to account for dynamical mismodeling, particularly due to the truncation of the onboard gravity model. This modeling approach allows the filter to remain robust to unmodeled accelerations and structural model simplifications, preventing overconfidence in the predicted state. Together, the combined modeling of measurement and process noise ensures filter consistency and enables the navigation system to maintain accurate performance across a range of sensing and computational scenarios.

The study also addressed an important challenge inherent in lunar optical navigation: the limited diversity of visual observations due to near-circular orbital configurations and the Moon’s slow rotation. These factors reduce the geometric variability of crater observations across consecutive images, potentially leading to filter overconfidence and inconsistency. The proposed framework mitigates this effect through adaptive crater selection and the inclusion of a process noise model that accounts for mismodeled accelerations, preserving filter consistency even under reduced measurement innovation and state observation variability.

Although the crater-based navigation approach presented in this work is applied to orbiting platforms, it can be extended to support landing mission scenarios by leveraging image-to-database feature matching to enable pinpoint landing accuracy on the lunar surface [11]. However, due to fundamental differences in dynamics, operational scale, and the nature of observable features, the sequential crater-based navigation framework is not directly transferable to surface-based robotic applications such as planetary rovers or robotic hoppers. In these cases, visual-LiDAR-inertial navigation techniques, the use of auxiliary local terrain features (e.g., keypoint-based landmarks), and alternative state estimation frameworks (e.g., particle filters [46,47]) are more appropriate for achieving robust and accurate localization in complex and dynamic environments.

Further developments of this work will investigate data-fusion techniques to overcome state correction gaps in shadowed areas. By combining crater-based measurements with auxiliary data from intersatellite links, altimeters [4], and signals from the Global Navigation Satellite System (GNSS) or future lunar communication and navigation services (LCNS) constellations [48], this multi-sensor approach is expected to significantly enhance navigation performance across various orbital segments, ensuring greater robustness under diverse environmental and lighting conditions.

The deployment of the navigation software on flight-representative, resource-constrained computing platforms will also be explored to characterize its computational efficiency and power consumption. This step is essential to prepare the system for future integration into deep-space missions involving small spacecraft and autonomous onboard navigation.