Maximum-Likelihood Aerial-to-Ground Landmark Matching for Drift Suppression in Lunar Visual–Inertial Navigation

Huseyin Duran; Ugur Murat Leloglu

doi:10.33012/navi.782

Abstract

Landmark matching from an aerial database to ground images (AtoG) remains underexplored. This paper proposes an enhancement to a visual–inertial system using AtoG landmark matching against a prior orbital database. The proposed algorithm employs a maximum-likelihood landmark association framework. The proposed approach integrates absolute position corrections derived from AtoG matching into two state estimation architectures, an extended Kalman filter and a pose-graph optimization framework. The method combines stereo-range measurements with a pretrained object detection network to identify and associate craters and rocks, avoiding sequential frame-to-frame feature tracking. Measurement uncertainty, including stereo-range variance and correlated orbital map errors, is incorporated into the likelihood computation. The algorithm is evaluated on a synthetic lunar traverse under low-solar-elevation conditions representative of high-latitude environments. Compared with standalone visual–inertial simultaneous localization and mapping as well as visual–inertial navigation system solutions, the accumulated drift is significantly reduced by fusing prior orbital information with onboard visual–inertial sensing.

Keywords

1 INTRODUCTION

Increased autonomy in localization using self-contained techniques is desirable in a wide range of applications where satellite navigation is unavailable or undesirable. Some examples of these applications include unmanned aircraft, autonomous surface and/or underwater vehicles, and planetary exploration vehicles. Exploring the surface of celestial bodies beyond Earth via robotic vehicles is a modern approach to solar system exploration. Self-localization of a space exploration vehicle is a key factor for enabling an extended traverse on the explored celestial body.

An exploration vehicle can be safely navigated on a planetary surface to a scientifically attractive destination in two ways. The first method is called blind driving, where the rover does not perform path planning to avoid obstacles and drives directly to the destination. The second method is called autonomous driving (also referred to as AutoNav in the literature), where the vehicle autonomously performs a hazard avoidance task and plans a safe path to the destination. There are also intermediate navigation methods that combine blind driving with hazard avoidance in a multi-mode framework. The trend in planetary rover navigation is toward autonomous motion, as the one-way signal travel time between Earth and Mars can reach up to 23 min. Communication delay due to distance is a fundamental limitation of Earth-based commands. Ground operations also require additional time because the control crew must plan the path. Moreover, communication losses caused by eclipses have also increased the demand for autonomy in space exploration. Thus, the need for autonomy in navigation is rapidly growing.

The lunar far-side rover Yutu-2 of the China National Space Administration's Chang'e 4 mission relied on stereo measurements to improve its dead-reckoning odometry and was operated both autonomously and remotely, i.e., it was tele-operated from Earth. However, absolute localization was generally performed by the ground crew because of the constraints inherent in its long-distance travel mode (Wang et al., 2020). The Sojourner rover (Golombek et al., 1997) was the first spacecraft to include onboard autonomous driving capabilities. The Sojourner rover relied on a technique called “heartbeat,” where the rover simply checks its distance to the lander, which serves as the base station. The Opportunity rover of the Mars Exploration Rovers used a localization strategy (Maimone et al., 2007; Volpe et al., 2000) that included selecting a fixed distinct surface shape to match two terrain maps (one map is obtained previously). Both maps are generated using an on-board stereo-vision system. This technique required the vehicle to stop after reaching a predetermined destination to image and process the selected shape again, where the localization problem was addressed using the maximum-likelihood estimation approach. These techniques had limited autonomy, which constrained operations. More recently, the Mars Science Laboratory has used visual odometry and autonomous navigation (AutoNav) to explore the Martian surface while also relying on Sun sensors and wheel displacement data to refine its positioning during long-distance travel. With stereo systems capable of detecting features as small as 3.3 cm at 100 m (Bell et al., 2021), the Curiosity rover explored significant ranges. The Tianwen-1 mission (Zou et al., 2021) also deployed an exploration rover equipped with similar navigation payloads. The engineering cameras, including the navigation cameras of the Mars 2020 Perseverance rover, have been comprehensively described by Maki et al. (2020), including mission objectives ranging from orbital entry and parachute descent to AutoNav surface operations.

Figure 1 provides a generic overview of the augmentation of a visual-inertial navigation system (VINS) and a simultaneous localization and mapping (SLAM) algorithm with the maximum-likelihood landmark association (MLLA) method. Our approach to solving the localization problem for a lunar rover over extended trajectories is based on two key components: (1) lunar landmark detection and ranging and (2) categorical matching to a prior landmark database. Aerial-to-ground (AtoG) landmark matching is performed between nadir satellite images and horizontal rover images. The overhead images are assumed to have a ground sampling distance (GSD) of 1 m; hence landmarks as small as 1–2 m can be detected with the help of shadows. The availability of such high-resolution (1-m GSD) aerial imagery for the exploration region is assumed, which represents a practical limitation of this work. Rover-level images are taken at a fixed, very low solar elevation (7.5°, to reflect a high-latitude lunar environment) and azimuth. The stereo distance measurement range and the field of view (FOV) form a 60° arc with a 150-m radius. We call this arc the detectable area. This area can be further expanded if the stereo cameras provide a complete loop of FOV coverage, resulting in a full circular detection region.

Figure 1

Demonstration of the VINS and SLAM solutions with their corresponding improved solutions, evaluated over the tested traverse on a synthetic surface

The ground-truth (GT) traverse length is 241 m. This path represents an externally commanded traverse path to an exploration destination. The improved solution represents the VINS/SLAM + MLLA implementation.

