## 4 Simulation Studies
### Generation of Membrane Weld Path  
In the case illustrated in Figure 11, as previously explained, when the robot is positioned at predefined locations, the welding sequence for edges, as determined in Figure 13, is established. The robot's motion path is generated using this Edge information. Predefined robot actions in the Edge information of the MDF are based on path points for the '(a) teaching membrane in Figure 17, where a human operator initially performed the welding task. In other words, when the robot welds the teaching membrane, it follows the Edge information in the MDF, replicating the actions taught.  
Figure 16: Joint velocity values of each manipulator joint when welding a membrane. The numbers in the legend are the joint indices of the manipulator, starting with 1 from the base and the index of the last joint is 6.  
Figure 15: Joint angle values of each manipulator joint when welding a membrane. The numbers in the legend are the joint indices of the manipulator, starting with 1 from the base and the index of the last joint is 6.  
When welding a non-reference membrane, the coordinates of the path points in the MDF's Edge information are adjusted based on the relative positional relationship between the reference membrane and the target membrane. The relative positional relationship with the reference membrane is determined based on the topology information of the LNG cargo hold. Since the membrane's size is a known value, the 2D relative position on the plane can be easily determined using the relative location from the reference membrane to the target membrane, utilizing the known sizes.  
Let us denote the robot's reference coordinate system as \(O_{base}\), the teaching membrane's reference coordinate system as \(O_{teaching}\), and the target membrane's reference coordinate system as \(O_{target}\). Each is depicted in Figure 17.  
\[P_{\text{teaching}}|_{O_{\text{base}}}=\left\{p_{1},p_{2},\cdots,p_{n}\right\} \right|_{O_{\text{base}}} \tag{1}\]  
\[P_{\text{teaching}}|_{O_{\text{teaching}}}=T_{\text{base}}^{\text{teaching}}(P_ {\text{teaching}}|_{O_{\text{base}}}) \tag{2}\]  
The Edge information for the teaching membrane is constructed based on the initial teaching points in \(O_{base}\) during the first teaching session, as represented in (1). In (1), \(p_{1},p_{2},\cdots,p_{n}\) represent each teaching point.  
To perform robot welding on the target membrane in \(O_{target}\), the teaching points in (1) must be expressed in the coordinate system of \(O_{target}\). To achieve this, the teaching points are first transformed using (2) to \(O_{teaching}\) coordinates. When defining the teaching points in \(O_{teaching}\) as in (3), it must match (4). This relationship is expressed in (5). The remaining step is to express (4) in the robot's reference coordinate system \(O_{base}\), which can be achieved using a simple homogeneous transform function in (6). In (6), \(T_{\text{target}}^{\text{teaching}}\) moves the points from \(O_{target}\) to \(O_{teaching}\), and \(T_{\text{teaching}}^{\text{base}}\) moves the points from \(O_{teaching}\) to \(O_{base}\). In the case of Figure 17, the relationship between _(a) teaching membrane_ and _(b) target membrane_ involves a simple upward horizontal shift, reflected in \(T_{\text{target}}^{\text{teaching}}\).  
\[P_{\text{teaching}}|_{O_{\text{teaching}}}=\left\{p_{1}^{\prime},p_{2}^{ \prime},\cdots,p_{n}^{\prime}\right\}|_{O_{\text{teaching}}} \tag{3}\]  
\[P_{\text{target}}|_{O_{\text{target}}}=\left\{p_{1}^{\prime},p_{2}^{\prime}, \cdots,p_{n}^{\prime}\right\}|_{O_{\text{target}}} \tag{4}\]  
\[P_{\text{teaching}}|_{O_{\text{teaching}}}=P_{\text{target}}|_{O_{\text{target}}} \tag{5}\]  
\[T_{\text{target}}^{\text{base}}=T_{\text{teaching}}^{\text{base}}T_{\text{ target}}^{\text{teaching}} \tag{6}\]  
Figure 17: Coordination relations among membranes and robot.