# Bi-Objective Function Optimization for Welding Robot Parameters to Improve Manipulability
## 2 Work Object Analysis and Trajectory Planning
### Trajectory Planning  
The desired position of the robot in the task space consists of Trans, Longi, Collar plate, and Outer hull plate intersecting each other, as shown in Figure 1. The size is defined based on the analysis in Section 2.1, with the dimensions at Longi, having a height of 400 mm and a width of 880 mm. The desired orientation, as shown in Figure 3, represents the posture conditions required for welding under 2F and 3F conditions. The numbers 2F and 3F distinguish between horizontal and vertical positions during welding, while the letters F and G represent Fillet and Groove welding, respectively [24]. The workpieces in this paper are grid structures, and since all components are joined vertically, they correspond to Fillet welding. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Dimension** & **Type** & **Median** & **Min** & **Max** \\ \hline \multirow{2}{*}{Longi. Height (Unit: mm)} & CONTAINER & 325 & 300 & 350 \\  & LNGC & 400 & 300 & 600 \\  & VLCC & 400 & 300 & 450 \\ \hline \multirow{2}{*}{Longi. Width (Unit: mm)} & CONTAINER & 845 & 723 & 987 \\  & LNGC & 880 & 582 & 1025 \\  & VLCC & 914 & 566 & 1036 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Magnitude of properties of U-cell block by ship type. Figure 3: Welding position code for horizontal and vertical fillet. Figure 2: Boxplot of height and width of U-cell block by ship type. Figure 4 delineates the forward angle technique and backward angle technique, as well as the work angle based on the welding joint criteria with respect to the welding direction. Forward and backward angle techniques are selectively used to form flat or relatively convex beads, depending on the shape and size of the gap during Groove or Filtel welding, respectively [25]. However, adjusting forward and backward angles in real-time to accommodate changing gap conditions during robot welding is practically infeasible. Therefore, from the robot's perspective, the most neutral position, with forward/backward angles at 0\({}^{\circ}\), becomes the basis for determining welding conditions. The theoretically optimal work angle to create symmetrical weld beads is 45\({}^{\circ}\). At the intersection point where the three faces of Longi, Plate, and Trans intersect, it is not feasible to position the welding torch at the ideal angle due to mechanical collisions. Therefore, the torch is given a backward angle to avoid collisions, while maintaining the work angle at 45\({}^{\circ}\). The torch is rotated until a position is reached where there are no collisions, and the backward angle becomes 0\({}^{\circ}\). The distance from the welding starting point to the point where the backward angle of the welding torch becomes 0\({}^{\circ}\) is referred to as the angle distance (AD). For vertical welding, the distance from the end of AD to the end of the welding line is called the welding distance (WD), while for horizontal welding, it is the distance from the end of the left AD segment to the start of the right AD segment. The robot in this paper is operated after being manually moved and installed by a person. Therefore, when generating work programs, an angle distance (AD) of typically 100 mm is set to accommodate for manual installation errors, robot origin calibration errors, fabrication errors of the workpiece, and other factors. Based on the aforementioned description, the standard welding posture conditions for the robot were as follows:  
1. For all straight segments excluding the AD segment, the travel angle was set to 0 degrees, and the work angle was fixed at 45 degrees. 2. In the AD segment, to avoid collisions, the travel angle was set to 45 degrees at the starting point, and the AD was interpolated such that at the end of the AD with a distance of 100 mm, the travel angle became 0 degrees. The value of AD 100 mm was determined using commercial simulation tools (RobCAD V7.0), which take into account torch interference.