## 2 Materials and Methods
### Algorithm for Determining the Correction of Track Axis Coordinates
#### 2.4.2 Calculation of the Corrected Coordinate Values  
Upon evaluating each receiver based on condition (10), it becomes possible to identify those that satisfy the predefined quality criteria. The coordinates of receivers condition (10) evaluates as TRUE form the input dataset for the algorithm responsible for computing the corrected positions of the central receivers AC and BC. Receivers for which the condition evaluates to FALSE are excluded from further analysis. The coordinates of the main receivers' antennas, AC and BC--positioned directly above the pivots of the platform's rotational axis--provide a precise and sufficient representation of the track axis location. The logical structure associated with the availability of individual receivers (i.e., those satisfying condition (10)) allows for the definition of nine possible configurations for computing the corrected coordinates of receiver AC, and likewise, nine configurations for receiver BC. The first three configurations pertain to scenarios in which either receiver AC or more than one receiver from group A is available. In these cases, the corrected coordinates are derived exclusively from the available receivers within group A. The remaining configurations (4 through 9) address situations in which only a single receiver from group A is available (i.e., either AL or AR). In such instances, the computation of the resulting position of the AC receiver's antenna necessitates the inclusion of at least one receiver from group B to determine the spatial orientation of the receiver layout, and consequently, the spatial orientation of the measurement platform itself. Below are the equations describing the variants for the pivot receiver AC. The relationships for receiver BC are analogous. Variant 1--All receivers within group A are available, whereas the availability of receivers within group B is optional. \[W_{1}: \begin{bmatrix}[W_{AC}\wedge W_{AL}\wedge W_{AR}\wedge(W_{BC}\lor W_{ BL}\lor W_{BR})]\lor\\ \lor[W_{AC}\wedge W_{AL}\wedge W_{AR}\wedge\
eg W_{BC}\wedge\
eg W_{BL}\wedge \
eg W_{BR}]\end{bmatrix}  \tag{11}\]  
Corrected AC receiver coordinates:  
\[Y_{\text{AC}1^{\prime}}=\frac{Y_{\text{AC}}+Y_{\text{AL}}+Y_{\text{AR}}}{3} \tag{12}\]  
\[X_{\text{AC}1^{\prime}}=\frac{X_{\text{AC}}+X_{\text{AL}}+X_{\text{AR}}}{3} \tag{13}\]  
Variant 2--Receivers AL and AR are available, while receiver AC is unavailable, and receivers from group B may or may not be available. Logical condition:  
\[W_{2}: \begin{bmatrix}[\
eg W_{AC}\wedge W_{AL}\wedge W_{AR}\wedge(W_{BC}\lor W _{BL}\lor W_{BR})]\lor\\ \lor[\
eg W_{AC}\wedge W_{AL}\wedge W_{AR}\wedge\
eg W_{BC}\wedge\
eg W_{BL} \wedge\
eg W_{BR}]\end{bmatrix}  \tag{14}\]  
Corrected receiver coordinates:  
\[Y_{\text{AC}2^{\prime}}=\frac{Y_{\text{AL}}+Y_{\text{AR}}}{2} \tag{15}\]  
\[X_{\text{AC}2^{\prime}}=\frac{X_{\text{AL}}+X_{\text{AR}}}{2} \tag{16}\]  
Variant 3--Receivers AC and AL, or AC and AR, are available, while receivers from group B may or may not be available. Logical condition:\[W_{3}: \begin{bmatrix}[W_{\text{AC}}\wedge\
eg W_{\text{AL}}\wedge W_{\text{AR}} \wedge(W_{\text{BC}}\lor W_{\text{BL}}\lor W_{\text{BR}})]\vee[W_{\text{AC}} \wedge W_{\text{AL}}\wedge\
eg W_{\text{AR}}\wedge(W_{\text{BC}}\lor W_{\text{BL }}\lor W_{\text{BR}})]\vee\\ \vee[W_{\text{AC}}\wedge\
eg W_{\text{AL}}\wedge\
eg W_{\text{AR}}\wedge(W_{ \text{BC}}\lor W_{\text{BL}}\lor W_{\text{BR}})]\vee[W_{\text{AC}}\wedge W_{ \text{AL}}\wedge\
eg W_{\text{AR}}\wedge\
eg W_{\text{BC}}\wedge\
eg W_{ \text{BL}}\wedge\
eg W_{\text{BR}}]\vee\\ \vee[W_{\text{AC}}\wedge\
eg W_{\text{AL}}\wedge W_{\text{AR}}\wedge\
eg W_{ \text{BC}}\wedge\
eg W_{\text{BL}}\wedge\
eg W_{\text{BR}}]\end{bmatrix}  \tag{17}\]  
Corrected receiver coordinates:  
\[Y_{\text{AC3}^{\prime}}=Y_{\text{AC}} \tag{18}\]  
\[X_{\text{AC3}^{\prime}}=X_{\text{AC}} \tag{19}\]  
Variant 4--Within group A, only receiver AL is available, whereas within group B, receiver BL is available.