## 8 Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Before**} & \multicolumn{2}{c}{**After**} \\ \hline
**Angle of Rotation** & **MAE** & **NRMSE** & **MAE** & **NRMSE** \\ \hline
10\({}^{\circ}\) & 1.0932 & 35.47\% & 0.6076 & 63.96\% \\
20\({}^{\circ}\) & 2.9180 & 20.29\% & 0.9420 & 75.27\% \\
30\({}^{\circ}\) & 3.2550 & 34.37\% & 1.0382 & 81.61\% \\
40\({}^{\circ}\) & 6.1717 & 16.69\% & 1.1045 & 84.62\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison of results validation before and after parameter estimation for link 1. Since the essence of the optimization of the model was the estimation of the model parameters using the chosen algorithm, it is important to analyze the estimated parameters themselves, which ensured the improvement of the results. These were the parameters of the manipulator together with the coefficients found in the friction components. In connection with this, it should be noted that since the goal was set in the validations--the best possible model outputs and results within the indicators. Therefore the arm parame  
Figure 30: The best model output after parameter estimation for the joint angle 40\({}^{\circ}\). Figure 29: The best model output before parameter estimation for the joint angle 40\({}^{\circ}\). ters, such as length, weight, etc., were also included in the estimation. Table 11 shows a comparison of the values of these parameters before estimation and after estimation. The weights of the arm parts \(m_{1}\) and \(m_{2}\) were changed by the estimation--the weight of the upper part of the arm was estimated to 0.9 kg extra, while the weight of the lower part of the arm was estimated to be 0.076 kg less. The parameter, Distance to the center of mass, for both links were increased by a value of 0.053 m for \(c_{1}\) and a value of 0.022 m for \(c_{2}\). The length of the arm on axis 1 is less by 0.063 m. Due to these changes, of course, the values of the moments of inertia \(I_{1}\) and \(I_{2}\) have also changed. The Viscous friction coefficient and Coulomb friction magnitude were set experimentally before the validations. However, after optimization, the values of the Viscous friction coefficient and Coulomb friction magnitude are smaller than the initial value of 0.35 (especially the value of \(f_{d2}\)). The parameters that were the subject of estimation and used for the subsequent optimization of the model were estimated in the Parameter Estimator. During the estimation process, this tool (among other things) offers the possibility to monitor how the values of the parameters are estimated (changed) depending on the iterations until all the values of the estimated parameters are stabilized. The result of this phase from the given toolbox can be seen in Figures 31 and 32. During the estimation, only one graph could be viewed, where the waveforms of all 11 estimated parameters were. However, since the masses \(m_{1}\) and \(m_{2}\) are represented by much higher values than the remaining parameters, the mass curves were exported to the second graph for better clarity. Based on the waveforms of Figure 32, it can be seen that the smallest changes in the parameter estimation occurred in the parameters \(c_{2}\) (Distance to the center of mass for link 2) and \(I_{2}\) (Moment of inertia for link 2). On the contrary, significant changes took place in the values of Coulomb friction magnitude \(f_{d1},f_{d2}\). In Figure 31, it is possible to observe how the weight estimate has changed. From the course of estimating the mass \(m_{2}\), it can be seen how the weight changed in the first iterations, but the result of the estimation in the last iteration was, in the end, very similar to the original value.