## 6 Computer Simulation Study  
### Simulation Setup  
The proposed scheme can be applied for a general n DOF CKCM manipulator. However, the implementation and application for a n DOF manipulator require massive computation effort and hardware complexity. Therefore, a 2 DOF CKCM robot manipulator that resembles a special case of the complete n DOF manipulation was designed and built for the purpose of testing of the results obtained in our projects. Hence, this article is devoted to investigate the simulation study of the obtained results on the 2 DOF manipulator. In this section, computer simulation will be conducted to study the performance of the above NFTSMC in comparison to other existing control schemes when they are employed to control the motion of a 2-DOF CKCM. The computer simulation study for the NFTSMC is described in the block diagram given in Figure 3 when \(n=2\) since this manipulator has two DOFs, and is designed in MATLAB/Simulink environment in Figure 4. When other existing control schemes are applied, then the block labeled as _Proposed_ in Figure 3 is replaced by their particular control schemes. For this particular manipulator, it is noted that the length of an actuator is denoted as its joint variable. To facilitate the analysis of tracking errors in Cartesian space, the actual joint variables \(\mathbf{q}\) and their joint velocities \(\mathbf{\dot{q}}\) of the CKCM are converted to their corresponding Cartesian variables by using the CKCM forward kinematic transformation, which is also a closed-form solution due to the number of DOFs of this manipulator. MATLAB-Simulink(r) is used to comparatively evaluate the performance of the developed NFTSMC (Syn-TDE-NFTSMC) in comparison with four other existing control schemes including PD-based control scheme (LINEAR), TDE-based LINEAR (TDE-LINEAR), TDE-based LINEAR with synchronization errors (Syn-TDE-LINEAR), and TDE-based SMC with synchronization errors (Syn-TDE-SMC) in tracking the same motion. A brief description of the above control schemes can be found in Appendix A. After conducting numerous simulations of the above control schemes, we selected the most optimal parameters for their best tracking performance. The parameters of the manipulator are listed in Table 1 while the control parameters of the control schemes are provided in Table 2.  
**Remark 2**. : _The parameters of the control scheme are tuned as: Tuning \(\overline{\mathbf{M}}\) and \(\boldsymbol{\alpha}\), diagonal matrices, by increasing the diagonal elements from small positive values, while checking the control performance by trial error. The selection of the other parameters of the proposed control scheme \(p_{1}\), \(p_{2}\), \(q_{1}\), \(q_{2}\), \(\mathbf{K_{1}}\), \(\mathbf{K_{2}}\), and \(\mathbf{K}\) are described in [28]. \(\mathbf{K_{sw}}\) can be selected from (28)._  
The Lagrangian dynamic equations of the above manipulator is given in [33] as follows:  
\[\boldsymbol{\tau}=\mathbf{M}(\mathbf{q})\mathbf{\ddot{q}}(t)+\mathbf{C}( \mathbf{q}\mathbf{\dot{q}})\mathbf{\dot{q}}(t)+\mathbf{G}(\mathbf{q})+\mathbf{ F}(\mathbf{q}\mathbf{\dot{q}}) \tag{46}\]  
with  
\[\boldsymbol{\tau}(t)=\left(\tau_{1}\ \tau_{2}\right)^{\mathsf{T}}\text{; } \mathbf{q}(t)=\left(q_{1}\ q_{2}\right)^{\mathsf{T}} \tag{47}\]  
where \(\tau_{i}\) denotes the joint force of the \(i\)th actuator, respectively, for \(i=1,2\). Figure 4: Control design in MatLab/Simulink environment. The inertia matrix, the Centrifugal and Coriolis forces, and the friction and the gravitational forces at two joints are given by the following:  
\[\mathbf{M}= \begin{bmatrix}m_{1}&0\\ 0&m_{1}\end{bmatrix} ,\ \mathbf{C}= \begin{bmatrix}0&\frac{m_{1}(q_{2}-q_{1})}{3v} \\ \frac{m_{1}(q_{2}-q_{1})}{3v}&0\end{bmatrix}