# Bi-Objective Function Optimization for Welding Robot Parameters to Improve Manipulability
## 5 Simulation Results
### Comparison of Initial and Optimized Parameters  
The optimal design variables are the population with the minimum of the two objective functions, and without any mathematical analysis, we analyzed the population of design variables within the range of variables with a horizontal axis objective function value of 20 and a vertical axis objective function value of 0.63 on the Pareto front in Figure 13. The design variables within this range were judged to have converged as the design variable sizes are almost similar as shown in Table 6.  
Compared to the initial parameters, the length of the \(L_{4}\) link from the pivot joint to the end-effect of the robot was shortened, but the length of the upper link \(L_{3}\) and lower link \(L_{2}\) was relatively longer. Among \(L_{5}\) and \(L_{6}\), the parameters for the placement position of the robot, \(L_{5}\), the front installation distance of the robot, was 0.548 m, which is 0.062 m shorter than the existing one, and \(L_{6}\), the side installation distance, was 0.009 m to the left of the middle installation distance of the member, but it is numerically close to zero. It can be interpreted that the more the robot is installed close to the center of the width of the cell, the more balanced work can be performed in the case of a robot with left-right symmetry. Figure 14: Pareto front with the set of optimal solutions derived through genetic algorithms. Figure 13: Computational procedure to find the optimal design variables. Figure 14 is a graph of the results of evaluating the manipulability performance for the three task objects (large, medium, and small) in Section 2.1 Table 1 for the optimal design variables [0.26, 0.349, 0.549, 0.317, 0.145, \(-\)0.015] in the first row of the optimal values in Table 6. The results were evaluated using Equations (19) and (20) to assess the isotropy and the volume of the manipulability ellipsoid. As shown in Figure 15, the optimal design parameters show significant improvements in the range covering the entire workload per ship type. For the linear velocity manipulability evaluation, the singularities in the optimal design parameters were eliminated compared to the initial parameters, and the volume of the ellipsoid was doubled. For the angular velocity manipulability evaluation, the initial design parameters resulted in some singularities in all the workpieces, and there are some areas where the volume of the ellipsoid changes rapidly. However, in the optimal design parameters, the singularities disappeared compared to the initial design parameters, and the volume of the ellipsoid decreased rapidly. Table 7 shows the mean and standard deviation of linear velocity manipulability and angular velocity manipulability to numerically verify the shape change and size change of the ellipsoid by workpiece size. The closer the mean of \(u_{1}(A)\) is to 1, the more isotropic the shape of the manipulability ellipsoid is, and the smaller the standard deviation, the smaller the range of change in the shape of the manipulability ellipsoid. The larger the mean of \(u_{2}(A)\), the larger the volume of the manipulative ellipsoid, and the smaller the standard deviation, the smaller the sudden change in volume. To analyze the relationship between the design variables and the objective function, a correlation analysis was performed as shown in Figure 16. The length \(L_{4}\) from the pivot joint to the end-effector and the distance \(L_{5}\) between Trans and the robot coordinate system are positively correlated. The distance \(L_{3}\) between the third axis of the robot and the pivot joint, the length \(L_{2}\) between the second and third axis of the robot, and the height \(L_{1}\) from the floor to the second axis are also positively correlated with the change in distance \(L_{5}\). This suggests that proportionally lengthening or shortening the link length depending on the positioning distance of the robot is beneficial for manipulability.