## 5 Simulations  
### Testing System Configuration: A Detailed Examination of the Setup  
The simulations were performed on a 3-DOF robot using MATLAB/SIMULINK R2021b. The mechanical components of the robot were designed using SOLIDWORKS2018 and subsequently integrated into the MATLAB/SIMULINK environment using the SIMSCAPE MULTIBODY LINK tool. This ensured that the simulation model of the robot was an accurate representation of the actual mechanical model. Figure 2 displays the robot model, and Table 1 lists the essential design specifications for the system, such as link dimensions, center of mass position, and inertia [13]. These specifications are incorporated into the SOLIDWORKS robot model. The proposed method does not rely on the mathematical model of the robot for control design. The robot's main objective is to follow a predefined trajectory accurately using its end-effector, which may involve complex motions and varying speeds. An example of such a desired trajectory is shown in the equation below, which describes the position of the end-effector as a function of time in a three-dimensional space, as shown in Table 2.  
Simulations were conducted to validate the proposed system under normal operating conditions of the robot, taking into account uncertainty components, such as friction forces and disturbances, as shown in Table 2. The performance of the developed method was compared with other model-based controllers, including SMC, TSMC, and fast TSMC (FTSMC). Additionally, it was compared to a PID controller, a well-known model-free controller, to demonstrate its superior performance. The objective of these comparisons was to highlight the advantages of the proposed method over existing control approaches. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Description** & **Link 1** & **Link 2** & **Link 3** \\ \hline Length (m) & \(l_{1}=0.25\) & \(l_{2}=0.7\) & \(l_{3}=0.6\) \\ \hline Weight (kg) & \(m_{1}=33.429\) & \(m_{2}=34.129\) & \(m_{3}=15.612\) \\ \hline \multirow{3}{*}{Center of Mass (mm)} & \(l_{1x1}=0\) & \(l_{2x2}=0.3477\) & \(l_{3x}=0.3142\) \\  & \(l_{1y}=0\) & \(l_{2y}=0\) & \(l_{2y}=0\) \\  & \(l_{41z}=-0.7461\) & \(l_{2z}=0\) & \(l_{2z}=0\) \\ \hline \multirow{3}{*}{Inertia (kg\(\cdot\)m\({}^{2}\))} & \(l_{1 xx}=0.7486\) & \(l_{2 xx}=0.3080\) & \(l_{2 xx}=0.0446\) \\  & \(l_{1 yy}=0.5518\) & \(l_{2 yy}=2.4655\) & \(l_{3 yy}=0.7092\) \\ \cline{1-1}  & \(l_{1 zz}=0.5570\) & \(l_{2 zz}=2.3938\) & \(l_{3 zz}=0.7207\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Essential design specifications for the manipulator. Figure 2: Designed model of a 3-DOF robot used in simulations [13]. To ensure fairness, control parameters from the original papers were utilized during the simulation of the controllers, considering the differences in controller structures. Parameters for the proposed controller were determined through experimentation. The design parameters of the developed algorithm are reported in Table 3.  
Assuming we possess knowledge of all the robot dynamics, including \(\tilde{M}(\alpha_{1})\), \(\tilde{C}(\alpha_{1},\alpha_{2})\), and \(\tilde{G}(\alpha_{1})\), with the exception of the uncertain term \(\mathcal{F}\), we can proceed to design various model-based controllers for the robot as outlined below. According to [15], the sliding surface of the SMC is constructed as follows:  
\[S=e_{2}+\mathcal{C}e_{1}\, \tag{46}\]  
where \(\mathcal{C}=\text{diag}(c_{i})\) with \(c_{i}>0\) for \(i=1,\ldots,n\). The control torque obtained from the output of the SMC is expressed as follows:  
\[\tau=\tilde{M}(\alpha_{1})\big{(}\dot{\alpha}_{2d}+\mathcal{C}e_{2}+\Lambda \text{sign}(S)+\Delta S\big{)}+\tilde{C}(\alpha_{1},\alpha_{2})\alpha_{2}+ \tilde{G}(\alpha_{1})\, \tag{47}\]  
where \(\Delta=\text{diag}(\delta_{i})\) with \(\delta_{i}>0\) for \(i=1,\ldots,n\), and \(|\mathcal{F}|\leq\Lambda\).