## 2 Construction and Characteristics of Digital Track Map
### Definition and Accuracy of Required Digital Track Map  
In order to meet the high-level position requirement, all absolute coordinates in the digital track map are referenced to the World Geodetic System--1984 Coordinate System (WGS84) reference system. The digital track map consists of a layered structure, consisting of a basic layer and an attribute layer, as depicted in Figure 1, where the basic layer is composed of three-dimensional coordinate points of the track's center line in the ECEF coordinate system. For digital track maps applied to the train's location algorithms, the logical relationships between infrastructure objects within the railway network and the absolute position of the track's center line (longitude, latitude, and altitude) are mandatory elements. Other information, such as the radius of curvature of the track and the slope, can be calculated from the coordinates of the track's center line and are therefore optional data. Railroads in China are laid on a standard gauge, with a distance of 1.435 m between two rails. The distance between two rails is 1.435 m. In order to prevent side conflicts between vehicles, the positioning error of the train position in the vertical rail direction (transverse direction) is required to be less than 1.5 m. According to the requirements of the train control system on the positioning accuracy and integrity of the train, in addition to the departure and Automatic Train Operation (ATO) automatic parking requirements of the longitudinal positioning accuracy of better than 10 m, the positioning accuracy of  
Figure 1: Definition of digital track map. the train interval tracking is \(\pm\)5 m + 5% s [17]. In addition, as a result of Airbus' accuracy analysis using a total error budget of 10 m for the localization algorithm, an upper bound of 0.1 m can be applied for the cross-track error. In practice, the digital map of the track is organized as a list of discrete center points, each with TrackEdge ID and geographic and offset information as shown in Equation (1). In Equation (1), the map information includes TrackEdgeAttribute_Type, TrackEdge_ID, AttributePoint_Offset, AttributePoint_Long, AttributePoint_Lat, AttributePoint_Alt, and other geographic information. All the geographic information can be downloaded from the CLUG project. \[ \begin{split}\text{map}=\{\text{TrackEdgeAttribute\_Type, TrackEdge\_ID, AttributePoint\_Offset}\\ \text{AttributePoint\_Long, AttributePoint\_Lat, AttributePoint\_Alt}\ldots\}\end{split}  \tag{1}\]  
To utilize map matching for the train's lateral and longitudinal error correction, it is necessary to establish the cross-track error between the maximum distance among the map's center points, the radius of curvature. Furthermore, the positions obtained from map matching will be used as observations in Kalman filtering, and the observations of Kalman filtering need to be set up with measurement noise \(R\) in order to form a complete closed loop of Kalman filtering. \[e_{2}=R-R\cos(\frac{l}{2R}) \tag{2}\]  
According to Figure 2, the maximum cross-track error \(e_{2}\) is a function of the radius of curvature \(R\) and the distance \(l\) from the discrete points on the curve [24]. \[e_{1}=l-2R\sin(\frac{l}{2R}) \tag{3}\]  
\(e_{1}\) is the difference between the length of the curve and the length of the line segment \(D\). The detailed summary of track's center line distances in different curve radii can be founded in [15]. Taking the maximum cross-track error of 0.1 m as the upper bound, the larger the radius of curvature of the track, the longer the distance \(D\) between the discrete points can be, and the smaller the amount of stored data is required for the map. Because the train track is mainly composed of straight lines, curves, and spiral curves, the cross-track error will be very small when the radius of the curvature is large enough or when it is a straight line; at this time, the distance \(D\) of the discrete points on the track map can be very large.