The proposed method includes the following features:

The MLLA algorithm operates on a categorical matching basis using a prior database generated from orbital data that incorporates bias and error correlations.
The corrections derived from the innovation are incorporated into two visual–inertial navigation algorithms, ORB-SLAM3 and OpenVINS, to evaluate performance using both batch optimization and filtering-based estimation approaches.
The stereo-range (SR) measurement error model and the orbital data error model are incorporated into both the MLLA and measurement update operations.
Measures to suppress false matches are implemented in two tested frameworks.

This paper investigates visual-inertial navigation using both an extended Kalman filter (EKF) and a factor-graph optimization (FGO) framework. The absolute measurements from the MLLA are integrated into both architectures to constrain the global positioning error.

This article is organized as follows. Section 2 presents a discussion of developments in navigation methods. A novel approach for improving visual–inertial navigation accuracy using the proposed matching algorithm is presented in Section 3. Section 4 presents the results of the proposed method. Finally, conclusions and future work are discussed in Section 5.

2 RELATED WORK

Estimating the pose of a moving camera is an active research area driven by rapid development of the mobile robotics industry (Cadena et al., 2017; Davison, 2003; Mur-Artal et al., 2015). This section summarizes pose and motion estimation approaches based on visual and visual-inertial techniques.

Although the motion of an autonomous vehicle can be estimated using various sensors, the measurements contain a variety of deviations, resulting in inaccurate position estimations. Dead-reckoning odometry can produce highly inaccurate estimates on surfaces where wheel slippage occurs. This limitation applies even to systems capable of measuring heading. If a system lacks such instrumentation, the localization drift is primarily caused by the unknown orientation. A comprehensive study on wheel-terrain interactions was performed by Iagnemma and Dubowsky (2004).

Other widely used odometry techniques include visual odometry and laser odometry. Active scanners generally outperform visual odometry in low-light environments (Crocetti et al., 2025; Kazerouni et al., 2022). However, power consumption is a primary concern in lunar applications. This requirement has also led scientists to design sensors that are operational in the dusty Martian environment with low power consumption, as presented by Apestigue et al. (2022).

Autonomous navigation in an unknown environment is generally divided into two tasks: mapping the environment and planning a path that ensures a safe trajectory, avoiding rollovers, becoming stuck, or tumbling. In contrast, this paper assumes that the lunar environment is known, which simplifies the autonomous navigation problem, as there are a vast number of terrain reconstructions derived from stereoscopic measurements and various lunar topographical data sets from missions such as Chang'e 3, the Selenological and Engineering Explorer, the Lunar Reconnaissance Orbiter, and the Apollo missions (Beyer et al., 2009; Broxton et al., 2009; Henriksen et al., 2017; Kim et al., 2011; Kokhanov et al., 2018; Mattson et al., 2012). Because the lunar environment changes little over time, data sets from these missions remain relevant.

2.1 Inertial Navigation

With recent developments in micro-electromechanical systems, inertial navigation systems (INSs) have become more compact. A more recent development in this field is the cold-atom inertial sensor (Bernard et al., 2022; Geiger et al., 2020). A typical INS consists of three orthogonal accelerometers, three orthogonal gyroscopes, and a computer to integrate the data to provide a dead-reckoned estimate of position and orientation with respect to a known starting position (Berrabah & Baudoin, 2011). This combination can be augmented by additional devices such as magnetometers, altimeters, etc. Such systems are widely used because they provide highly accurate measurements during sharp changes in vehicle state. The accuracy of the state variables degrades significantly over time during the traverse. In contrast, such measurements are essentially required because the inertial measurement unit (IMU) provides accurate measurements for a sudden acceleration change in both translational and rotational motions. Note that wheel odometry explicitly measures no motion in the stationary case. However, wheel odometry measurements are affected by ground-contact point interactions.

An INS is advantageous because it provides accurate state information in sharp acceleration changes over a short period. Additionally, an INS is a self-contained navigation technique (Berrabah & Baudoin, 2011) and is not affected by environmental conditions such as the weather or illumination. The most common disadvantage of an INS is the rapid decrease in accuracy caused by error accumulation.

2.2 Visual Navigation

Visual odometry obtained from calibrated and ordered image sequences recovers the motion of the moving camera in an unknown environment. Feature-based processes can be described as estimating motion by keeping track of the relative translation and rotation of the same features extracted from consecutive image frames in a sequence. The metric scale for this relative pose solution from monocular visual odometry is unknown. An inertial aid is commonly used to solve for the metric scale. Nistér et al. (2006) accurately solved for this parameter using a stereo system. Howard (2008) achieved sub-meter accuracies using a stereo visual odometry algorithm. Optimization-based approaches, such as bundle adjustment (BA), are often superior to filtering-based approaches in frame-to-frame visual odometry applications. First, they handle nonlinearities better by iteratively minimizing a nonlinear least-squares problem. Second, previous state estimates are optimized simultaneously, resulting in less error accumulation over time. However, BA does not entirely eliminate error accumulation.

Our algorithm provides corrections to accumulated errors, ensuring higher accuracy and robustness for extended traverses. Hence, our solution reduces positioning drift in systems that accumulate error, without incurring the high computational complexity typical of SLAM algorithms (Mourikis & Roumeliotis, 2007).

2.3 Aerial-to-Ground Navigation

Robotic lunar navigation research is extensive, with numerous studies centered on crater-based navigation. However, most studies have focused on the ascent and landing phases. The work by Matthies et al. (2022) and the LunarNav project (Daftry et al., 2023) represent recent advances in crater-based lunar surface navigation (CLSN).

