## 4 Simulation Results
### Quantitative Analysis  
In order to further compare and validate the performance of the proposed controllers, the following statistical indices are used:  
1. Maximum absolute value of the error for each joint. \[\Delta q_{i_{max}}=\max_{j=1,\ldots,M}(|\Delta q_{i}(j)|).\] (39)
2. Root mean square (rms) values of the error and input torque for each joint. \[\Delta q_{i_{rms}}=\sqrt{\frac{1}{M}\sum_{j=1}^{M}\|\Delta q_{i}(j)\|^{2}}.\] (40) \[\tau_{i_{rms}}=\sqrt{\frac{1}{M}\sum_{j=1}^{M}\|\tau_{i}(j)\|^{2}}.\] (41)
3. Percentage change in the rms values of the error and input torque for the proposed control approaches compared with the PD control. \[\%\textit{change}\ \Delta q_{i_{rms}}=\frac{(\Delta q_{i_{rms}})_{ proposed\ controller}-(\Delta q_{i_{rms}})_{PD\ controller}}{(\Delta q_{i_{rms}})_{PD\ controller}}\times 100\%,\] (42) \[\%\textit{change}\ \tau_{i_{rms}}=\frac{(\tau_{i_{rms}})_{ proposed\ controller}-(\tau_{i_{rms}})_{PD\ controller}}{(\tau_{i_{rms}})_{PD\ controller}}\times 100\%\] (43)  
Figure 6: Performance in terms of drifts and systematic errors for the second proposed approach compared with the PD approach: (**a**,**b**) desired and actual displacements of joint 1 and joint 2, respectively; (**c**,**d**) angular displacement errors of joint 1 and joint 2, respectively.  
where \(i\) is the joint number and \(M\) is the number of sampling steps of the simulation.  
Tables 2 and 3 summarize the performance for the both proposed approaches compared with the PD control. The indices in (39)-(41) for both joints are obtained from the results in Figures 3 and 5.  
Table 4 presents the percentage changes in (42) and (43) for each one of the proposed controller compared with the PD controller. Moreover, the time for evaluating control actions of both controllers compared with the PD controller are presented in Table 5.  
From the above tables, which compare the performance of each control approach with the PD control, we deduce the following points:  
1. The first approach reduces the average tracking errors of both joints by about 97% with almost the same control efforts.
2. The second approach reduces the average tracking errors and control efforts of both joints by about 98% and 81%, respectively.
3. The proposed controllers have higher computation time than that in the standard PD controller.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Indexes** & **First Approach** & **Second Approach** \\ \hline \% \(change\)\(\Delta q_{1 rms}\) & \(-96.40\) & \(-98.47\) \\ \% \(change\)\(\Delta q_{2 rms}\) & \(-97.01\) & \(-99.01\) \\ \% \(change\)\(\tau_{1_{\text{max}}}\) & \(-0.45\) & \(-68.38\) \\ \% \(change\)\(\tau_{2_{\text{max}}}\) & \(+0.33\) & \(-96.565\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Percentage changes of the errors and input torques for the proposed control approaches compared with the PD control.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Indexes** & **PD Control** & **First Approach** \\ \hline \(\Delta q_{1 max}\)(deg) & 2.0981 & 0.0878 \\ \(\Delta q_{2 max}\)(deg) & 1.5781 & 0.0590 \\ \(\Delta q_{1 rms}\)(deg) & 1.0455 & 0.0374 \\ \(\Delta q_{2 rms}\)(deg) & 0.8487 & 0.0253 \\ \(\tau_{1_{\text{max}}}\)(Nm) & 4.3128 & 4.2933 \\ \(\tau_{2_{\text{max}}}\)(Nm) & 1.5666 & 1.5719 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Performance summary for the first roposed approach compared with the PD control.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Indexes** & **First Approach** & **Second Approach** \\ \hline \% \(change\)\(\Delta q_{1 rms}\) & \(-96.40\) & \(-98.47\) \\ \% \(change\)\(\Delta q_{2 rms}\) & \(-97.01\) & \(-99.01\) \\ \% \(change\)\(\tau_{1_{\text{max}}}\) & \(-0.45\) & \(-68.38\) \\ \% \(change\)\(\tau_{2_{\text{max}}}\) & \(+0.33\) & \(-96.565\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Performance in terms of computation time for the proposed control approaches compared with the PD control.