## 2 Materials and Methods
### Calculation of Corrections--Algorithm  
The initial step in determining the cant is the rough (automatic) analyses of the obtained measurement signal. Based on the GNSS signal analysis, the track curvature at any given point and the azimuth of the tangent to the track axis at that place are determined. The track curvature radius can be roughly determined based on an analysis of horizontal versines. Since this method is affected by numerous uncertainties, it is regarded exclusively as a preliminary  
Figure 3: The distribution of acceleration in different movement patterns. (**a**) a case of movement along a circular curve without a cant, (**b**) a case of movement along a tangent section with a cant, (**c**) a case of movement along a circular curve with a cant. solution bringing the measured signal closer to the actual track axis shape. To this end, for each of the measured points, a 20 m long virtual chord is introduced (10 m on each side of the point) and the value of the horizontal ver  
for this point is calculated (Figure 4). Considering that the length of a chord sections \(l_{1}\) and \(l_{2}\) may generally be different from each other, the following relationship is used to calculate the arc radius:  
\[s=R-\sqrt{R^{2}-l_{1}\cdot l_{2}} \tag{9}\]  
where:  
\(s\) - Versine [m],  
\(l_{1}\), - Length of the curve chord part [m]. \(l_{2}\)  
After expanding of this expression into a Maclaurin series and its conversion to calculate the radius, the corresponding relationship is expressed by the following Formula:  
\[ \begin{array}{ll}R=&\frac{\sqrt{\left(X_{p(i-n)}-X_{p(i_{\text{max}})}\right) ^{2}+\left(Y_{p(i-n)}-Y_{p(i_{\text{max}})}\right)^{2}}\cdot\sqrt{\left(X_{p( i_{\text{max}})}-X_{p(i+n)}\right)^{2}+\left(Y_{p(i_{\text{max}})}-Y_{p(i+n)} \right)^{2}}}{2\cdot\sqrt{\left(X_{p(i_{\text{max}})}-X_{p(i)}\right)^{2}+ \left(Y_{p(i_{\text{max}})}-Y_{p(i)}\right)^{2}}}+\ldots\\ &\ldots+\frac{\left(\left(X_{p(i-n)}-X_{p(i_{\text{max}})}\right)^{2}+\left(Y _{p(i-n)}-Y_{p(i_{\text{max}})}\right)^{2}\right)\cdot\left(\left(X_{p(i_{\text {max}})}-X_{p(i+n)}\right)^{2}+\left(Y_{p(i_{\text{max}})}-Y_{p(i+n)}\right)^{ 2}\right)}{8\cdot\left(\left(X_{p(i_{\text{max}})}-X_{p(i)}\right)^{2}+\left( Y_{p(i_{\text{max}})}-Y_{p(i)}\right)^{2}\right)}\end{array}  \tag{10}\]  
where:  
\(X_{p}\) - The northern coordinate in the Cartesian coordinate system [m],  
\(Y_{p}\) - The eastern coordinate in the Cartesian coordinate system [m],  
\(i\) - Measurement index,  
\(n\) - Natural number,  
\(P(aux)\) - Auxiliary point. The first operation to be carried out to calculate the auxiliary point coordinates is to calculate the equation of the straight line passing through the two points that determine the chord and of the straight perpendicular line passing through the point at which the curvature radius is being determined:  
\[m_{p(i-n)p(i+n)}=\frac{X_{p(i-n)}-X_{p(i+n)}}{Y_{p(i-n)}-Y_{p(i+n)}} \tag{11}\]  
\[b_{p(i-n)p(i+n)}=X_{p(i-n)}-\frac  
and the auxiliary point coordinates are as follows:  
\[Y_{P(aux)}=\frac{b_{P(i)P(aux)}-b_{P(i-n)P(i+n)}}{m_{P(i)P(aux)}-m_{P(i-n)P(i+n)}} \tag{15}\]  
\[X_{P(aux)}=m_{P(i-n)P(i+n)}\cdot\frac{b_{P(i)P(aux)}-b_{P(i-n)P(i+n)}}{m_{P(i)P (aux)}-m_{P(i-n)P(i+n)}}+b_{P(i-n)P(i+n)} \tag{16}\]  
where:  
\(m\): - Straight line inclination, \(b\): - Absolute term [m]. Then, for each pair of coordinates measured by the satellite method, the time interval between the previous and the next measurement is determined (with an accuracy of 0.01s). Later on, for this time interval, a mean value of the measured acceleration is measured (the acceleration is measured with a frequency of 100 Hz, therefore there are an average of 10 single acceleration measurements per a pair of coordinates). Since half of the analyzed acceleration signal is then introduced into calculations for the next point, the method in question is identical to the analysis of a signal in a time series using a moving average.