Crater detection from rover-level images or point clouds obtained from cameras or lidar has been investigated by Matthies et al. (2022). This approach was further extended in the work of Daftry et al. (2023), where a state estimate was recovered from matched craters using a prior database derived from orbital reconnaissance images. Daftry et al. (2023) presented a comprehensive study on lunar crater detection and matching for extended traverses, adopting a particle filter to associate detected craters with those mapped from orbit. To compensate for false negatives, the prior probability in the particle filter was tuned.

AtoG image-matching techniques that employ feature-based approaches are highly sensitive to aerial image resolution, which can lead to rapid uncertainty growth (Ebadi & Agha-Mohammadi, 2018; Pro et al., 2024). Rough terrain can also provide localization cues, as shown in the work of Volpe et al. (2000), where terrain elevation features were matched against a global map. By incorporating elevation map estimation errors, a probability density function was computed and used for localization. Feature-based methods have also been applied to AtoG image matching. Ebadi and Agha-Mohammadi (2018) showed that such approaches can gradually reduce localization errors along the rover traverse.

Unlike crater-based navigation approaches designed primarily for ascent and descent, this work performs image-to-map navigation. The proposed method extends CLSN to include both craters (with and without rims) and rocks for surface vehicle localization. The MLLA framework explicitly accounts for map uncertainty. The approach is also implemented within both FGO and EKF architectures.

3 SYSTEM OVERVIEW

3.1 Data and Assumptions

It is assumed that an autonomous rover, equipped with a VINS, is tasked with moving from a known starting point to a given destination on the lunar surface. We further assume that a high-resolution orthophoto of the area, together with a digital elevation model (DEM), is available for identifying lunar landmarks. From these data, a prior landmark map can be constructed, containing the characteristics and categories (e.g., elliptical or circular shapes, rimmed or flat craters) of rocks and craters. The rough topography of the surface is assumed to be known.

As shown in Figure 2, a three-dimensional (3D) surface model of a representative area on the lunar surface is built analogously based on actual satellite images. It is important to note that strict accuracy with respect to the real surface is not required for this model; rather, it serves as a functional representation for experimental purposes and for testing the algorithm.

FIGURE 2

Example of an overhead image: (a) representative area near the lunar south pole; (b) realistic surface imitation with 7.5° Sun elevation and large shadows

The 3D model does not reflect the topography of any specific region on the lunar surface.

The rover follows a predefined path determined during the mission planning stage within this modeled environment. The synthetic scene spans 250 m × 250 m, with rocks sparsely distributed across the surface. The rock distribution is modeled according to a Trask equilibrium. This equilibrium distribution is implemented using terrain texture modeling. The terrain texture is represented as a multi-layer splat map. Rocks are placed on the surface according to the weights of the texture layers, ensuring consistency with the Trask equilibrium distribution. Eight different rock prefabs are generated in varying shapes and textures, as shown in Figure 3. Arbitrary sizes are assigned as rocks are generated on the surface.

FIGURE 3

Lunar regolith prefabs distributed on the terrain with arbitrary shapes

The placement is determined by the weight of the terrain texture layer and rock surface correlation.

Pose data and color, depth, and semantic images are recorded synchronously. Poses obtained from the simulation are post-processed to generate IMU measurements via inverse kinematics. Ground-truth detections are produced using a dedicated semantic camera in the scene, which employs a Unity custom shader (CS) and a custom rendering pass (RP). The visibility of semantic objects is determined via a depth-buffer (Z-buffer) visibility test. SR measurements are generated using a dedicated depth camera configured with a similar CS and RP setup. As shown in Figure 4, additional post-processing is applied to the recorded data. One of these operations involves assigning ground-truth bounding boxes by identifying and matching the unique colors encoded by the CS. These colors are semantically associated with individual landmarks, enabling precise correspondence between rendered pixels and annotated objects.

FIGURE 4

Example of simulation and visual–inertial data generation

Inertial data generation includes post-processing of the position and orientation information. Producing the ground-truth labels before detector training involves post-processing of the color and semantic-labeled images recorded by the custom RP.

A synthetic visual-inertial data set is generated under the given assumptions. These data are collected in Unity on a laptop equipped with an NVIDIA RTX A2000 graphics processing unit, a 20 core i9 – 12900H central processing unit, and a 64-GB random access memory rendering the frames with a size of 640 × 480 at 50 Hz. The cameras have square pixels, and the pixel pitch P is $24 * 10^{- 6} m$ . The horizontal FOV is 60° wide. The rover width is 1.5 m, and the IMU-camera extrinsics for the left camera are given by $t_{BC}^{B} = {[\begin{array}{l} 0.5 & 1.25 & 0.075 \end{array}]}^{⊤} m$ .

3.1.1 Measurement Models

In this section, the modeled error characteristics of the IMU and SR measurements are presented.

The IMU measurements are simulated using inverse kinematics, with additive noise and bias terms applied. The SR measurements are simulated using a range-dependent bias model.

