## 1 Introduction  
Today, trains are one of the most common options for transporting passengers and freight. Rising oil prices will inevitably force us to use rail transport even more. Studies on the improvement of railway track components contribute greatly to the construction of better and, above all, more comfortable railway lines, which also ensure greater safety [1]. This study focused on the measurement of vibration and consequent deformation of railway concrete sleepers, which are a key element in the railway track structure ensuring railway safety. Prestressed concrete sleepers (PCSs) are a key structural element that support the railway system and absorb the varying load from trains running on the track [2]. Rail sleepers, which are one of the most important elements of the railway superstructure, are very important in maintaining the geometry of the track and transferring the loads and vibrations from the rails to the infrastructure [3]. The vibrations absorbed by the rails and, through them, by the rail sleepers can be harmonic or inharmonic. Harmonic vibrations occur in the case of periodic anomalies due to rail irregularities or inappropriate wheel geometry, while inharmonic vibrations occur when the distance between sleepers changes, switches are positioned or the track beam is deformed [4]. In addition to quality sleepers, a suitable subgrade is also important for taking dynamic loads, as the stability of railway tracks is a key safety and operational requirement [5, 6, 7]. Research to date has shown that improving the railway system should start with accurate analysis and monitoring of vibrations and deformations on track elements. Such monitoring reliably leads to correct decisions related to track maintenance and railway safety. Properly carried out monitoring of the dynamic response of railway sleepers can greatly assist engineers involved in the analysis of the causes of damage--which are usually due to vibration, particularly at resonant frequencies--and of the procedures for improvements in design [8]. It is important that such measurements use simple techniques that do not impede traffic on the railway line, and the most appropriate are undoubtedly non-contact measurement techniques. Increasing traffic load, together with higher speeds and axial loads, requires special measures for maintenance or adequacy control, such as non-contact monitoring. One of the important issues in this respect is the early detection, identification and analysis of possible deformations in all elements of the railway line. This is very important, especially for railway lines that are in almost continuous use and where it is very difficult to find traffic-free periods for sophisticated geodetic measurements [9; 10; 11]. Over the last fifteen years, the development of geodetic instruments, sensors and global positioning systems (GPSs) and navigation satellite systems (GNSSs) has allowed for the wider integration of these types of instruments in motion and vibration monitoring systems [12]. Dynamic tests have recently become more widespread. They are considered to be the most comprehensive monitoring techniques for bridge structures, allowing a global assessment of the structure [13]. The first investigations and projects for the determination of displacements and deformations in construction structures using geodetic measurements appeared in the 1990s [14]. The first applications of GPS measurement systems for the determination of displacements and deformations in dams appeared at the end of the 1980s [15]. After the first dynamic monitoring projects using geodetic instruments--primarily GNSSs--research has focused on sensor integration and monitoring system operability. In recent years, however, the use of GNSSs has been replaced by the use of robotic total stations (RTS), which allow us to continuously monitor a geodetic target in motion at a frequency of up to 30 Hz.