# Real-Time Geo-Localization for Land Vehicles Using LIV-SLAM and Referenced Satellite Imagery
## 3 Materials and Methods
### AKF Based on MC and OAGS  
(1) Model: The system state \(\mathbf{x}\) of the proposed AKF is given as below:  
\[\mathbf{x}=\left[\mathbf{p}^{T},\mathbf{v}^{T},\mathbf{a}^{T}\right]^{T} \tag{7}\]  
where the three sub-vectors \(\mathbf{p}^{T},\mathbf{v}^{T},\mathbf{a}^{T}\) denote the position, velocity, and acceleration in a horizontal direction, respectively. The process model of the proposed AKF is a typical motion model, which can be expressed as follows:  
\[\overset{\vee}{x}_{t}=F\hat{\mathbf{x}}_{t~{}-~{}1}+N_{a},~{}~{}~{}N_{a}\sim\text{ N}(0,\sigma_{a}) \tag{8}\]  
where \(\overset{\vee}{x}_{t}\) and \(\hat{x}_{t}\) represent the prior estimation and posterior estimate of the state at time \(t\), \(F\) gives the transition of the system state over time, and the process noise \(N_{a}\) is attributed to uncertainties in acceleration. The position observations are passed by the cross-modal matching algorithm, while the LIV-SLAM submodule provides the measurements of velocity, leading to the measurement model as follows:  
\[\mathbf{z}=H\mathbf{x}_{m}+N_{m} \tag{9}\]  
where \(x_{m}\) contains the measurement of position and velocity, \(H\) represents an identity matrix, and the measurement noise \(N_{m}\) consists of position noise and velocity noise, defined as follows:  
\[N_{m}\sim \begin{pmatrix}N\left(0,\sigma_{\mathbf{p}_{m}}\right)\\ N\left(0,\sigma_{\mathbf{v}_{m}}\right)\end{pmatrix}  \tag{10}\]  
(2) Challenges: Since the observations are obtained from two independent submodules, there will be data latency and synchronization issues. Both the position and the velocity observations are subject to a certain delay due to the processing time before being input into the filter. Additionally, there is a discrepancy in update frequencies between position and velocity observations. Error position observation is another challenge. Although our cross-modal matching algorithm can adapt to image distortions to a certain extent, localization may degrade or even fail in the presence of significant scene changes or heavy occlusions or during transitions into indoor environments. (3) Motion Compensation (MC): In the proposed method, historical observational data and the estimated navigation states output by the AKF are retained. The position observations are compensated using the past velocity observations, while the velocity observations are compensated based on historical acceleration estimations. This compensation mechanism enables the Kalman filter to process asynchronous data in a temporally consistent manner, ensuring fixed-frequency operation while minimizing the impact of observational delays, as shown in Figure 5. The effectiveness of MC is validated in the comparison of filters in ablation experiments (Section 4.3). (4) Observation-Aware Gain Scaling (OAGS): To evaluate the confidence of position observations, we leverage the three indicators as follows:  
* Matching coefficient: This refers to the value of the similarity metric defined in (6), which is an indicator of matching reliability. A higher matching coefficient means a higher confidence level for a position observation, and vice versa. * Matching inconsistency: After the initial matching, the sub-patches of the projection image will be re-searched in corresponding sub-searching areas. The matching inconsistency \(T\) is calculated as follows: \[T=\sum_{i}\|t_{i}-t\|\] (11) where \(t\) is the translation estimated with the projection image, and \(t_{i}\) is the translation estimated with a sub-patch. A large matching inconsistency typically suggests that the associated position observation is incorrect. * Prediction deviation: We evaluate the deviation between the predicted state of the Kalman filter and the position observation obtained from the cross-modal matching. A large deviation is highly indicative of the position observation being erroneous.