## 4 Conclusions and Discussion  
Various approaches and powerful learning algorithms of great value have been introduced in recent decades; however, the integration of the various approaches in ANN optimization has allowed researchers to improve performance and generalizability in ANNs. The results of this work revealed that the proposed systematic and experimental approach is a useful alternative for the robust design of ANNs since it allows simultaneously considering the design and the noise variables, incorporating the concept of robustness in the design of ANNs. The RDANN methodology used in this work was initially proposed in the field of neutron dosimetry, so it was adapted for implementation in the field of robotics, allowing us to improve the performance and generalization capacity in an ANN to find the solution to the inverse kinematics in the Quetzal manipulator. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Trajectory** & **Calculated Final Position (cm)** & **Calculated Measure** & **Predicted Position (cm)** & **Measured Distance (cm)** & **Error \%** \\ \hline
6 & \([-0.93,\,-3.81,\,59.34]\) & 4.9862 & \([5.51,\,4.22,\,58.91]\) & 5.2 & 4.2 \\ \hline
13 & \([-4.56,\,-19.98,\,52.47]\) & 22.78 & \([-8.78,\,-17.90,\,52.97]\) & 22.5 & 1.2 \\ \hline
161 & \([-11.32,\,-35.76,\,22]\) & 55.13 & \([-27.17,\,-25.88,\,18.96]\) & 54.5 & 1.1 \\ \hline
216 & \([-22.27,\,-26.95,\,28]\) & 48.97 & \([-27.18,\,-23.14,\,28]\) & 51.5 & 5.1 \\ \hline
236 & \([-14.97,\,-25.94,\,25.77]\) & 47.33 & \([-28.22,\,-13.01,\,20]\) & 51 & 7.7 \\ \hline
317 & \([14.32,\,-27.56,\,8.46]\) & 62.25 & \([14.58,\,-24.46,\,4.94]\) & 63.5 & 2 \\ \hline
663 & \([4.36,\,-18.87,\,46.35]\) & 25.16 & \([5.12,\,-19.25,\,46.33]\) & 26 & 3.3 \\ \hline
988 & \([30.75,\,-12.49,\,33.98]\) & 43.70 & \([26.63,\,-20.93,\,29.13]\) & 45.5 & 4.1 \\ \hline
1025 & \([-12.41,\,1.87,\,57.09]\) & 13.63 & \([-11.11,\,4.18,\,58.39]\) & 14 & 2.7 \\ \hline
1216 & \([23.41,\,24.49,\,37.90]\) & 41.82 & \([19.59,\,28.94,\,34.78]\) & 43.5 & 4 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c c} \hline \hline
**Average** & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Trajectory comparison. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Iterations** & **Training Time** & **Total Tested Networks** & **Prediction Error** \\ \hline BPNN trial and error & Often millions & Several hours & 100 in several hours & Undetermined \\ \hline PSO conventional [49] & 5000 & Not specified & 10 & 0.00913 \\ \hline PSO [11] & 100 & 03:51:38 & 10 in 39 h & 0.0456 \\ \hline This study & 10,000 & 00:17:30 & 36 in 178 min & 0.0171 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison of results with other methods. The time spent during the network design process was significantly reduced compared with the conventional methods based on trial and error. In the methods that are generally proposed by the previous experience of the researcher, the design and modeling of the network can take from several days to a few weeks or even months to test the different ANN architectures, which can lead to a relatively poor design. The use of the RDANN methodology in this study allowed the network design to be carried out in less time, with approximately 13 h of training, due to the orthogonal arrangement corresponding to the 36 training sessions performed using a conventional AMD Ryzen 7 5700 series processor with an integrated graphics card. Depending on the complexity of the problem, the use of this methodology allows handling times ranging from minutes to hours to determine the best robustness parameters in the network architecture. Therefore, it is possible to streamline the process and reduce efforts, with a high degree of precision in network performance.