## 3 Measurement Constraints for BIW Inspection
### Spatial Constraints for BIW Inspection  
The Laser Radar must avoid collisions with both the ground and the BIW during station operation. Within the inspection coordinate system, the ground is defined as a plane with a constant \(Z\)-coordinate of \(z_{G}\). Considering the physical dimensions of the  
Figure 7: Interference checking of laser beam. Laser Radar, a safe distance (h\({}_{\text{s}}\)) from the ground is necessary (Figure 8). Consequently, the collision-avoidance constraint is formulated as follows:  
\[\text{J}_{0}=\text{z}_{\text{G}}+\text{h}_{\text{s}}-\text{S}_{\text{k}}(\text{ z})\leq 0 \tag{25}\]  
To avoid the Laser Radar colliding with the BIW, four orthogonal datum walls are assumed around the BIW. The station needs to be outside of the virtual walls (Figure 9). Within the inspection coordinate system, four walls are constructed as modular fixturing assemblies comprising the following:  
\[\left\{ \begin{array}{l}\left(\text{x},\text{W}/2,\text{z},1\right)\,w_{1}\\ \left(\text{x},-\text{W}/2,\text{z},1\right)w_{2}\\ \left(\text{L}/2,\text{y},\text{z},1\right)w_{3}\\ \left(-\text{L}/2,\text{y},\text{z},1\right)w_{4}\end{array} \right. \tag{26}\]  
So, the constraints are as follows:  
\[\text{J}_{1}=\left(\text{W}/2-\text{S}_{\text{k}}(\text{y})\leq 0\cup\left( \text{S}_{\text{k}}(\text{y})+\text{W}/2\leq 0\right)\right. \tag{27}\]  
\[\text{J}_{2}=\left(\text{S}_{\text{k}}(\text{x})-\text{L}/2\geq 0\cup\left( \text{S}_{\text{k}}(\text{x})+\text{L}/2\geq 0\right)\right. \tag{28}\]  
The collision constraints are combined as follows:  
\[\text{J}=\text{J}_{0}|(\text{J}_{1}\cup\text{J}_{2}) \tag{29}\]  
The Laser Radar is mounted on a robot, so the Laser Radar station must be positioned within the robot's reachable area. To ensure that the station remains within this accessible region, the robot's workspace boundary must be defined. A 6-DOF industrial robot is chosen for radar mounting (Figure 10a). It can be simplified to a linkage model (Figure 10b), with local coordinate systems defined at corresponding joints. Parameters about industrial robot's arms are listed in Table 1.  
Figure 8: Collision constraint with ground. Figure 9: Collision constraint with BIW. The transformation matrix between adjacent coordinates is as follows:  
\[\mathrm{T_{i}^{i-1}}= \begin{bmatrix}\mathrm{c}\mathrm{\wp}_{\mathrm{i}}&-\mathrm{ s}\mathrm{\wp}_{\mathrm{i}}\mathrm{c}\mathrm{\alpha}_{\mathrm{\alpha}_{\mathrm{i}}}& \mathrm{s}\mathrm{\wp}_{\mathrm{i}}\mathrm{s}\mathrm{\alpha}_{\mathrm{\alpha} _{\mathrm{i}}}&\alpha_{\mathrm{i}}\mathrm{c}\mathrm{\wp}_{\mathrm{i}}\\ \mathrm{s}\mathrm{\wp}_{\mathrm{i}}&\mathrm{c}\mathrm{\wp}_{\mathrm{i}} \mathrm{c}\mathrm{\alpha}_{\mathrm{i}}&-\mathrm{c}\mathrm{\wp}_{\mathrm{i}} \mathrm{s}\mathrm{\alpha}_{\mathrm{i}}&\alpha_{\mathrm{i}}\mathrm{s}\mathrm{ \wp}_{\mathrm{i}}\\ 0&\mathrm{s}\mathrm{\alpha}_{\mathrm{i}}&\mathrm{c}\mathrm{\alpha}_{\mathrm{i} }&\mathrm{d}_{\mathrm{i}}\\ 0&0&0&1\end{bmatrix} ,\mathrm{i}=1,2,\ldots,6 \tag{30}\]  
In the matrix, s means the sine function and c means the cosine function. The subscript is the angle between the axes of adjacent joints. For example, s\(\mathrm{\wp}_{\mathrm{i}}\) means sin(\(\mathrm{\wp}_{\mathrm{i}}\)) and c\(\mathrm{\alpha}\)i means cos(\(\mathrm{\alpha}_{\mathrm{i}}\)). The robot's coordinate does not coincide with the inspection coordinate. The transformation matrix between them is as follows:  
\[\mathrm{T_{o}^{M}}= \begin{bmatrix}0&1&0&\mathrm{x}_{0}\\ -1&0&0&\mathrm{y}_{0}\\ 0&0&1&\mathrm{z}_{0}\\ 0&0&0&1\end{bmatrix}  \tag{31}\]  
The optimized radar's posture on the station is ensured by the last wrist of the robot. Point is the center of P robot's last wrist (Figure 10a). Therefore, for any planned station, it cannot be outside of the \(P\) region that it can reach. In the inspection coordinate system, the reachable region of \(P\) is denoted as \(\mathbf{\Omega_{F}}\):  
\[\mathrm{\Omega_{F}}= \begin{bmatrix}\mathrm{x_{F}}\\ \mathrm{y_{F}}\\ \mathrm{z_{F}}\\ 1\end{bmatrix}