## 2 Materials and Methods
### The Cant Estimation Method  
The cant is one of the most important design parameters on railway and tram lines. It is defined as a difference of rails elevation in a particular track cross section. Along the circular part of arc the nominal cant value is constant, while in adjacent parts the cant value changes. These sections of differ cant value are called cant transitions [20]. On curves with a cant, it is possible to achieve higher speeds, because a resultant value of centrifugal acceleration is lower. Therefore, even on high speed railways, the horizontal arcs are designed with a cant and cant transitions between arcs and tangents. From a point of view of presented measurements, cant affects the accuracy of positioning the track axis in the horizontal plane what is shown in Figure 1. Therefore, the measured coordinates should be corrected by taking into account a transverse shift. The correction of this shift is calculated according following Formula: [21]:  
\[\Delta Y=\frac{e}{2}\cdot(1-\sqrt{1-\left(\frac{D}{e}\right)^{2}})+\frac{D \cdot X_{a}}{e} \tag{1}\]  
where:  
* Transverse correction [mm],
* Centreline spacing of the rails [mm],
* Cant [mm],
* Measuring device mounting height [mm]. The most commonly used method of cant determination is a static measurement using a manual gauge which is equipped with a level or inclinometer. Such a device is calibrated to allow a directly  
Figure 1: Cross-section of measured track with a cant: e—distance between rails axes; D—cant; \(\alpha\)—cant angle. reading the cant in millimeters. The second common method is based on dynamic measurements and is used in measuring trolleys or trains [1; 3; 4; 9; 11]. Due to additional accelerations and dynamic effects which affect the measurement during the vehicle run, the effective inertial measurement is relatively difficult. Therefore, to determine the cant, the measuring vehicles for railway track inventory are equipped mainly with advanced gyroscopes. There are two common types of gyros:  
* Gyrostat, consisting of rotating elements which retain their original position using the gyro effect. With such a device, it is possible to directly determine the angle of rotation of the gyro casing [22]. * Vehicle's angle of inclination in relation to the plane [rad], \[t\]
- Time [s], \[\omega\]
- Angular velocity [rad/s]. Gyroscopic measurements have one fundamental disadvantage: the measurement result is dependent on the previous condition of the device, because the angle of rotation is not measured regarding to the global reference system, but to the earlier position of the gyro itself. For this reason, the device must be calibrated before measurement, but the measurement error increases with the test time [23; 24; 25]. Therefore, in many standards for railways, the requirements for cant determination are not strict (accuracy within 5 mm) [26; 27; 28]. The main disadvantage of a gyroscope is a drift, i.e., a systematic measurement error involving the accumulation of the measurement's result and a certain value (offset or bias) [29; 30]. The offsets can also change over time. Therefore, the value obtained from the gyroscope should be considered as a resultant value of such components as:  
\[\omega_{g}=\omega(t)+\omega_{d}(t)+W(t) \tag{3}\]  
where:  
* Actual angular velocity [rad/s],
* Angular velocity read from the gyroscope [rad/s],
* Drift [rad/s],
* White noise [rad/s]. In the research, a series of tests were performed to check the repeatability of angle measurements. In the case in question, a triaxial MEMS gyroscope was used. The tilt angle of the platform was measured and the obtained calculation results were compared. The tests showed very high convergence of single angular measurement. However, at the same time, rapid accumulation of measurement uncertainty due to the overlapping of measurement errors has been reported. The readout difference of 2 degrees was recorded after only four tilts of the scaling device (tests were made with different rotational speeds).