## 3 Results and Discussion
### Simulation Results of the End Effector Positioning  
In this section, simulation results based on the FSBSC system, in which the controller's parameters are optimized by GWO-WWOA technique of the previous section are presented. The XYZ reference positions are given as \(x_{r}=300\) mm, \(y_{r}=\)\(-195\)\(+19.5t\) mm, \(z_{r}=-100\sin(0.016(y_{r}\)\(+150))+120\) mm. Figure 5 shows the trajectory of the end effector command position. For comparison with the proposed FSBSC system, the conventional BSC system was designed.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Joint** & **Optimized Values of Each Parameter** \\ \hline \multirow{6}{*}{Joint 1} & \(m_{11}=-1.9,\ m_{12}=-0.4,\ m_{13}=0,\ \ m_{14}=0.4,\ m_{15}=0.98\) \\  & \(m_{21}=-1.9,\ m_{22}=-0.9,\ m_{23}=0,\ \ m_{24}=0.41,\ m_{25}=0.95\) \\  & \(\sigma_{11}=0.58,\ \sigma_{12}=0.49,\ \sigma_{13}=0.62,\ \sigma_{14}=0.49,\ \sigma_{15}=0.59,\) \\  & \(\sigma_{21}=0.47,\ \sigma_{22}=0.45,\ \sigma_{13}=0.49,\ \sigma_{14}=0.61,\ \sigma_{15}=0.48,\) \\  & \(\Delta m_{1}=0.014,\ \Delta m_{2}=0.05\) \\  & \(m_{11}=-0.9,\ m_{12}=-0.4,\ m_{13}=0,\ \ m_{14}=0.4,\ m_{15}=1\) \\  & \(m_{21}=-0.9,\ m_{22}=-0.5,\ m_{23}=0,\ \ m_{24}=0.4,\ m_{25}=0.9\) \\  & \(\sigma_{11}=0.63,\ \sigma_{12}=0.62,\ \sigma_{13}=0.45,\ \sigma_{14}=0.64,\ \sigma_{15}=0.54,\) \\  & \(\sigma_{21}=0.54,\ \sigma_{22}=0.52,\ \sigma_{13}=0.47,\ \sigma_{14}=0.51,\ \sigma_{15}=0.5,\) \\  & \(\Delta m_{1}=0.02,\ \Delta m_{2}=0.02\) \\  & \(m_{11}=-1.9,\ m_{12}=-0.9,\ m_{13}=0,\ \ m_{14}=0.47,\ m_{15}=0.90\) \\  & \(m_{21}=-1.9,\ m_{22}=-0.9,\ m_{23}=0,\ \ m_{24}=0.40,\ m_{25}=1.0\) \\  & \(\sigma_{11}=0.45,\ \sigma_{12}=0.51,\ \sigma_{13}=0.45,\ \sigma_{14}=0.65,\ \sigma_{15}=0.47,\) \\  & \(\sigma_{21}=0.61,\ \sigma_{22}=0.65,\ \sigma_{13}=0.45,\ \sigma_{14}=0.64,\ \sigma_{15}=0.48,\) \\  & \(\Delta m_{1}=0.01,\ \Delta m_{2}=0.01\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Optimized values of the parameters of IT2-FLS via GWO-WWOA.  
Figure 4: The ITAE values for each joint obtained from WOA, GWO, and GWO-WOA, and WWOA techniques. (**a**) link 1. (**b**) link 2. (**c**) link 3.  
Simulation results for the Cartesian positioning command of the end-effector given in Figure 5 are presented in Figure 6. Figure 6a shows the 2D reference positioning command of the end effector. For the end effector positioning command shown in Figures 5 and 6a, the joint position command was generated as shown in Figure 6b by the inverse kinematics. The positioning tracking results in the joint 1 are presented in Figure 6c,d, where the tracking error of the proposed FSBSC system is very small compared to that of the BSC system. The tracking outputs and errors are presented in Figure 6e-h, where the proposed FSBSC systems shows the results outperforming that of the BSC system.  
Transforming the joint tracking results into that of the end-effector, the tracking output of the end-effector can be represented as shown in Figure 6i, where the better control performance of the proposed FSBSC system over the conventional BSC system is viewed as a 3D configuration. For the FSBSC system, ITAE values for each optimization technique are presented in Table 3, where the GWO-WWOA has the best performance over other three techniques.  
In simulation, although a few control gains of the BSC system were twice as that of the FSBSC system, the control performance of the former system is poorer than the latter. Figure 6j,k present the control inputs of the BSC and FSBSC systems, where the level of the control input of the FSBSC system is lower than that of the BSC system owing to higher control gains of the BSC system. Therefore, the simulated results prove that the performance of the proposed FSBSC system is much enhanced compared to the conventional B  
Figure 6: _Cont._