## 3 Factors Affecting Accuracy and Accuracy-Improvement Methods
### Effect of the Stability in Output Beam Direction of the LD  
Laser beam drifts have been proved to be one of the critical error sources in the LD-based measurement system, especially for the long-distance measurement. In our previous works [11], an active error compensator, which consisted of an optical mount (OM), a mirror and two mini-sized PZT actuators was proposed to compensate the angular drift of the LD. However, it only considered the angular drift near the sensor head. The laser beam drifted far away from the sensor head and the linear drift was not considered. To address the above problems, an improved active error compensator (IAEC) was proposed and integrated into the LD-based system for simultaneously compensating the linear drift and the angular drift of the LD in the long-distance measurement. The principle of the IAEC is shown in Figure 6, which was composed of a feedback unit 1, a feedback unit 2 and a Backpropagation Neural Networks-based PID (BPNN-based PID) controller. The feedback unit 1 consisted of QD5 and a mechanism 1. The feedback unit 2 consisted  
Figure 4: Relationship between the position offset and the laser beam diameter. Figure 5: Effect of the position offset on the measurement accuracy (**a**) without the angular error, (**b**) with the angular error and (**c**) with the straightness error. of QD6, FL6 and a mechanism 2. The mechanism 1 and the mechanism 2 had an identical construction, which consisted of an optical mount (OM), a mirror (M) and two mini-sized PZT actuators (P). Compared with previous research [11; 23; 24], the improved active error compensator was not only simple in structure, but could also compensate both the linear drift and the angular drift of the laser source for the long-distance measurement. The inputs of the BPNN and the PID controller were the outputs of QD5 (\(\delta_{\alpha}\) and \(\delta_{\beta}\)) and QD6 (\(\theta_{\alpha}\) and \(\theta_{\beta}\)), which represented the linear drift and the angular drift of the LD-based system. The parameters of the PID controller (\(K_{i}\), \(K_{p}\), \(K_{d}\)) were obtained in real-time by the neural network training of the BPNN. The angles of OM1 and OM2 were fast tuned by controlling the mini-PZTs' voltages (\(V_{i}\), \(i\) = 1 to 4) using the BPNN PID controller. M1 and M2, mounted on OM1 and OM2, were then employed to adjust the propagation direction of the laser beams so that the laser spots could always remain at the centers of QD5 and QD6, from which the linear drift and the angular drift of the LD could be actively compensated. The outputs of QD5 (\(\delta_{\alpha}\) and \(\delta_{\beta}\)) and QD6 (\(\theta_{\alpha}\) and \(\theta_{\beta}\)) were, respectively, composed of four parts, namely outputs caused by the angular drift (\(\alpha_{0}\) and \(\beta_{0}\)), outputs caused by the linear drift (\(\varepsilon\) and \(\eta\)), outputs caused by the rotation of M1 (\(\alpha_{1}\) and \(\beta_{1}\)) and outputs caused by the rotation of \(M2\) (\(\alpha_{2}\) and \(\beta_{2}\)), as shown in Figure 7. For the sake of being brief, only the related optical elements and the intersection point between the optical element and the laser beam are shown in Figure 7.  
In the ideal case, the incident laser beam \(I_{0}\) travels along the \(Z\)-axis with a direction vector of \(\left[ \begin{array}{cc}0&0&-1\end{array} \right]^{T}\). However, the direction vector is varied to be \(\left[ \begin{array}{cc}\alpha_{0}&\beta_{0}&-1\end{array} \right]^{T}\) when the angular drift (\(\alpha_{0}\) and \(\beta_{0}\)) exists about the \(X\) and \(Y\) axes, as shown in Figure 7a. The normal vector of the reflection planes of M1, M2 and BS1 can be expressed by:  
\[n_{M_{1}}=\left[ \begin{array}{cc}N_{M1x}&N_{M1y}&N_{M1z}\end{array} \right] =\ \left[ \begin{array}{cc}\frac{\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\end{array}  \right]^{T}, \tag{11}\]  
\[n_{M_{2}}=\left[ \begin{array}{cc}N_{M2x}&N_{M2y}&N_{M2z}\end{array} \right] =\ \left[ \begin{array}{cc}-\frac{\sqrt{2}}{2}&0&\frac{\sqrt{2}}{2}\end{array}