## 2 Measuring System GEPARD by Raytec  
The GEPARD device is primarily used for measuring the linearity, parallelism, and perpendicularity of linear guides but also for measuring position deviations and coaxiality. It works with a laser beam as a reference standard for linearity. The device consists of a laser transmitter and a laser receiver. The laser beam is simply aimed parallel to the linear guide, targeting the laser receiver. The latter can detect subtle changes in the position of the incident beam on the receiving sensor in a very simple way. The linearity of the path is measured by moving the receiver along the linear trajectory and continuously measuring the beam deflections [17]. Compared to conventional solutions, a huge advantage of the device is that it enables measuring deviations in both directions perpendicular to each other simultaneously, even in dynamic mode. This mode enables the user, for example, to align the guiding with the final curves being evaluated in real-time. A significant advantage is the wireless interconnection of all components. The laser receiver communicates with the computer through Bluetooth technology. Both the laser transmitter and the receiver are powered by batteries [18]. The main part of the receiver is a sensitive position detection matrix (PSD) and the amplifying and evaluating electronics. The GEPARD (type 5) matrix is 10 mm \(\times\) 10 mm in size. The laser beam hits the matrix through a ground opening, so the transverse range of measurement in the X- and Y-axes is 5 mm \(\times\) 5 mm, Figure 1. [19]. To measure linearity, place the laser transmitter at the beginning of the measured guiding and guide the transmitter beam to target the laser receiver position sensor. To set the beam, use micrometric screws. In the software, select the position-measuring mode. In this mode, the area of the beam position sensor will appear on the personal computer screen, indicating the place where the beam falls. Using micrometric screws, we make the place of the beam incidence more precise, as close to the geometric center of the sensor as possible, in the range of the measured length [20]. The laser receiver gradually moves along the measured path to selected points, and at each point, the position of the incident laser beam is measured in the x- and y-axes. The measured data appear on the computer screen and are plotted in two ready-made charts [21], with axis x showing the receiver distances from the first measured point and axis y showing the measured deviations. The values measured on axis x are plotted in the first graph, and those measured on axis y in the second, Figure 2[22]. Using the least squares method, the software lays a straight line, which is called the regression line, over the measured points. The least squares method is a mathematical and statistical method of approximating pairs of measured data [_xy_]. Approximation means the closest representation of the real value. The simplest case of using the least squares  
Figure 1: Laser receiver—transmitter with PSD. Figure 2: Software GEPARD, v 6.30.1.  
method is an approximation of the measured data using the straight line \(\hat{y}=\text{kx}+\text{q}\). The resulting sum of the areas of all the squares must be the smallest possible:  
\[S=\sum_{i=1}^{n}\left(\Delta y_{i}\right)^{2} \tag{1}\]  
Upon inserting the dependence function for the straight line, we obtain the following:  
\[S=\sum_{i=1}^{n}\left[y_{i}-\left(\text{kx}_{i}+\text{q}\right)\right]^{2} \tag{2}\]  
We are looking for the k and q factors so that all the \(\Delta y_{i}\) deviations are the smallest possible. A criterion is that the sum of the second power of all deviations must have the least value. We use the second power to obtain a positive value. Thus, in terms of geometry, we deal with the square [23]. For the user to see an exact deviation of lengthwise measurements from zero or the reference line, the individual measured values must be mathematically processed to compensate for deflection (a) and the tilt angle (\(\alpha\)).