## 2 Detection Principles and Self-Deflection Analysis
### Analysis of Borescope Trolley Attitude Deflection  
By utilizing the alignment and ranging capabilities of the dot laser, it is possible to correct the three-axis position of the borescope trolley, but three-axis attitude cannot be rectified. Meanwhile, due to phenomena such as wear and aluminum deposition, the bore surface becomes uneven, causing the borescope trolley to pitch, yaw, and roll during its movement. Roll deviation can be mitigated by registering the profile laser point clouds, whereas pitch and yaw deviations cannot be mitigated. Therefore, it is necessary to analyze their impact. In addition, since the manners in which pitch and yaw affect the trolley's detection are similar, only the influence of pitch will be discussed.  
Assuming that the borescope trolley experiences a pitch deviation as illustrated in Figure 4, where the rear wheel drop depth is \(h_{1}\), the front wheel lift height is \(h_{2}\), and the distance between the front and rear wheels is \(L\), the pitch angle \(\theta\) is given by  
\[\theta=\arcsin\frac{h_{1}+h_{2}}{L} \tag{9}\]  
The trolley can be considered a rigid body, so the ring laser plane also undergoes a deflection angle of \(\theta\). From this, the vertical coordinates \(y^{\prime}_{\text{w}}\) of the deflected point clouds in the world coordinate system, with the dot laser spot center \(p\) as the origin, satisfy  
Figure 3: Schematic of bore straightness extraction.  
\[y^{\prime}_{\rm w}=\frac{y_{\rm w}}{\cos\theta} \tag{10}\]  
where \(y_{\rm w}\) is the vertical coordinate of the undeflected point clouds in the world coordinate system.  
Before and after deflection, the world coordinate system centered at \(p\) and the registered coordinate system satisfy  
\[\left\{\begin{array}{l}y_{\rm w}=y_{\rm r}-y_{\rm r,p}\\ y^{\prime}_{\rm w}=y^{\prime}_{\rm r}-y^{\prime}_{\rm r,p}\end{array}\right. \tag{11}\]  
where \(y_{\rm r}\) and \(y^{\prime}_{\rm r}\) are the vertical coordinates of the laser point clouds in the registered coordinate system before and after deflection, respectively, and \(y_{\rm r,p}\) and \(y^{\prime}_{\rm r,p}\) are the vertical coordinates of point p in the registered coordinate system before and after deflection.  
According to (10) and (11),  
\[y^{\prime}_{\rm r}=\frac{y_{\rm r}-y_{\rm r,p}}{\cos\theta}+y^{\prime}_{\rm r,p} \tag{12}\]  
Regardless of whether deflection occurs, the vertical coordinate of the center point of the bore profile in the registered coordinate system should be 0. That is, when \(y_{\rm r}=0\), then \(y^{\prime}_{\rm r}=0\); therefore,  
\[y^{\prime}_{\rm r,p}=\frac{y_{\rm r,p}}{\cos\theta},\quad y^{\prime}_{\rm r}= \frac{y_{\rm r}}{\cos\theta} \tag{13}\]  
Consequently, the surface deformation detection deviation \(E_{\rm 8,y}\) and the arch straightness detection deviation \(E_{\rm v}\) caused by trolley attitude pitch satisfy  
\[E_{\rm 8,y}=\delta^{\prime}_{\rm y}-\delta_{\rm y}=y_{\rm r}\left[\cos^{-1} \left(\arcsin\frac{h_{1}+h_{2}}{L}\right)-1\right] \tag{14}\]  
\[E_{\rm v}=v^{\prime}-v=y_{\rm r,p}\left[1-\cos^{-1}\left(\arcsin\frac{h_{1}+h _{2}}{L}\right)\right] \tag{15}\]  
From (14) and (15), it can be seen that \(h_{1}\) and \(h_{2}\) have the same effect on \(E_{\rm 8,y}\) and \(E_{\rm v}\), so only the influence of \(h_{2}\) needs to be analyzed. Figure 5 shows the variation in \(E_{\rm 8,y}\) and \(E_{\rm v}\) with the height of the front wheel lift-off, where \(y_{\rm r}=40\) mm and \(y_{\rm r,p}=30\) mm. It can be observed that the detection deviations of surface deformation and arch straightness exhibit positive and negative growth, respectively, with an increase in \(h_{2}\). Additionally, increasing the wheelbase \(L\) can effectively reduce the magnitude of this growth.  
Figure 4: Schematic of detection deviations caused by the borescope trolley attitude pitch.