## 1 Introduction  
It is well known that speed and position are very important for an intelligent railway vehicle system. The accurate speed and position of a train is the key element of the automatic protection system and control system (ATP&ATC) of a train [1, 2, 3, 4]. By the end of 2022, the operation mileage of China's high-speed railway exceeded 40,000 km, and the total railway operation mileage exceeded 150,000 km. However, the current speed measurement and positioning methods of trains mainly rely on trackside equipment and onboard equipment [5, 6, 7]. The typical onboard equipment is the odometer (OD), which is used to estimate the speed and traveled distance of the train at the same time [8, 9, 10]. Considering the uncertainty and complexity of the train operation environment and the requirements of high reliability, the OD-based method has certain limitations, such as slipping in acceleration and deceleration and wear of train wheel diameter. Therefore, the use of balises to correct the speed measurement and traveled distance error of the odometer has become the standard configuration of the current train control system [1, 3, 11]. To satisfy the Safety Integrity Level (SIL) requirement in intelligent railway vehicles, which is \(10^{-9}\) h for SIL 4, the average laying distance of the transponder is 2 km, whether in the Chinese Train Control System (CTCS) [3, 4] or the European Train Control System(ETCS) [12]. Because the basile is exposed outdoors, it is easily damaged, and the high-density layout, operation, and maintenance costs are not a small burden, especially in sparsely populated areas in Western China [13]. According to the Certifiable Localization Unit with the Global Navigation Satellite System (GNSS) in the railway environment (CLUG) project funded by the European Union (EU), one of the key characteristics of the proposed onboard location unit is a multi-sensor fusion engine that utilizes a digital track map. In the CLUG project, digital track maps play a pivotal role as primary inputs to both the navigation engine and the integrity engine, which collectively constitute the foundational components of the onboard localization system. In the train positioning system, the primary method of multiple sensor fusion is based on the integration of the global position from GNSS and dead reckoning primarily through inertial navigation with OD [4; 14]. Due to the complexity of the train operating environment compared to civil aviation applications, trains pass through areas with serious GNSS signal blockages and multi-path phenomena, such as dense forests, canyons, urban areas, as well as long tunnels with no GNSS signal at all [15]. In such cases, dead-reckoning systems without corrective global position signal are prone to divergence. Fortunately, the center-line coordinates in the digital track map are based on coordinates in the Earth-Centered Earth-Fixed (ECEF) system and naturally have global properties. Therefore, digital track map matching is a good substitute for when GNSS signals are unavailable [16]. Map matching is a very important source of information for train location because the train is constrained to the track and because train location is essentially one-dimensional compared to the 3D navigation of road vehicles, as long as the position of the train relative to the starting point is determined [17]. In addition to being able to improve the positional accuracy of train positioning, the use of map-matching calculations can also correct errors such as gyro drift in inertial navigation systems and scale factors in odometers. Map-matching algorithms exhibit a spectrum ranging from elementary search techniques to those using more complex mathematical techniques [18], including Kalman Filters [19] and Hidden Markov Models (HMMs) [20]. A number of different algorithms have been proposed for map matching across various application domains, each of which carries its own set of merits and demerits [21].