A Inertial Measurement Unit

The IMU measurement model is given in Equation (1):

$\begin{array}{l} {\overset{―}{a}}_{k}^{B} = a_{k}^{B} + R_{G, k}^{B} \times g^{G} + b_{a, k} + ϵ_{a, k} \\ {\overset{―}{ω}}_{k}^{B} = ω_{k}^{B} + b_{g, k} + ϵ_{g, k} \end{array}$ 1

where the noise terms of the accelerometer and gyroscope measurements are given as $ϵ_{a, k} \sim N (0, σ_{a}^{2})$ and $ϵ_{g, k} \sim N (0, σ_{g}^{2})$ , respectively, with covariances of $Σ_{a} = σ_{a}^{2} I_{3}$ and $Σ_{g} = σ_{g}^{2} I_{3}$ . Additionally, the random walk biases are modeled as ${\dot{b}}_{a, k} = ϵ_{b a, k}$ and ${\dot{b}}_{g, k} = ϵ_{b g, k}$ , where $ϵ_{b a, k} \sim N (0, σ_{b a}^{2})$ and $ϵ_{b g, k} \sim N (0, σ_{b g}^{2})$ . Table 1 shows the parameter values used in the simulations.

View this table:

TABLE 1 IMU Parameters Used in the Simulations

The true linear acceleration, $a_{k}^{B}$ , and angular rate, $ω_{k}^{B}$ , are computed using the inverse kinematics formulation described by Groves (2015), neglecting planetary rotation and other negligible effects. The notation $\overset{―}{(\cdot)}$ denotes measured quantities throughout the remainder of the paper. g^G is the gravity vector in the world frame (G), and $R_{G, k}^{B}$ is a rotation matrix that transforms from the world to body frame.

The sampling rate of the IMU is 200 Hz.

B Stereo-Range Measurement

Modeling the variance of stereoscopic depth estimation requires a systematic treatment. Sibley et al. (2007) modeled this error in two components. The first component is a range-dependent statistical bias, which is corrected as follows:

$\hat{Z} = \bar{Z} - \frac{1}{2} σ_{d}^{2} \frac{\partial^{2} \bar{Z}}{\partial d}$ 2

Here, the correction is obtained by linearizing around the depth estimate. $\bar{Z}$ is the SR measurement, and $σ_{d}^{2}$ is the disparity variance. The range variable is defined as $\bar{Z} = \frac{B f}{d}$ , where d is the disparity. This formulation is used to compensate for the statistical bias. This study reports an estimator whose variance approaches the Cramér-Rao lower bound for depth estimation, $Σ_{3 D} \geq I^{- 1}$ , where I is known as the information matrix and $Σ_{3 D}$ is the stereo measurement covariance.

Assuming uncorrelated two-dimensional (2D) covariance matrices for the left and right cameras and a one-pixel disparity noise $(σ_{d} = P)$ , the information matrix is given as follows:

$I_{33} = \frac{4 f^{2}}{{\bar{Z}}^{4}} \frac{1}{P^{2}}$ 3

where P is the pixel pitch.

The corresponding standard deviation of the range estimate then converges as follows:

$σ_{z z_{-}} \geq \frac{{\bar{Z}}^{2}}{2 f} P$ 4

To account for measurement uncertainty along the other camera axes, σ_zz can be conservatively scaled by a relaxation factor. Moreover, approximating the landmark center by its minimum bounding box (MBB) introduces additional error into the optical measurements. This error can also be incorporated in the same manner. Note that measurement errors along the other axes are neglected, because the depth axis has a significantly larger covariance. For the given camera parameters, the variances along the other two axes are equal and given by $σ_{11, 22} = \frac{1}{\sqrt{2} f} P$ .

The active detection region in Figure 5 highlights the effective operating range of the stereo measurements. While the theoretical model predicts a decreasing σ_zz- at short distances, the minimum practical detection range is approximately 1.6 m owing to system geometry and sensor constraints, while the maximum effective SR is 150 m. The stereo depth uncertainty increases rapidly beyond this range. In this regime, the overall error becomes increasingly dominated by the SR measurement error rather than the map error. Detections outside this interval are increasingly rejected by the geometric and statistical gating stages owing to rapidly growing depth uncertainty, making stereo-based association progressively unreliable.

FIGURE 5

Depth standard deviation σ_{zz_} as a function of distance for the stereo measurement model

The shaded region indicates the active detection range defined by the admissible standard deviation bounds. The quadratic growth of σ_{zz_} with distance results in increasing depth uncertainty at long ranges, which limits the effective operational range of stereo-based gating.

3.2 Overview of the Algorithm

The proposed algorithm relies on predetermined characteristics of the landmarks in the exploration region and a pretrained landmark detection framework. A flowchart of this approach is presented in Figure 6. The contents of the prior database used in the simulations are presented in the Appendix. In the simulations, craters are modeled as circular features, represented by their center positions and radii.

FIGURE 6

Flowchart of the system (ML: maximum likelihood)

A stereo camera is used to estimate the distance to each detected landmark. A pretrained detector provides the class and size of each landmark, along with a bounding box for each detection. The MBB is used as the 2D image point of the observed landmark. Partially observed landmarks, i.e., those for which at least one edge of the bounding box aligns with the image boundary, introduce errors in stereo-based size estimation and in the image-plane center position of the landmark. Therefore, landmarks that are not fully visible in the image are excluded from the matching process. In the case of a rimmed crater, the rover must see it over a hill for a correct detection.

The matching operation transforms the estimated state covariance and the detection area into a search region via a convolution step known as the Minkowski sum (Kjeldsen, 2008). The pseudocode for this process is provided in the Appendix. The categorical information provided by the stereo camera and the detector is used for a preliminary selection of potential matches covered by the search region. The computed likelihood for each of these landmarks assigns the match from the prior map.

The likelihood function of a 2D multivariate Gaussian distribution is directly defined by its probability density function:

$L (x ∣ θ) = f (x ∣ μ, Σ)$ 5

where μ and Σ denote the mean vector and covariance matrix, respectively.

