## 3 Measurement Constraints for BIW Inspection  
### Position Constraints of Station Planning  
In the context of radar station configuration, the feasible region for a feature to be measured is determined by both the inspection distance and inspection angle. L\({}_{i}\) and l\({}_{i}\) are the maximum and minimum allowed inspection distances. \(\alpha_{i}\) is the maximum allowed inspection angle of the feature to be measured. The feasible region corresponding to F\({}_{i}\) is denoted as A\({}_{i}\) (Figure 3a). The shared feasible region among multiple features is shown in Figure 3b. Figure 2: Key definitions in inspection by Laser Radar. In the context of inspection operations, minimizing the number of Laser Radar stations is advantageous, as station switching is time-consuming. A Laser Radar station shared among multiple features can reduce the overall number of stations needed. However, conducting measurability checks between each station and feature significantly increases the planning time. To address this, reducing the feasible region of the planning area can eliminate unnecessary checks. Identifying shared feasible regions efficiently is crucial. An attributed adjacent graph (AAG) is employed to record intersection relationships. However, with a large number of features, the AAG becomes too intricate for intersection searches. Therefore, a clustering algorithm is applied to classify features before constructing the AAG, improving the process's efficiency and manageability. The clustering steps for the features to be measured are as follows (Figure 4):  
1. A local coordinate system is assigned for each measuring feature. The origin is the center point (\(\mathbf{F_{i}}\)). The \(\mathbf{Z}\) axis is along the normal vector pointing outward from the material surface at \(\mathbf{F_{i}}\);
2. The K-means++ algorithm is used to cluster feasible regions. The number of features in each cluster is constrained to a manageable size;  
Figure 4: Clustering for features to be measured. Figure 3: (**a**) The feasible region for a feature; (**b**) The shared region for multiple features. 3. In the local coordinate system of the feature, a point denoted as C\({}_{i}\) is defined: \[\text{C}_{i}=[0,0,0.61\text{SL}_{i},1]^{\text{T}}\] (1)
4. The transformation matrix from the local coordinate to the inspection coordinate is denoted as T. The point (C\({}_{i}\)) in the inspection coordinate system is denoted as C\({}_{i}\)\({}^{\prime}\): \[\text{C}_{i}^{\prime}=\text{T}\cdot\text{C}_{i}\] (2)
5. The distortion function in clustering is J, and \(\mu_{\text{c}}\) is the centroid of a cluster: \[\text{J}=\sum_{i=1}^{\text{m}}\lVert\text{C}_{i}^{\prime}-\mu_{\text{c}} \rVert^{2}\] (3)
6. The clustering result is denoted as G\({}_{i}\). In each G\({}_{i}\), the intersections of the feasible regions are checked to construct the AAG\({}_{i}\). It is difficult to express the shared area of feasible regions precisely. Therefore, the feasible region should be simplified first. The inspection distance is \(\text{l}_{i}\leq\text{D}_{i}\leq\text{L}_{i}\). The inspection angle is \(0\leq\theta_{1}\leq\alpha_{i}\). The feasible region (A\({}_{i}\)) could be simplified into a group of spherical regions (Figure 5). The approximation steps for feasible regions are as follows:  
1. The feasible region is simplified into a group of basic spherical regions and supplementary spherical regions which are denoted as follows: \[\text{B}_{i,j}=\left(\text{c}_{i,j},\text{r}_{i,j}\right)\] (4) where \(\text{c}_{i,j}\) and \(\text{r}_{i,j}\) denote the sphere's center coordinates and radius, respectively. The number of basic spherical regions is m: For the first spherical region, \(\text{c}_{i,1}\) and \(\text{r}_{i,1}\) are expressed as follows: \[\text{c}_{i,1}=[0\ 0\ \frac{\text{l}_{i}}{1-\text{sin}\alpha_{i}}1]\] (5)  
Figure 5: Discretized spherical regions for feasible region of features to be measured.