## 4 Numerical Simulation Studies
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Control Algorithm** & **Control Parameters** & **Control Parameter Values** \\ \hline SMC & \(z,G,g,u\) & \(20,10,0.01,0.7\) \\ S-SMC & \(\lambda,z,K,G,g,u\) & \(0.9,20,10,10,0.01,0.7\) \\ NFTSMC & \(G,g,\Gamma_{1},\Gamma_{2},\gamma_{1},\gamma_{2},u\) & \(10,0.01,10,5,1.4,1.28,0.7\) \\ S-FOTSMC & \(\lambda,G,g,\Gamma_{1},\Gamma_{2},\gamma_{1},\gamma_{2}\) & \(0.9,10,0.01,16,6.5,0.5,0.67\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the control algorithms. Figure 2: The 3D CAD model of the parallel manipulator. \[\left\{ \begin{array}{l}f_{n\alpha 1}=(5/2)\dot{\theta}_{\alpha 1}+(1/2)\text{sgn} \Big{(}\dot{\theta}_{\alpha 1}\Big{)}\\ f_{n\alpha 2}=(5/2)\dot{\theta}_{\alpha 1}+(1/2)\text{sgn}\Big{(}\dot{\theta}_{ \alpha 2}\Big{)}\\ f_{n\alpha 3}=(5/2)\dot{\theta}_{\alpha 3}+(1/2)\text{sgn}\Big{(}\dot{\theta}_{ \alpha 3}\Big{)}\end{array} \right. \tag{23}\]  
Figures 3-6 show the trajectory position tracking, the tracking errors of the end-effector in the X-direction and in the Y-direction, and comparison of tracking errors in active joint space, respectively. Synchronization errors and cross-coupling errors in active joint space are shown in Figure 7. The end-effector of the manipulator has the initial position as \((0.5284,0.3681)\) and this end-effector is driven to follow a circular path. It should be noted that the initial positions of the robotic system were selected according to the workspace of the robotic system. That means these initial positions must be satisfied inside the workspace. From Figures 3-6 it is seen that the tracking path generated by SMC has the biggest discrepancy compared to the prescribed path; it has the worse tracking result among the four methods. S-SMC and NFTSMC produce better results than SMC but NFTSMC provides better performance and faster error convergence than both S-SMC and SMC. It is noteworthy that the proposed system is developed by applying the combination of synchronization control and FOTSMC. Accordingly, the tracking path generated by the proposed system has the smallest discrepancy compared to the prescribed path, and it has the fastest convergence rate to the prescribed path among the four control methods. From Figures 4-6, it is observed that the tracking errors generated by S-SMC are smaller than the tracking errors generated by SMC. However, S-SMC provides a worse tracking error than NFTSMC. Especially since the proposed system provides the smallest tracking errors compared to SMC, NFTSMC and S-SMC. The tracking accuracy under the proposed method has been significantly improved in comparison with NFTSMC and S-SMC. The control input signals for all control types, including SMC, S-SMC, NFTSMC, and the suggested method are illustrated in Figure 8. From Figure 8, it is clear that SMC, S-SMC, and NFTSMC provide a continuous control signal because these three methods use a BLA to eliminate chattering. However, it is seen that the chattering has not been completely removed in the torque input of the NFTSMC. Additionally, the use of a BLA leads to the reduce robustness and accuracy of the control method. While the suggested method uses an integral of the switching control term to reject chattering without an accuracy reduction. Figure 3: The desired trajectory and real trajectory of the end-effector when the parallel manipulator tracks a circular trajectory. Figure 4: The tracking errors of the end-effector in the X-direction. Figure 5: The tracking errors of the end-effector in the Y-direction. Figure 6: The tracking errors in the active joint space (**a**) at Joint 1, (**b**) at Joint 2, and (**c**) at Joint 3. Figure 7: The Synchronization Error and Cross-Coupling Error in the active joint space (**a**) at Joint 1, (**b**) at Joint 2, and (**c**) at Joint 3. **Case 2. **_When the parallel manipulator tracks a linear trajectory._  
Figure 9 shows the results of tracking a linear trajectory.