## 5 Simulation and Experimental Results
### Second Simulation Experiment  
In this part of the simulations, an edge-case experiment is conducted to evaluate the proposed approach. To solve the inverse kinematics, a desired end-effector position is proposed within the workspace to ensure it is reachable. However, some desired positions cannot be achieved with certain orientations. Therefore, appropriate orientation selection is crucial. This issue makes the inverse kinematics problem challenging to solve. In these edge cases, gains \(k_{t}\) and \(k_{R}\) in Equation (11) can be adjusted to give priority to minimize the error between the desired and the current end-effector position, rather than orientation errors. The edge-case experiments involve defining a reachable desired position with an unreachable desired orientation for the [PERSON] robot. The proposed approach is run independently 100 times for the same pose, and the best fitness value is retained for statistical analysis to compare all considered algorithms. Additionally, position and orientation errors between the desired and current end-effector poses are analyzed (see Equation (11)). The experimental setup is conducted as follows: gains are set to \(k_{t}=1.5\) and \(k_{R}=0.25\) to give priority to minimize the position error. Moreover, \(G=300\), \(N=30\), \(\gamma=1000\), \(C_{R}=0.9\), and \(F=0.6\). Finally, the desired end-effector pose is defined as  
\[\mathbf{T}_{d}= \begin{bmatrix}\mathbf{R}_{d}&\mathbf{t}_{d}\\ \mathbf{0}&1\end{bmatrix} = \begin{bmatrix}0&0&-1&0.5\\ 0&-1&0&0.4\\ -1&0&0&0.6\\ 0&0&0&1\end{bmatrix} \]  
Figure 8 presents the statistical results of the best fitness value achieved by each algorithm. As shown, DE-H outperformed the others, demonstrating a smaller data distribution and lower fitness values. It is worth noting that the best fitness values are around 0.15 due to the significant orientation error. Consequently, position and orientation errors are now compared independently for statistical analysis. Table 8 presents the position error results. The DE-H algorithm outperformed the others, achieving the lowest mean and standard deviation (std) values. Even the worst position error obtained by DE-H is better than the mean errors of BA-H, PSO-H, and IWO-H. Although the best position error is achieved by ICA-H, its performance is considered an outlier due to the larger data distribution observed in Figure 8.  
Figure 7: Comparison of standard and hybrid algorithms in solving inverse kinematics of the KUKA iiwa robot. (**a**) Standard algorithms. (**b**) Hybrid algorithms. In contrast to the position errors, the orientation errors reported in Table 9 are significantly higher, as the desired orientation is considered unreachable. As expected, DE-H reported lower mean, standard deviation (std), and worst-case orientation error values compared to those in reachable poses. Although the best orientation error was achieved by PSO-H, its data distribution, as shown in Figure 8, is considerably larger than that of DE-H.  
We present another edge-case experiment involving four randomly selected unreachable desired positions, each with a corresponding reachable desired orientation, for the [PERSON] robot. Table 10 presents the position and orientation error results for each case, while Figure 9 illustrates the performance of the proposed approach in handling these unreachable poses. Coordinate frames in red represent the unreachable poses and coordinate frames in blue the achieved ones. Notice that all coordinate frames are adjusted with respect to orientation errors; however, position errors are not. The results demonstrate that the proposed approach successfully solves the inverse kinematics problem by minimizing position errors, even when the desired orientation is unreachable. Figure 8: Comparison of hybrid algorithms for solving the inverse kinematics of the [PERSON] robot in an orientation-unreachable end-effector pose.