3.3 Innovation

The state uncertainty grows with time and distance. The positioning measurement is available once a landmark has been matched. Using the probability density function in Equation (5), the landmark with the highest likelihood is selected as a match, as illustrated in Figure 7.

FIGURE 7

Landmark-matching framework illustrating the likelihoods of objects from the same class distributed over a region with a sparse representation

For a relatively dense placement of rocks, size measurement (using SR) of the detected landmark becomes an informative metric.

An illustration of maximum-likelihood matching is shown in Figure 6. The sector in front of the rover represents the detectable region, whereas the full circle surrounding the rover corresponds to the 3σ position confidence ellipse. Their Minkowski sum lies on the outermost boundary, forming the search region, which is outlined by a green line. The red rays pointing upward toward the probability density function illustrate the maximum-likelihood process. Only the likelihoods of the rocks that have a similar attribute are computed by MLLA and shown by red rays.

1 ALGORITHM

Algorithm for matching of landmarks from vertical and horizontal views

Omission and commission errors are likely to occur in a pretrained detector network, potentially resulting in false matches from the MLLA algorithm. To address this issue and suppress false matches, appropriate formulations are implemented in the optimization and filtering frameworks. The matching algorithm is described as follows.

The matching algorithm utilizes the class information provided by a pretrained network, the size information from stereo measurements, and the current pose estimate and its covariance. The 3σ ellipse is obtained in the first line, and the principal components of the confidence ellipse are computed in the second line. The search region is formed using the current heading estimate and the ellipse via the Minkowski sum in the sixth line. Finally, the landmark class and the search region are used to iterate over the database.

3.4 Pretrained Detector

We evaluate the YOLO26n object detector (Sapkota et al., 2025) fine-tuned on our custom data set. The detector is trained using ground-truth-labeled landmark classes, flat craters, rimmed craters, and rocks.

The metrics are evaluated under low-illumination conditions with a Sun elevation of 7.5°, reflecting a high-latitude environment. The detector maintains accuracy even for overlapping bounding boxes, as shown in Figure 8(a). The normalized confusion matrix is shown in Figure 8(b). Training is conducted for 100 epochs with an input image size of 640 × 640 and a batch size of 16. Training converged after approximately 30 epochs. The best validation performance is achieved at epoch 29, from which all reported metrics are obtained.

FIGURE 8

Detection performance and validation results: (a) overlapping detections under low polar Sun angles and long shadows; (b) normalized confusion matrix for the custom data set

Table 2 shows the mean average precision (mAP) metrics of the pretrained YOLO26n model on the Moon data set. It achieves a mAP at an intersection over union (IoU) threshold of 0.5 ([email protected]) of 98.7%, and a mAP averaged over IoU thresholds ranging from 0.5 to 0.95 ([email protected]:0.95) of 89.7%, demonstrating strong detection performance despite its lightweight architecture. Its compact architecture makes YOLO26n well suited for real-time deployment.

View this table:

TABLE 2 Detection Performance of the Pretrained YOLO26n Model on the Moon Data Set at the Optimal Validation Epoch (Epoch 29)

Detector omission and commission errors are mitigated through a two-stage consistency check. First, a geometric consistency test based on reprojection is applied in both the EKF and FGO frameworks to reject geometrically inconsistent landmark associations. Second, statistical outlier rejection is performed using a chi-square test in the EKF and a robust Huber loss function within the pose-graph optimization.

3.5 Optical Measurement

The absolute position measurement model is derived from the YOLO26 detections and the SR measurements.

Figure 9 shows the steps of the optical measurement framework. Note that we simplified the problem by assigning the 2D keypoint as the MBB. This approximation introduces error in the measurements. We incorporate this error in the covariance model derived in this section.

FIGURE 9

Optical measurement framework

The position and covariance updates after a match are relatively straightforward:

$\begin{array}{l} \overset{―}{p} = x_{p} + \hat{Z} h - {\hat{R}}^{⊤} t_{BC}^{B} \\ Σ = ({\hat{R}}^{⊤} R_{C}^{B}) Σ_{-} {({\hat{R}}^{⊤} R_{C}^{B})}^{⊤} \end{array}$ 6

where the vector x_p is the position of the matched landmark. $\hat{Z}$ is the bias-corrected SR measurement to the landmark, and h is a unit vector from the landmark to the vehicle. $t_{BC}^{B}$ is the translation from the body to the camera in the body frame.

The corrections derived from the innovation procedure are illustrated in Figure 10. The covariance updates, denoted as Σ−, must be transformed to the body frame before being sent to the estimator, as the measurements are performed in the camera frame. If a notation for the updated position p – is introduced, it corresponds to $x_{p} + \hat{Z} h$ :

$Σ_{-} = [\begin{array}{l} {(σ_{x x_{-}})}^{2} & 0 & 0 \\ 0 & {(σ_{y y_{-}})}^{2} & 0 \\ 0 & 0 & {(σ_{z z_{-}})}^{2} \end{array}] + Σ_{map}$ 7

FIGURE 10

Position and covariance update showing a 1° bounded attitude uncertainty and an SR that forms a cone in 3D Updates illustrated are in the camera frame.

