## 3 Main Results
### The Designed Control Methodology
\tag{36}\]  
To clearly present the results within the simulation period and to facilitate easier comparison, the averaged tracking error i:  
\[E_{j}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\left\|e_{j}(k)\right\|^{2}\right)} ;\,\,j=1,2,3 \tag{37}\]  
where \(Z\) is the number of simulation steps. To demonstrate the superiority of the designed controller, the average control error is calculated over two different simulation periods (10 s and 30 s). The averaged tracking errors are reported in Table 2.  
The designated trajectory configuration and real trajectory under three control methods at the first three joints are displayed in Figure 2. It can be seen from Figure 2 that all three controllers appear to have a similar tracking control performance. However, they have different convergence times in the following order: the designed controller has the fastest convergence time among all three control methods, and NFTSMC has faster convergence time than the normal SMC. Figures 3 and 4 show the position control errors and the velocity control errors, respectively. It can be seen from Figure 3 and Table 2, the position control errors of the designed control scheme are relatively small compared to those of the other control methods, in the order of \(10^{-7}rad\). The position control errors of the NFTSMC are in the order of \(10^{-6}rad\). SMC provides the largest position control errors of the three control methods, in the order of \(10^{-4}rad\). From Figure 4, it is seen that the designed control method also has the smallest velocity control errors among all the three control methods. The control torque for all three control manners, including SMC, NFTSMC, and the designed FTSMC, are displayed in Figure 5. It can be recognized from Figure 5, SMC and NFTSMC have discontinuous control torque because of using the high-frequency control law. Meanwhile, the designed system has \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Error Control System** & \(E_{1}\) (10 s) & \(E_{2}\) (10 s) & \(E_{3}\) (10 s) & \(E_{1}\) (30 s) & \(E_{2}\) (30 s) & \(E_{3}\) (30 s) \\ \hline SMC & 0.03102 & 0.01946 & 0.03113 & 0.01077 & 0.00671 & 0.01082 \\ NFTSMC & 0.02322 & 0.01284 & 0.02208 & 0.00774 & 0.00428 & 0.00737 \\ Proposed Controller & 0.01939 & 0.01037 & 0.01793 & 0.00646 & 0.00345 & 0.00597 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The averaged tracking errors under the input of the controllers. smooth control torque with a significant elimination of the chattering phenomena. To achieve this goal, the suggested controller applies STCL to substitute the high-frequency control law in removing chattering behavior. Figure 2: The designated trajectory configuration and real trajectory under three control methods: (**a**) at Joint 1, (**b**) at Joint 2, and (**c**) at Joint 3. Figure 3: The position control errors: (**a**) at Joint 1, (**b**) at Joint 2, and (**c**) at Joint 3. Figure 4: The velocity control errors: (**a**) at Joint 1, (**b**) at Joint 2, and (**c**) at Joint 3. Response time of the sliding mode manifolds, including SMC, NFTSMC, and designed FTSMC, are shown in Figure 6.  
Figure 5: The control input signals: (**a**) Sliding Mode Control (SMC), (**b**) Non-singular Fast Terminal Sliding Mode Control (NFTSMC), and (**c**) designed Fast Terminal Sliding Mode Control (FTSMC).