## 4 Experiment and Discussion  
A series of experiments were carried out to verify the effectiveness of the above-motioned active methods. A laboratory-built prototype of the LD-based system was constructed to measure the \(\delta x\), \(\delta y\), \(\theta x\) and \(\theta y\) of a linear stage, as shown in Figure 9. A compact and low-cost LD (DA635, Huanic, Xi'an, China), which had an X-directional diameter \(d_{x}\) of 5 mm, an ellipticity e of 0.945 and a wavelength of 635 nm, was adopted as the laser source of the system. A comparison unit, with a construction similar to the detector, was integrated into the measurement system to carry out a group of comparison experiments. The \(s_{1}\), \(s_{2}\) and \(s_{3}\) shown in Figure 9 represent the distance between the sensor head and the feedback unit 2, and the detector and the comparison unit, respectively. Figure 8: The flowchart of the BPNN-based PID controller. Figure 9: Experimental setup of the LD-based measurement system with the IAEC. Firstly, the effect of the position offset \(\Delta z\) on the laser spot characteristic was investigated. The relationship between the sensitivity of the photodetector and the diameter of the laser beam can be expressed by:  
\[K_{\theta x}(\Delta z)=\frac{8}{1.22\pi}\frac{d_{x\Delta z}}{\lambda}\text{ and }K_{\theta y}(\Delta z)=\frac{8}{1.22\pi}\frac{d_{y\Delta z}}{\lambda}, \tag{19}\]  
where \(K_{\theta x}(\Delta z)\) and \(K_{\theta y}(\Delta z)\) represent the \(X\)- and \(Y\)-directional sensitivities of the QD, respectively. As can be seen, if \(K_{\theta x}(\Delta z)\) equals \(K_{\theta y}(\Delta z)\), \(d_{x\Delta z}\) will be equal to \(d_{y\Delta z}\), which means the elliptical laser spot is shaped to be a circle laser spot. Therefore, \(\Delta z\) can be confirmed by the evaluated \(K_{\theta x}(\Delta z)\) and \(K_{\theta y}(\Delta z)\) using a group of calibration experiments. Figure 10 shows the calibrated \(X\)-directional and \(Y\)-directional sensitivities of the QD under various \(\Delta z\). As can be seen, when \(\Delta z\) equals 48.73 \(\upmu\)m, \(K_{\theta x}(\Delta z)\) is approximately equal to \(K_{\theta y}(\Delta z)\). As mentioned above, the LD exits at a divergence angle \(\alpha\), which will affect \(\Delta z\). Therefore, a commercial beam-profiling camera was used to detect the diameter of the LD at various measurement positions. When \(s_{2}\) was set to be 50 mm and 1500 mm, the \(X\)-directional diameter (\(d_{x}\)) of the LD was evaluated to be 5.0 mm and 5.4 mm, respectively. The variation of \(\Delta z\) was thus evaluated to be 4.51 \(\upmu\)m according to the simulation results. The variation of \(\Delta z\) was small enough that the effect of the divergence angle of the LD along the measurement direction on the measurement accuracy of the geometric errors could be ignored. Therefore, the position offset \(\Delta z\) was fixed to be 48.73 \(\upmu\)m in the following experiments. The capability of the designed IAEC and the stability of the LD-based measurement system were then tested. Firstly, the traditional PID controller was used to compensate the laser beam drifts for comparison. Figure 11 shows the outputs of QD5 and QD6 when the IAEC was activated with the traditional PID controller. s1, the sampling time and the sampling frequency were set to be 1800 mm, 5 h and 100 Hz, respectively. The Butterworth filter and the moving average method were applied to reduce the effect of the noise. It can be seen from the figure that the stability ranges of QD5 and QD6 were within 2.0 \(\upmu\)m and 5.0 arcsec, respectively. The BPNN-based PID controller was then used to compensate the laser beam drifts. A neural network with three, five and three neurons in the input layer, the hidden layer and the output layer, respectively, was applied. A tangent hyperbolic was selected as an activation function of the hidden layer and the output layer of the BPNN-based PID  
Figure 10: Relationship between \(\Delta z\) and the detector sensitivity. controller.