In Equation (7), σ_{xx_} and σ_{yy_} are equal because the cross-section geometry is circular in Figure 10. These terms are given as follows:

$σ_{x x_{-}}, σ_{y y_{-}} = \frac{\hat{Z} \tan (0.5 π / 180)}{3}$ 8

to align the 3 – σ confidence to the circular cross-section. Σ_ is the covariance update in the camera frame, as shown in Figure 10. The current work focuses on a low-solar-elevation condition representative of high-latitude lunar environments. Under this assumption, a conservative fixed 1° root mean square (RMS) Gaussian model is adopted to reflect degraded illumination geometry provided externally by an optical unit. The covariance is modeled as diagonal, implicitly neglecting cross-correlations between state components. This simplifying assumption is adopted for computational efficiency, and a full assessment of statistical consistency under this model is beyond the scope of the present work. Comprehensive work addressing statistical consistency and high-integrity error bounding, especially under heavy-tailed and time-correlated measurement errors, has been presented by Gallon et al. (2026) and Li et al. (2026).

Another type of error arises from landmark-mapping inaccuracies in the orbital data. To account for these errors, we define a map covariance, Σ_map, as follows. The horizontal and vertical errors are assumed to be uncorrelated, with the vertical component determined by the DEM data, whereas the horizontal error is correlated and derived from the orbital data. These errors contribute additively in the measurement covariance because the database provides prior information:

$Σ_{map} = [\begin{array}{l} {(σ_{{map}_{x}})}^{2} & ρ σ_{{map}_{x}} σ_{{map}_{y}} & 0 \\ ρ σ_{{map}_{x}} σ_{{map}_{y}} & {(σ_{{map}_{y}})}^{2} & 0 \\ 0 & 0 & {(σ_{{map}_{z}})}^{2} \end{array}]$ 9

where $σ_{{map}_{z}}$ is determined by the DEM data and the horizontal error is determined by the orbital data. We assume correlated horizontal uncertainty and adopt representative values¹ for the components of Σ_map, taking into account the claimed ground-track geolocation uncertainties and orbit/pointing knowledge described by Barker et al. (2021), where an RMS vertical uncertainty of approximately 0.30–0.50 m is demonstrated for regional scales.

The likelihood function accounts for uncertainty in both the camera position and the landmark-mapping error. The map uncertainty contributes additively.

Estimating the orientation of the unit vector h requires a sophisticated approach. The 2D pixel position of a feature approximated to the center of the detected landmark (MBB) is projected to the 3D camera frame using the forward projection function:

$x = K^{- 1} x^{'} = [\begin{array}{l} x \\ y \\ 1 \end{array}]$ 10

where $x^{'} = {[\begin{array}{l} u & v & 1 \end{array}]}^{⊤}$ gives the 2D pixel coordinates and K⁻¹ is the forward projection function for the perspective case. We have the following:

$v = \frac{x}{‖ x ‖} = [\begin{array}{l} v_{x} \\ v_{y} \\ v_{z} \end{array}]$ 11

which is the unit vector from the landmark to the camera in the camera frame, also termed the bearing vector. We then obtain the following:

$h = - {\hat{R}}^{⊤} R_{C}^{B} v$ 12

The explained update method in this section can be used when there are fewer than six landmarks available. If six or more matches are available in a frame, the camera pose can be estimated using a perspective-n-point (PnP) solution, where a maximum-likelihood PnP approach (Urban et al., 2016) can provide the metrics required for the integration.

3.6 Pose-Graph Optimization and EKF Integration

To evaluate the impact of the proposed matching algorithm on SLAM/VINS performance, we integrated it into two state-of-the-art estimation frameworks. ORB-SLAM3 Campos et al. (2021) is a visual-inertial SLAM system that formulates state estimation as an FGO problem, in which camera poses and landmarks are represented as graph vertices connected by visual reprojection factors and inertial constraints are modeled using pre-integrated IMU factors. The system employs BA to jointly optimize camera poses and map points under visual and inertial measurements. ORB-SLAM3 performs local sliding-window BA over a set of recent keyframes and their associated landmarks, while global consistency is enforced via pose-graph optimization during loop-closure events. Overall, ORB-SLAM3 relies primarily on batch optimization over local and global graph structures, rather than purely incremental filtering approaches.

OpenVINS (Geneva et al., 2020) is an open-source VINS that follows a filtering-based estimation approach. It builds on the multi-state-constraint Kalman filter framework (Mourikis & Roumeliotis, 2007) by extending it to an EKF-SLAM formulation and incorporates the first-estimates Jacobian technique (Huang et al., 2009) to address spurious observability issues, particularly those related to the direction of gravity. Unlike batch optimization methods, OpenVINS relies on an EKF, in which the system state (including the IMU pose, velocity, sensor biases, and a selected set of landmark parameters) is propagated using inertial measurements and updated with visual observations. To maintain computational efficiency, OpenVINS applies a sliding-window strategy through state marginalization, retaining only a limited number of keyframe poses and features in the filter state. An incremental online estimation is performed rather than joint batch optimization.

In these two state-of-the-art frameworks, absolute updates are incorporated by articulating the problem as a pose-graph optimization in ORB-SLAM3 and as an EKF measurement update in OpenVINS. The measurement provides a global position for both implementations.

3.6.1 Pose-Graph Optimization

To address the state estimation starting from an initialized state using the FGO architecture of ORB-SLAM3, we present the following representation for the visual–inertial FGO:

The factor-graph formulation in Figure 11 works well with the high-grade² IMU model used in this study. However, the bias estimation of the visual-inertial BA is sensitive to the initialization error.

FIGURE 11

Factor graph representation for the initialized visual–inertial system

The pose-graph optimization for incorporating absolute position measurements is formulated as follows. To avoid blocking real-time operation owing to frequent global optimization, a batch of n successive absolute measurements is accumulated before the pose-graph optimization is executed.

