## 2 Materials and Methods
### Measurement Platform Configuration and Correction Principle  
To evaluate the potential for uncertainty reduction and reliability enhancement in a mobile railway track axis positioning system, a six-antenna, multi-receiver configuration was implemented on the measurement platform. Two primary antennas were mounted directly above the bogie pivot points, while four auxiliary antennas were arranged at \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Point Id & 5830F00170\_20210609\_083941749 \\ Start Time & 9 June 2021 08:39:42 \\ GPS Time & 2161 290381.750 \\ WGS84 Cartesian X [m] & 3,531,069.0187 \\ WGS84 Cartesian Y [m] & 1,191,245.5035 \\ WGS84 Cartesian Z [m] & 5,158,894.6301 \\ WGS84 Latitude [\({}^{\circ}\)] & 54.33952279\({}^{\circ}\)N \\ WGS88 Longitude [\({}^{\circ}\)] & 18.64239643\({}^{\circ}\)E \\ WGS84 Ellip. Height [m] & 34.6315 \\ Easting [m] & 6,541,778.6156 \\ Northing [m] & 6,023,434.1565 \\ Ortho. Height [m] & 5.3597 \\ CQ 3D [m] & 0.0112 \\ CQ 2D [m] & 0.0063 \\ CQ 1D [m] & 0.0092 \\ PDOP & 2.1 \\ HDOP & 1.2 \\ VDOP & 1.7 \\ GDOP & 2.9 \\ GPS SVs & 6/9 \\ GLONASS SVs & 4/5 \\ Galileo SVs & - \\ Beidou SVs & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sample record of GNSS RTK/RTN measurement. known offsets to form a rectangular array (Figure 2). Synchronous GNSS measurements from this array of six receivers, each with precisely surveyed horizontal offsets, provide redundant observations that improve the accuracy of the pivot-point antenna positions. Antennas AC and BC, which are mounted directly above the bogie pivots (Figure 2a), serve as the primary receivers, while the remaining four act as auxiliary receivers. Antenna positioning was performed with tachymetric precision: longitudinal beams affixed to the platform supported specialized mounts featuring holes for the centering screws of geodetic prisms. Prior to antenna installation, prism locations were measured with a tachymeter using reference prisms. Following adjustment, the antenna array achieved a precisely surveyed spatial configuration, with coordinate uncertainty below 1 mm. The proposed method employs an analytical solution to correct the coordinates of the primary receivers mounted above the bogie pivots. To simplify the correction algorithm, the actual antenna positions are constrained to an ideal rectangular layout (Figure 2b), reducing the number of parameters required for computing the \(X\) and \(Y\) adjustments. The pairwise distances between antennas are then represented in a symmetric distance matrix \(L\), as defined by Equation (1):  
\[\mathbf{L}= \begin{bmatrix}0&\cdots&L_{ij}\\ \vdots&\ddots&\vdots\\ L_{ij}&\cdots&0\end{bmatrix} ,i=1,2,\cdots,6\,j=1,2,\cdots,6 \tag{1}\]  
where \(\mathbf{L}=\sqrt{\left(X_{i}-X_{j}\right)^{2}+\left(Y_{i}-Y_{j}\right)^{2}}\). \(X_{i},Y_{i}\)--coordinates of the prism mounted on the mounts during the tachymetric adjustment of the antenna positions on the platform. The determined control distances can be divided into so-called \"short\" distances, corresponding to the transverse dimensions (\(L_{4}\)-\(L_{9}\)), and \"long\" distances characteristic of the longitudinal dimensions (\(L_{1}\)-\(L_{3}\)) and (\(L_{10}\)-\(L_{15}\)) (cf. Figure 3). Using the notation from Figure 3a, we obtain the matrix \(L\), which is a symmetric matrix, and the rectangular layout  
Figure 2: Concept of GNSS receiver arrangement: (**a**) Arrangement of GNSS receivers on the measurement wagon, (**b**) geometric layout of the GNSS antenna configuration: antennas AC and BC define the baseline between the bogie pivot points. in which the antennas are arranged leads to a reduction in the values of individual lengths, which can be grouped as follows:  
\[L_{1}=L_{2}=L_{3}=L_{1} \tag{2}\]  
\[L_{4}=L_{5}=L_{6}=L_{7}=L_{\rm II} \tag{3}\]  
\[L_{8}=L_{9}=L_{\rm III} \tag{4}\]  
Taking into account the track's inclination, the distances between the GNSS receivers determined in two-dimensional Euclidean space require adjustment for their reduction.