## 6 Discussion  
The proposed approach was applied in three simulation experiments involving the Puma-560, Baxter, and KUKA iiwa robots. The inverse kinematics were successfully solved  
Figure 16: Time history with a cubic polynomial timing law for Experiment 3. (**a**) Joint positions. (**b**) Joint velocities.  
Figure 17: KUKA YouBot final joint configuration for Experiment 3. (**a**) Real-world final joint configuration. (**b**) Joint motion results measured by encoders.  
in each case, demonstrating the applicability of the system across different configurations. Among the tested algorithms, differential evolution (DE) consistently outperformed BA, PSO, and ICA. While the performance of DE and IWO exhibited similar performance across experiments, DE reported the lowest dispersion of the data.  
When comparing the experimental results with and without hybridization, a significant improvement in the solution of the inverse kinematics is evident across all algorithms. The Jacobian step enhances the exploitation capabilities of the algorithm; algorithms like PSO, ICA, and BA exhibit stagnation, which indicates weak exploration capabilities, whereas DE and IWA demonstrate a strong exploration performance.  
DE-H reported better fitness value results in all tests, achieving the lowest mean and standard deviation (std) values. This indicates that DE-H achieved the smallest error in both position and orientation with precision, along with the smallest data distribution, reflecting more consistent results. In contrast, ICA and PSO performed poorly in all tests, achieving the worst results in terms of mean and std values. Additionally, the performance of IWO-H was very similar to DE-H. However, DE-H outperformed all the other algorithms by achieving lower mean and std values.  
The evaluation of the standard deviation (\(5.8348\times 10^{-3}\)) indicates that the method achieves sufficient precision in both position and orientation, making it suitable for applications such as medical procedures, where fine adjustments are critical.  
The proposed approach was also applied in real-world experimentation, where the KUKA YouBot robot was considered as a case study. The application on the real robot demonstrates that the inverse kinematics results are acceptable, confirming that the constrained problem is solved despite the DE algorithm not being specifically designed for constrained problems.  
A drawback of the conventional Jacobian pseudoinverse is that the inversion of J cannot always be guaranteed. Singular configurations cause J to become rank deficient, leading to system instability. In this work, a Jacobian pseudoinverse step is used to generate a candidate solution. In the presence of singularities, unfeasible solutions are avoided in the Jacobian pseudoinverse operation, as only improved joint solutions replace the current ones. This mechanism effectively prevents singularities.  
Future work will focus on solving inverse kinematics for a mobile manipulator [16], incorporating the Jacobian step for fine-tuning. Additionally, we aim to extend the use of inverse kinematics to trajectory tracking and address grasping problems on platforms equipped with dual manipulators [35; 36].