Let $T^{i} \in SE (3)$ denote the pose of the i-th keyframe, parameterized by rotation $R^{i} \in SO (3)$ and translation $t_{c w}^{i} \in R^{3}$ . The camera center expressed in the world frame is given by the following:

$c_{i} = - {(R^{i})}^{⊤} t_{c w}^{i}$ 13

Given absolute position measurements $p_{i} \in R^{3}$ , the relative translational residual between two consecutive keyframes i and j is defined as follows:

$r_{i j} = (c_{j} - c_{i}) - (p_{j} - p_{i})$ 14

This formulation constrains only the relative translational drift while preserving the global gauge freedom of the map.

The corresponding cost term is added to the pose graph:

$\sum_{(i, j) \in B} ρ ({‖ r_{i j} ‖}_{Σ_{i j}^{- 1}})$ 15

where B is the set of all keyframes providing an absolute measurement. Σ_ij denotes the measurement covariance, and ρ (⋅) is the Huber robust kernel.

Before insertion into the pose graph, each absolute measurement is validated through a geometric consistency check based on the reprojection error. Subsequently, a Huber norm is applied during optimization to further suppress the influence of potential outliers.

3.6.2 EKF Integration

The measurement residual for an EKF implementation of a pose measurement update is defined as follows:

$δ z = [\begin{array}{l} δ θ \\ δ p \end{array}] = [\begin{array}{l} 2 vec (q_{meas} \otimes q_{pred}^{- 1}) \\ p_{meas} - p_{pred} \end{array}] \in R^{6}$ 16

where vec(⋅) extracts the imaginary (vector) components of a quaternion.

The corresponding measurement Jacobian, in an OpenVINS-compatible implementation, is as follows:

$H = [\begin{array}{l} I_{3} & 0_{3} & 0_{3} & 0_{3} & 0_{3} \\ 0_{3} & - I_{3} & 0_{3} & 0_{3} & 0_{3} \end{array}]$ 17

where each block corresponds to the relevant state components in the EKF state vector.

For scenarios in which only a position measurement is available, only the position residual δp and the corresponding part of H are used. A chi-square gating is applied as a measure for false detections.

4 RESULTS

The experimental setup consists of simulated sensor measurements collected over a 241-m traverse on the previously presented synthetic surface. An average base velocity of 1 m/s is selected to ensure sufficient parallax between consecutive frames (or keyframes) while maintaining a constrained computational load. The implementation of the absolute measurement update in both evaluated frameworks is designed to support multi-threaded asynchronous operation. The lightweight YOLO26n detector exhibits a latency ranging from 4.1 ms to 4.7 ms on the given data set. When compared with the 200-Hz IMU measurement rate, the detection process can be considered computationally efficient and suitable for real-time integration.

The RMS error (RMSE) results are obtained for 100 runs, and the mean for each path is presented in Table 3. The improvement over the VINS/SLAM solution is investigated. The RMSE is reduced from 6.06 m to 3.53 m with a 41.75% improvement over ORB-SLAM3 and from 12.01 m to 0.41 m with a 96.53% improvement over OpenVINS. The MLLA method yields a significant reduction in position error when integrated within the EKF framework. In contrast, the pose-graph implementation demonstrates superior positioning performance within the SLAM framework. This difference can be understood by examining how each framework formulates and solves the state estimation problem.

View this table:

TABLE 3 RMSE of the VINS and SLAM Solutions with Their Corresponding Improved Solutions

In the factor-graph architecture of ORB-SLAM3, the IMU biases are primarily estimated through the pre-integration factors and visual constraints within a local BA window. Consequently, bias observability is largely driven by short-term motion dynamics, which causes the IMU biases to be primarily observable through sudden changes in position and orientation.

Figure 12 illustrates the accelerometer bias estimates obtained with ORB-SLAM3 and the MLLA-enhanced ORB-SLAM3. As can be observed, the introduction of pose-graph optimization improves the accelerometer bias estimation. The refined estimates exhibit behavior similar to that of the overall SLAM solution, which can be attributed to the aforementioned local sliding-window structure employed by the algorithm. Pure global position constraints do not directly provide heading information, unlike in a loop-closure situation. With a global position measurement, the yaw remains unobservable, in contrast to a loop-closure situation where a relative pose constraint is introduced. Consequently, no significant changes are observed in the orientation estimation of ORB-SLAM3 and the MLLA-enhanced ORB-SLAM3, as shown in Table 3.

Accelerometer bias estimates of the SLAM system in a pose-graph implementation of the MLLA compared with standalone ORB-SLAM3 bias estimation through BA (mg: milligravity)

FIGURE 12 Accelerometer bias estimates of the SLAM system in a pose-graph implementation of the MLLA compared with standalone ORB-SLAM3 bias estimation through BA (mg: milligravity)

As shown in Figure 13, the EKF architecture of OpenVINS can continuously apply global position anchoring within its state propagation and update cycle, which allows the filter to directly fuse the MLLA measurements as absolute constraints on the state.

FIGURE 13

Position error with EKF implementation of the MLLA method

The empirical standard deviation is obtained from 100 Monte Carlo (MC) runs. The EKF standard deviation is obtained from the mean-propagated covariance of the filter.

Unlike the pose-graph approach, where global information is indirectly incorporated through optimization over a sliding window, the EKF enforces these corrections at every update step, immediately influencing the state estimate and the IMU bias terms, as shown in Figure 14 and Figure 15. As a result, the biases become observable not only through short-term motion excitation but also through long-term drift relative to the anchored global reference. This tighter coupling between global measurements and the state vector explains the more pronounced reduction in position error observed when MLLA is integrated within the EKF framework.

