## 3 Laser Point Cloud Registration
### Coarse Registration  
The bore of an electromagnetic launcher typically consists of a pair of arc-surface rails and a pair of flat-surface insulators, where the arc-surface rail can be divided into concave and convex types. Therefore, coarse registration can leverage the arc features of the rail to extract the deflection angle \(\alpha\) and offset vector (\(\Delta x\), \(\Delta y\)), in order to rotate and translate the laser point clouds of the bore profile to a position close to the standard bore profile. Taking the concave rail bore shown in Figure 6 as an example, first, the deflection angle \(\alpha\) of profile point clouds can be obtained by identifying the arc centers of the upper and lower side rails before registration, and then connecting them.  
The arc centers of the upper and lower side rails can be determined using the least-squares-based circle fitting method. Taking the upper side rail as an example, the coordinates of its center (\(x_{\text{p,1}}\), \(y_{\text{p,1}}\)) satisfy  
\[\begin{bmatrix}x_{\text{p,1}}\\ y_{\text{p,1}}\end{bmatrix}=\begin{bmatrix}-0.5&0&0\\ 0&-0.5&0\end{bmatrix}(A^{\text{T}}A)^{-1}A^{\text{T}}\Bigl{[}-\Bigl{(}x_{\text {p}}^{2}+y_{\text{p}}^{2}\Bigr{)}\Bigr{]} \tag{16}\]  
where \(A=[x_{\text{p}}\,y_{\text{p}}\ 1]\).  
Figure 5: Relationship between detection deviation and front wheel lift height under different wheeleses: (**a**) Surface deformation detection deviation; (**b**) Arch straightness detection deviation.  
Figure 6: Schematic of coarse registration principle.  
Similarly, the arc center coordinates (\(x_{\rm p,2}\), \(y_{\rm p,2}\)) of the lower side rail can be calculated, thereby obtaining the deflection angle \(\alpha\) as follows:  
\[\alpha=\arctan\left(\frac{x_{\rm p,1}-x_{\rm p,2}}{y_{\rm p,1}-y_{\rm p,2}}\right) \tag{17}\]  
Then, the rotationally corrected laser point clouds (\(x_{\rm pr}\), \(y_{\rm pr}\)) satisfy  
\[\begin{bmatrix}x_{\rm pr}\\ y_{\rm pr}\end{bmatrix}=\begin{bmatrix}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}x_{\rm p}\\ y_{\rm p}\end{bmatrix} \tag{18}\]  
Next, the offset vector (\(\Delta x\), \(\Delta y\)) of the bore profile is solved, which mainly involves solving for the center point of the laser point clouds (\(x_{\rm pr}\), \(y_{\rm pr}\)). Here, the fitting function is replaced by a parabola equation:  
\[\begin{bmatrix}x_{\rm pr}^{2}&x_{\rm pr}&1\end{bmatrix}\begin{bmatrix}p_{1}&p_{ 2}&p_{3}\end{bmatrix}^{\rm T}=y_{\rm pr} \tag{19}\]  
Based on (19), by least-squares fitting the vertex coordinates (\(x_{\rm pr,1}\), \(y_{\rm pr,1}\)) and (\(x_{\rm pr,2}\), \(y_{\rm pr,2}\)) of the upper and lower parabola curves, the following can be obtained:  
\[\begin{bmatrix}\Delta x\\ \Delta y\end{bmatrix}=-\frac{1}{2}\Bigg{\{}\Bigg{[}\begin{array}{c}x_{\rm pr,1}\\ x_{\rm pr,2}\end{array}\Bigg{]}+\begin{bmatrix}\begin{array}{c}y_{\rm pr,1}\\ y_{\rm pr,2}\end{array}\Bigg{]}\Bigg{\}} \tag{20}\]  
Translating (\(x_{\rm pr}\), \(y_{\rm pr}\)) according to (\(\Delta x\), \(\Delta y\)), the laser point clouds (\(x_{\rm pt}\), \(y_{\rm pt}\)) of the bore profile after coarse registration are obtained:  
\[\begin{bmatrix}x_{\rm pt}\\ y_{\rm pt}\end{bmatrix}=\begin{bmatrix}x_{\rm pr}\\ y_{\rm pr}\end{bmatrix}+\begin{bmatrix}\begin{array}{c}\Delta x\\ \Delta y\end{array}\end{bmatrix} \tag{21}\]