## 4 Results and Discussion  
First, performance of the tested linear encoder was checked under different scanning speeds. The maximal traversing velocity specified by a vendor of the encoder was 1 m/s. The Lissajous curves of several signal periods at 100, 500, and 1000 mm/s are shown in Figure 5.  
Figure 4: Schematic view of experiment setup. From these graphs, it is clear that encoder performance started to differ when scanning speed was increased. The Lissajous curve changed its size and shape. Higher SDE was introduced. Moreover, several tested signal periods slightly varied from one another. This added additional uncertainty that had to be evaluated. For a more detailed investigation, the reading head of the tested encoder was driven at traversing velocities from 100 to 1100 mm/s with a step of 100 mm/s. The output electric signals were recorded by using a sampling frequency of 250 MHz, and the SDE was determined according to the methodology described in a previous section. Six different signal periods were randomly chosen to calculate the average value of the SDE at each speed level. As an example, the graphical representation of the electric signals and their combination in the X-Y plot when the reading head was driven at 300 mm/s is shown in Figure 6.  
Calculated average SDE at this speed is shown in Figure 7. FFT analysis showed that the first, second, and third harmonics had the greatest influence on the error amplitude. Therefore, all three first harmonics are shown in the same graph to present the impact of each. The total magnitude of the error was \(\pm\)0.185 \(\upmu\)m. The second harmonic, which was repeated twice per period, created the biggest error value. Figure 5: Lissajous curves of several signal periods at different scanning speeds: 100, 500, and 1000 mm/s. Figure 6: Electric signals A and B and their representation in the X–Y plot. Encoder was working at 300 mm/s traversing velocity. Similar data processing was made for all tested speeds. Calculated total error values were plotted into one graph, as shown in Figure 8. Results showed a strong linear relationship between scanning speed and total SDE. More than 85% of the experimentally determined error values could be mathematically associated with the increasing traversing speed of the encoder (R-squared = 0.8581). The low \(p\)-value confirmed that the applied linear-regression model was statistically significant (\(p\)-value = 0.00004 \(<\) 0.05). The biggest error was \(\pm\)0.52 \(\upmu\)m at 1100 mm/s. Two magnitudes at 900 and 1000 mm/s were quite distinct and distinguished from the linear relation. To understand the physical meaning of the error that occurs during the scanning process, the magnitude and behavior of the first three error harmonics were analyzed. Their dependency on the tested encoder's scanning speed is shown in Figure 9.  
Figure 8: Total SDE dependency on scanning speed. Data points and their statistical parameters. Figure 7: Average SDE and its decomposition into three first harmonics when scanning speed is 300 mm/s. The biggest part of the SDE at each speed was caused by the second harmonic. This harmonic demonstrates a strong linear relationship (R-square = 0.96) to traversing velocity. Its size proportionally increased with increasing speed. This kind of error behavior is a result of signal-amplitude variation, phase shift, or their combination. The first harmonic of the error demonstrates a nonlinear relationship. Amplitude values started to increase when the encoder reached a speed above 500 mm/s. An extreme jump of values was noticed in the range of 900-1000 mm/s. In this region, magnitude relatively dropped from 0.26 to 0.01 \(\upmu\)m. This harmonic is a result of the emerged electric signals offset from a zero level. The variation of the third harmonic meaning was not large in comparison with others. Its amplitude reached the highest value of 0.06 \(\upmu\)m. This magnitude was noticed at a scanning speed of 900 mm/s, as with the first harmonic.