FIGURE 14

Velocity and attitude errors for EKF implementation of the MLLA method: (a) velocity error; (b) orientation error

FIGURE 15

IMU bias estimation errors with EKF implementation of the MLLA method: (a) gyroscope bias estimation error; (b) accelerometer bias estimation error

In the EKF-based VINS framework, global position measurements directly improve the observability of gyroscope biases. Unlike in a sliding-window BA with pose-graph implementation, where global position constraints only indirectly influence orientation, the EKF continuously couples position, orientation, and sensor biases through its propagation model and cross-covariances. As the system integrates global position information at every update step, orientation errors induced by gyroscope biases manifest as discrepancies between the predicted and measured positions. These discrepancies create observable innovations that the filter uses to correct both the orientation and the underlying gyroscope biases. Consequently, the EKF implementation allows long-term global position measurements to gradually reduce bias uncertainty, rendering gyroscope biases observable even in trajectories where short-term motion alone would be insufficient.

4.1 Map Error Sensitivity

To evaluate robustness against uncertainty in orbital landmark geolocation, we varied the assumed isotropic map covariance σ_map from 0.1 m to 3.0 m. For each value, simulations were performed under identical trajectory and noise conditions.

Figure 16 shows that the performance gradually degrades as σ_map increases. For $σ_{map} \leq 0.5 m$ , the system achieves maximum drift reduction. For $σ_{map} \geq 1.5 m$ , the method approaches baseline VINS performance, as expected when map measurements become weakly informative. In contrast, the pose-graph SLAM framework shows less sensitivity to map error growth.

FIGURE 16

Position error with EKF implementation of the MLLA method

The empirical standard deviation is obtained from 100 runs. The EKF standard deviation is obtained from the mean-propagated covariance of the filter.

Importantly, both estimators remain stable across the entire tested range, demonstrating robustness to realistic map uncertainty levels. This result indicates that whereas EKF-based VINS strongly benefits from accurate global map information, the SLAM framework is relatively insensitive to moderate increases in map uncertainty.

5 CONCLUSIONS AND FURTHER WORK

A positioning method based on an AtoG landmark-matching approach has been presented in this study. A match to the detected landmark from a previously provided catalog is assigned by relying on the VINS/SLAM uncertainty and the category and characteristic of the landmark. To demonstrate the proposed method, a lunar rover is shown to traverse an experimental distance of 241 m. Compared with the VINS and SLAM solutions with RMSEs of 12.01 m and 6.06 m, respectively, the overall localization is greatly improved, i.e., with RMSEs of 0.41 m and 3.53 m, respectively. The results qualitatively highlight the influence of estimator structure when absolute measurements are integrated in this simulated, lunar-like environment. Continuous filtering approaches appear to benefit from persistent global anchoring, whereas pose-graph methods may require additional constraints, such as relative pose or loop-closure information, to achieve comparable improvements in state consistency. Within the EKF framework, global position measurements are directly fused into the state at each update step, enhancing coupling between position and other states through cross-covariances and contributing to reduced drift in the presented simulations. In contrast, in the factor-graph formulation of ORB-SLAM3, global position information is indirectly incorporated through optimization within a sliding window. While this approach improves the overall positioning performance and supports bias refinement, global position constraints alone do not explicitly introduce heading observability in this formulation, and only limited improvements in orientation accuracy are observed in the simulated experiments.

The work presented in this paper covers implementations from various disciplines. Further improvements to this work can be outlined as follows. An automation algorithm could be developed to automatically generate the prior regional map. Additional improvements to the detector architecture could be explored, such as investigating alternative networks, optimizing feature extraction modules, and refining training strategies to enhance detection accuracy and robustness under challenging illumination or occlusion conditions. A weighting algorithm for extracting the 2D keypoint from the bounding box object could also be developed.

HOW TO CITE THIS ARTICLE:

Duran, H., & Leloglu, U.M. (2026). Maximum-likelihood aerial-to-ground landmark matching for drift suppression in lunar visual-inertial navigation. NAVIGATION, 73. https://doi.org/10.33012/navi.782

CONFLICT OF INTEREST

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

APPENDIX

Crater Data File

The crater data file contains one record per crater, with the following fields listed in order:

size: Scalar value representing the crater size (diameter)
position_x: X coordinate of the crater center
position_y: Y coordinate of the crater center
position_z: Z coordinate of the crater center
has_rim: Binary indicator specifying the presence (1) or absence (0) of a crater rim

Rock Data File

The rock data file contains one record per rock, with the following fields listed in order:

size_x: Extent of the rock along the X axis
size_y: Extent of the rock along the Y axis
size_z: Extent of the rock along the Z axis
position_x: X coordinate of the rock center
position_y: Y coordinate of the rock center
position_z: Z coordinate of the rock center

Helper Algorithms

Here, we present an explicit algorithm for the convolution of polygonal shapes, known as the Minkowski sum algorithm:

2 ALGORITHM

Minkowski sum algorithm

A point is determined to be within an ellipse based on the following constraints:

3 ALGORITHM

Algorithm to select landmarks inside the confidence ellipse

ACKNOWLEDGMENTS

HD thanks Roketsan for supporting this research.

Footnotes

↵1 $^{1} σ_{{map}_{x}} = 0.5 m, σ_{{map}_{y}} = 0.5 m, σ_{{map}_{z}} = 0.5 m, ρ = 0.7$
↵2 Bias repeatability: accelerometer, 1 milligravity (mg); gyroscope, 1°/h.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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