## 1 Introduction  
The integration of the Global Navigation Satellite System and the Inertial Navigation System has long provided a reliable solution for navigation in various dynamic environments [1; 2]. GNSS offers high-precision location information [3], while the INS can maintain short-term navigation capabilities without external signals. However, the performance of this integrated system is limited by the availability of GNSS signals, especially in urban environments with high buildings, tunnels, and similar settings where GNSS signals are prone to interference or interruptions. Additionally, while the INS can provide location information without GNSS signals, its errors accumulate rapidly over time, leading to decreased accuracy with prolonged use. Thus, maintaining the continuity and accuracy of the navigation system in the absence of GNSS signals has become a focal point of navigation technology research. In integrated navigation systems, data fusion algorithms play a central role, especially in synchronizing and merging data from different sensors. Traditionally, the Kalman Filter (KF) [4] and its derivatives, such as the Extended Kalman Filter (EKF) [5; 6] and the Unscented Kalman Filter (UKF) [7; 8], have been crucial for estimating system states. These algorithms significantly enhance the system's positioning accuracy and robustness across various environments by effectively integrating positioning data from different navigation systems. However, as the application environments become increasingly complex, these filtering techniques show certain limitations in handling complex nonlinear dynamics, error accumulation, and global optimization needs. Traditional filtering methods tend to perform estimations in local regions, where errors may accumulate over long periods or during large-scale operations. To address these challenges, factor graph algorithms have recently attracted widespread attention from researchers [9]. Factor graphs offer a novel perspective and approach, integrating various sensor data through a globally optimized graph model to comprehensively update and optimize system states. Unlike traditional sequential update filtering methods, factor graphs use historical information to establish more complex constraints and significantly reduce error accumulation through iterative optimization processes, thereby enhancing long-term accuracy and stability [10]. This method not only performs excellently in highly nonlinear systems but also maintains high performance in complex navigation environments. Factor graphs can achieve more consistent global state estimates compared to KF, EKF, and UKF through global optimization. [PERSON] et al. [11] conducted a performance comparison of factor graph-based and EKF-based integrated navigation. The factor graph, using multiple iterations for robust estimation, achieved better navigation performance than the EKF. [PERSON] et al. [12] integrated PDR step lengths and GNSS using a factor graph for pedestrian navigation. The relinearization and full utilization of time-correlated states and measurements in factor graphs contribute to achieving better estimates than KF. [PERSON] et al. [13] conducted indoor integrated navigation experiments with drones to evaluate the performance of EKF and factor graphs in various environments. The factor graph utilizes more historical data to construct constraints and obtains the optimal state by solving nonlinear optimization. [PERSON] et al. [14] applied a factor graph for INS/USBL integration in underwater navigation and positioning. Due to multiple iterations and the application of large amounts of data, its performance is generally superior to that of KF. Studies [11; 12; 13; 14] compared factor graphs with KF and its derivatives across various test scenarios, with experimental results showing that factor graphs provided superior positioning accuracy, confirming their potential for navigation and positioning applications in complex scenarios.