## 4 Numerical Simulation Studies  
To confirm the effectiveness of the suggested algorithm, the simulated examples are performed for a 3-DOF planar parallel robot on SOLIDWORKS and the SimMechanics of MATLAB. First, The SOLIDWORKS software was used to construct the parallel manipulator model which has the 3-D computer-aided design (CAD) type. Where each mechanical element of the robot is separately built then linked by the joints. Second, the SimMechanics link plug-in was used to export an XML file of the robot mode. Then, this XML file is imported to Simulink. Therefore, the CAD assembly is of geometry files and linked the main body in SimMechanics. Third, sensors, joint actuators were set up to the mechanical system, and external disturbance and fictions were applied to test robustness. Finally, the torque commands from the control method block were applied to this robot system. The kinematic illustration of the robotic system and its 3D CAD model are shown in Figures 1 and 2, respectively. The robot parameters are stated in Table 1 and the selection parameters of the control algorithms are stated in Table 2. We used MATLAB/SIMULINK software for all numerical simulation studies, solver ode3 (Bogacki-Shampine), and the sampling time was set to \(10^{-3}\) s.  
To analyze the effectiveness of the suggested approach, the approach was applied to a prescribed trajectory tracking control for a 3-DOF planar parallel robotic manipulator, and its tracking results were compared with those of SMC, synchronization SMC (S-SMC), and NFTSMC. These control methods for comparison have been briefly stated in Appendices A-C.  
Figure 1: The kinematic illustration of the robotic system. The simulations were carried out with respect to 2 cases when the parallel manipulator tracks a circular trajectory and a linear trajectory. **Case 1. **_The desired trajectory of the end-effector for the position tracking when the parallel manipulator tracks a circular trajectory:_  
\[\left\{ \begin{array}{l}x=0.49+0.03\cos(\pi t/3)\\ y=0.37+0.03\sin(\pi t/3)\\ \phi=\pi/2\end{array} \right. \tag{22}\]  
We assume that the effect of the friction force on the passive joints is much smaller than that on the active joints. Thus, in order to simplify the dynamic model, only the friction forces on the active joints are considered. The friction models of the system, including the viscous friction and the Coulomb friction torques, are only assumed to test the robustness of the control system. Because it is difficult to exactly calculate these friction terms, in this paper, we do not measure the physical value of friction. Therefore, the following friction forces at each active joint were modeled: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Robot Parameters** & **Description** & **Value** & **Unit** \\ \hline \(l_{1}\) & The lower part length of each leg & 0.4 & m \\ \(l_{2}\) & The upper part length of each leg & 0.6 & m \\ \(l_{3}\) & The dimension of the motion platform & 0.2 & m \\ \(l_{c1}\) & Distance from the joint to the mass center of each lower leg & 0.2 & m \\ \(l_{c2}\) & Distance from the joint to the mass center of each upper leg & 0.3 & m \\ \(m_{i1}\) & Mass of each lower leg & 5.12 & kg \\ \(m_{i2}\) & Mass of each upper leg & 7.39 & kg \\ \(m_{p}\) & Mass of the motion platform & 3.84 & kg \\ \(l_{i1}\) & Inertia moment of the lower of \(i^{\text{th}}\) leg & \(91\times 10^{-3}\) & kg\(\cdot\)m\({}^{2}\) \\ \(l_{i2}\) & Inertia moment of the upper of \(i^{\text{th}}\) leg & \(267\times 10^{-3}\) & kg\(\cdot\)m\({}^{2}\) \\ \(l_{p}\) & Inertia moment of the motion platform & \(65\times 10^{-3}\) & kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The robot parameters.