## 1 Introduction
Yildiz Technical University developed a multisensor railway track geometry surveying system, which integrated the linear variable differential transformers, inclinometer, GNSS receiver, total station, and an adaptive Kalman filter model to estimate the geometric parameters of tracks [16; 17]. Relative measurement track inspection equipment, also known as track checkers, which use mobile or static laser string measurement methods, can only detect the internal geometric parameters of tracks [18; 19]. Combined with odometer output mileage information, the track internal geometry parameters are measured using the chord surveying method [20; 21]. In addition, Xi'an Safeway developed a rail track inspection system based on a dual antenna Global Positioning System (GPS)-aided approach [22; 23]. The inertial measurement method using the dual gyroscope scheme is unable to realize accurate modeling or compensate for gyroscope error, Earth rotation angular velocity, and the implied angular velocity, which results in a reduced measurement accuracy. These problems with the pure gyroscope system are partially corrected by the dual-antenna GPS-aided inertial navigation system, but the measurement accuracy of motion parameters is low. Based on high-accuracy absolute solutions provided by RTK, Wuhan University integrated an inertial navigation system with a geodetic surveying apparatus and designed a modular system (TGMT) based on aided INS, which can be configured according to different surveying tasks. With a GNSS/INS configuration, this system offers good repeatability of measurements of track irregularities in both horizontal and vertical directions, with an error of less than 0.3 mm [24]. However, with increasing baseline length, the correlation of atmospheric errors decreases, resulting in a decrease to RTK positioning accuracy. A novel track measurement system based on GNSS/INS and multisensors was developed by Tsinghua University and obtained 2 mm accuracy for both lateral and vertical deviations of the track, and 1 mm horizontal accuracy and 1.5 mm vertical accuracy for the deformation point [25]. To obtain high-accuracy absolute coordinates, the system not only relies on a GNSS CORS station, but also needs to stand still for 5 min every 150 m, which inevitably reduces its efficiency and increases the cost. To reduce costs and improve the measuring efficiency, a multisensor fusion system consisting of multiconstellation global navigation satellite systems (GNSS)-based precise point positioning, an inertial navigation system, odometer, and track gauge was proposed by the German Research Centre for Geosciences, which was further enhanced using a Rauch-Tung-Striebel smoother. This system reached a measurement accuracy of approximately 0.25, 0.59, 0.65, and 0.48 mm for the super elevation, horizontal, vertical, and gauge components, respectively [26]. Since 2013, we have been engaged in the research and development of a high-speed railway track measurement system based on GNSS/INS multi-sensors. The system has realized high efficiency and a 0.47 mm measurement accuracy of the internal geometric parameters of tracks, but its engineering application algorithm is still based on the classical Kalman filter method [27; 28; 29]. The selection of the filter method and the design of the filter are highly important for the accuracy and stability of the inertial/GNSS/odometer integrated navigation system. In this study, to address the issue of poor fault tolerance performance resulting from the use of a centralized filter, and the large computational effort necessitated by the high-dimensional characteristic of the multisensor filtering system [30], a federated filter was designed to realize the advantages of an inertial/GNSS/odometer. Then, a forward-backward smoothing method was developed to apply to the federated filter results, to further improve the integrated accuracy and corresponding track measurement system.