# A Practical Positioning Method in End-Plate Surface Distance Measurement with Nano-Meter Precision
## 4 Accuracy Enhancing Methods
### Optical and Mechanical Aspects
#### 4.1.1 Vibration  
Among the mechanical error sources, the vibration is a leading factor that affects the repeatability. The vibration isolation is a necessary step that is relatively easy to do. Generally, it is effective to eliminate the vibration coming from outside the system when the measurement system is mounted in a whole on the isolation platform. For the vibration generated from inside the system, it is not easy to eliminate vibration by isolating only. The vibration elimination in this case is studied. Figure 6 is the layout of experiment system.  
Figure 5: Theoretical white light interference signal.  
Figure 6: The layout of positioning unit.  
A kinetic characteristic of the system is expressed by Equation (4).  
\[M\ddot{\varphi}+C\dot{\varphi}+K\varphi=Q \tag{4}\]  
where \(C\) is the system damping matrix; \(K\) is the system stiffness matrix; \(Q\) is the systematic general force matrix; \(M\) is the mass matrix; \(\phi\) is the modal coordinates describing the flexible deformation of the system; and \(\overset{\bullet}{\phi}\) is the first derivative and second derivative of \(\phi\).  
Based on Equation (4), the equation for solving the natural frequency of the system can be obtained as Equation (5).  
\[\det(p^{2}I-M^{-1}K)=0 \tag{5}\]  
where \(p\) is the natural frequency of the system and \(I\) is the unit matrix.  
As seen from Equation (5), the natural frequency of the system is determined by the mass matrix \(M\) and the stiffness matrix \(K\). \(M\) and \(K\) are related to the structural and motion parameters of the system, respectively. For structural parameters, if the height of the light beam (the height of step mounting) is decreased, the stiffness of the system will increase, and the same goes for natural frequency of the corresponding system. The repeatability experimental results confirmed that when the height of step mounting was decreased from 50 mm to 30 mm, the repeatability was better than 20 nm. For motion parameters, the moving speed \(V\) is one of the adjustable parameters to obtain the better signal. The experiment result showed that when the moving speed \(V\) was 50 \(\upmu\)m/s, the best signal was obtained. The positioning signal at different moving speeds is presented in Figures 7-9. In the case of a relative lower speed at 30 \(\upmu\)m/s, it was easy to be affected by different frequencies of vibration. In case of a relative higher speed at 80 \(\upmu\)m/s, although the signal SNR was good, the symmetrical characteristics of the signal were poor, which also affected the signal to reduce the positioning repeatability.  
The moving stability of table is another factor to affect repeatability. Figure 10 shows the length interference signals of different motion tables. Automatic interference comparator (AIC) is a specialized length measurement with a high precision. The uniformity and smoothness of the interference signal are very good because the motion slide is driven by hydraulics [6; 7]. The vibration from the driven unit is negligible. The motion driven part used in this paper is a step motor, and the vibration from the motor is inevitable. Then, if the driven mode changes, the uniformity and smoothness of the motion will be improved. Estimating from the measurement experience of AIC, at least 5 nm of repeatability will be reduced.  
Figure 7: Positioning signal at moving speed of 30 \(\upmu\)m/s.  
Figure 8: Positioning signal at moving speed of 50 \(\upmu\)m/s.  
Figure 10: Signal difference in different driving modes. AIC is automatic interference comparator.  
Figure 9: Positioning signal at moving speed of 80 \(\upmu\)m/s.