## 6 Parameter Estimation of 2-DOF Planar Arm
### Results of Validation after Parameter Estimation  
_Link 1_  
The results of validation after the estimation of parameters were assessed based on the same statistical criteria as in the validation before estimation. Table 7 provides an overview of the MAE and NRMSE indicators for link 1. From the results, it can be seen that based on the MAE indicator, the model at the initial joint angle of 10\({}^{\circ}\) (MAE = 0.6076) is the best, while based on the NRMSE indicator, it is the model at an initial joint angle of 40\({}^{\circ}\), with NRMSE = 84.62%.  
The improvement of the model results is obvious also from the graphic curves. Figures 21-24 show the model outputs after parameter estimation for the joint 10\({}^{\circ}\)-40\({}^{\circ}\) for link 1. The model output (blue solid line) approaches the required mean response (red dashed line) with increasing initial joint angle, which also indicates an increase in the value of the NRMSE indicator in Table 7. Before estimating the parameters, the model output had the problem of achieving mean response amplitudes and also the peaks of the model output amplitudes that were shifted for the time axis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Notation** & **Value and Unit** & **Notation** & **Value and Unit** \\ \hline m\({}_{1}\) & 6.813 kg & I\({}_{1}\) & 0.264 kg:m\({}^{2}\) \\ m\({}_{2}\) & 3.361 kg & I\({}_{2}\) & 0.012 kg:m\({}^{2}\) \\ l\({}_{1}\) & 0.552 m & \(\mathrm{f_{c1}}\) & 0.335 N\(\cdot\)m\(\cdot\)s\(\cdot\)rad\({}^{-1}\) \\ c\({}_{1}\) & 0.346 m & \(\mathrm{f_{c2}}\) & 0.279 N\(\cdot\)m\(\cdot\)s\(\cdot\)rad\({}^{-1}\) \\ c\({}_{2}\) & 0.212 m & \(\mathrm{f_{d1}}\) & 0.203 N\(\cdot\)m \\  & & \(\mathrm{f_{d2}}\) & 0.076 N\(\cdot\)m \\ \hline \hline \end{tabular}
\end{table}
Table 6: The list of model parameters after parameter estimation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Angle of Rotation** & **MAE** & **NRMSE** \\ \hline 10\({}^{\circ}\) & 0.6076 & 63.96\% \\
20\({}^{\circ}\) & 0.9420 & 75.27\% \\
30\({}^{\circ}\) & 1.0382 & 81.61\% \\
40\({}^{\circ}\) & 1.1045 & 84.62\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: Results of validation after parameter estimation for link 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Angle of Rotation** & **K** & **Notation** & **Parameter Estimation Bounds** \\ \hline 10\({}^{\circ}\) & 2100 & m\({}_{1}\) & 5–7 kg \\
20\({}^{\circ}\) & 3182 & m\({}_{2}\) & 3–5 kg \\
30\({}^{\circ}\) & 4546 & l\({}_{1}\) & 0.5–0.7 m \\
40\({}^{\circ}\) & 5028 & c\({}_{1}\) & 0.2–0.4 m \\  & & c\({}_{2}\) & 0.13–0.25 m \\  & & l\({}_{1}\) & 0.25–0.45 kg\(\cdot\)m\({}^{2}\) \\  & & l\({}_{2}\) & 0.009–0.017 kg\(\cdot\)m\({}^{2}\) \\  & & f\({}_{c1}\) & 0.00001–20 N\(\cdot\)m\(\cdot\)s\(\cdot\)rad\({}^{-1}\) \\  & & f\({}_{c2}\) & 0.00001–20 N\(\cdot\)m\(\cdot\)s\(\cdot\)rad\({}^{-1}\) \\  & & f\({}_{d1}\) & 0.00001–20 N\(\cdot\)m \\  & & f\({}_{d2}\) & 0.00001–20 N\(\cdot\)m \\ \hline \hline \end{tabular}
\end{table}
Table 5: The list of the number of samples and the parameters’ spaces.  
Figure 23: Model output after parameter estimation for the joint angle 30\({}^{\circ}\)—link 1.  
Figure 21: Model output after parameter estimation for the joint angle 10\({}^{\circ}\)—link 1.  
Figure 22: Model output after parameter estimation for the joint angle 20\({}^{\circ}\)—link 1.