# Monitoring of Preload Variation of Linear Guide Positioning Stage Using Artificial Neural Network
## 3 Dynamic Characteristics of Positioning Stage
### Relationship between Stage Rigidity and Vibration Level  
Figure 10 illustrates the typical vibration signals measured at the center of the stage during the traveling movement from 0 to 1000 mm. To illustrate the difference in the vibration levels measured at different placement of sensors, we first conducted several tests on the stage assembled with specific preload conditions. For each case, the stage was driven along a stroke of 1000 mm at different feed rates of 2000, 4000, 6000, and 8000 mm/min. Each test was repeated thrice. The root mean square (RMS) values of the vibration signals in three directions (X, Y, and Z axes) were calculated to compare the vibration levels measured under different conditions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Stage Condition** & **Stage Rigidity (N/\(\mu\)m)** & **Preload Amount of the Sliding Blocks** \\ \hline Case I & 2020 & All the four blocks were preloaded at \\  & & highest level (Z2) \\ Case II & 1696 & All the four blocks were preloaded at \\  & & medium level (Z1) \\ Case III & 1176 & All the four blocks were preloaded at \\  & & medium level (Z0) \\ Case IV & 1809 & Three blocks with high preload and one \\  & & with low preload \\ Case V & 1598 & Two blocks with high preload and the \\  & & other two with low preload \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Preload conditions of stage for motion tests. Figure 10: Vibration signals measured at the center of the stage under different preloaded conditions: (**a**) case I—highest rigidity; (**b**) case II—medium rigidity; (**c**) case III—lowest rigidity; (**d**) case IV—high-medium rigidity; (**e**) case V—medium-low rigidity. Figure 9: Measurement of the vibration of the stage in feeding motion using three-axial accelerometers: (**a**) sensors at mid-points and center of stage; and (**b**) sensor at corner of stage. Figure 11 shows the vibration RMS values of stages with different rigidities under different motion speeds, in which the vibrations were obtained from sensors at different placements, (a) and (b), as illustrated in Figure 9. As shown in Figure 11a,b, for each stage preload condition, the vibration induced in motion increased with an increase in the feed rate. A higher feed rate generated higher vibrations. The influence of stage rigidity on vibration was further examined. Unfortunately, this tendency is not observed in Figure 11a, where the vibration sensors were mounted on the center point of the edge and top surface of the stage. This implies that the sensor placements did not capture the main vibration modes generated from the movement of the positioning stage. However, in another case in Figure 11b, the vibrations of the stage were clearly affected by the preload state of the stage. There was a negative relationship between the stage rigidity and vibration level, with a correlation coefficient R of around 0.74-0.79. A stage equipped with high-preloaded linear guides induced lesser vibration than the lower preloaded stage. Comparisons of the results in Figure 11a,b show that the sensor placement will affect the measurement of the vibration of the stage as well as the significance between the stage rigidity or linear guide preload and vibration. Therefore, this experiment implies that the sensor placement on the corner of the stage is more favorable for assessing the vibration of the stage in subsequent investigations. According to the study [41], vibrations associated with the angular motion of the stage can be described in terms of the angular displacement of the stage with respect to the linear axes, X, Y, and Z axes, as shown in Figure 12a, in which the angular motion (\(\phi\)) about the X axis is rolling vibration, the angular motion (\(\theta\)) about the Y axis is pitching vibration, and the angular motion (\(\psi\)) about the Z axis is the yawing vibration. Considering the contribution of the natural modes to the vibration of the feeding stage in motion, the angular vibrations were introduced in a subsequent analysis.