## 3 Development of Wireless Laser Displacement Monitoring Equipment
### Measurement Principle of Wireless Laser Displacement Monitoring Equipment  
Based on wireless transmission architecture, the principles of laser pulse ranging and phase ranging were combined for different application situations. Through the improvement of the algorithm principle, monitoring equipment that meets the needs of the deflection monitoring of short and medium-span bridges was developed. A schematic diagram of the laser displacement measurement is shown in Figure 4. The laser displacement monitoring equipment is installed at the position of the bridge pier or abutment, and the reflector with enhanced radiation is installed at the measurement point. The displacement of the beam at the measuring point is calculated through the conversion of laser ranging, sensor installation angle and reflector installation angle. The model of this algorithm is shown in Figure 5. The reflective device is arranged at the position where the deflection of the measured bridge is large, and the three-dimensional coordinates (\(x_{0}\), \(y_{0}\), \(z_{0}\)) of the laser emission point O are read by the total station. The three-dimensional coordinates of point A (the laser reflection) are (\(a_{0}\), \(b_{0}\), \(c_{0}\)), and the other three noncollinear points A\({}_{0}\) (\(x_{1}\), \(y_{1}\), \(z_{1}\)), B\({}_{0}\) (\(x_{2}\), \(y_{2}\), \(z_{2}\)), and C\({}_{0}\) (\(x_{3}\), \(y_{3}\), \(z_{3}\)) (on the reflective device) are read as well. The laser monitoring device reads the initial distance \(l_{0}\) from the laser monitoring device to the reflective device, and after the reflective device settles \(d\), the device reads the distance between the laser monitoring device and the reflective  
Figure 4: Schematic diagram of displacement measurement with laser displacement sensor. Figure 3: Waveform and timing diagram of the pulsed laser ranging system. device \(l_{1}\). In the model used, \(\alpha\) is the initial inclination angle between the laser monitoring equipment and the horizontal plane, \(\beta\) is the initial inclination angle between the reflective device and the horizontal plane, point A is the initial reflection point position of the laser monitoring equipment on the reflective device, A' is the corresponding position of the reflection point after the reflective device moved, and B is the reflection point position of the reflective device. By the law of sine, we can obtain Formulas (6) and (7):  
\[\frac{d}{\sin\left(\alpha+\beta\right)}=\frac{l_{0}-l}{\cos\beta} \tag{6}\]  
\[d=\frac{\left(l_{0}-l\right)\sin(\alpha+\beta)}{\cos\beta} \tag{7}\]  
The inclination angle (\(\alpha\)) (Formula (8)) between the laser monitoring equipment and the horizontal plane is:  
\[\alpha=\arctan\left(\frac{\left|z_{0}-c_{0}\right|}{\sqrt{\left(x_{0}-a_{0} \right)^{2}+\left(y_{0}-b_{0}\right)^{2}}}\right) \tag{8}\]  
According to the coordinates of the three noncollinear points (\(A_{0}(x_{1},y_{1},z_{1})\), \(B_{0}(x_{2},y_{2},z_{2})\), \(C_{0}(x_{3},y_{3},z_{3})\)) initially read by the laser monitoring device (see Figure 6), the normal vector of the plane can be expressed as Formulas (9) and (10):  
\[n=A_{0}B_{0}\times A_{0}C_{0}=(a,b,c) \tag{9}\]  
\[a=(y_{2}-y_{1})(z_{3}-z_{1})-(y_{3}-y_{1})(z_{2}-z_{1})\]  
\[b=(z_{2}-z_{1})(x_{3}-x_{1})-(z_{3}-z_{1})(x_{2}-x_{1})\]  
Figure 5.0  
The angle (\(\beta\)) between the reflective device and the horizontal plane is Formula (11):  
Figure 5.1  
Assuming that the accuracy of laser ranging is \(\Delta l\) and the accuracy of displacement measurement is \(\Delta d\), the displacement conversion accuracy is expressed in Formula (13):  
Figure 5.2  
It can be seen from the above formula that the smaller the value of \(k\) is, the higher the displacement accuracy of the conversion. Therefore, the smaller \(\alpha\) and \(\beta\) are, the smaller k and \(\Delta d\) are. When \(\beta\) approaches \(0^{\circ}\), \(\cos\beta\) approaches 1. At this time when \(\Delta l\) is certain, \(\Delta d\) is only related to \(\alpha\).