## 4 Discussion  
### The Feasibility of the Proposed ATDC-SM in Two-Link Robot Manipulator  
To determine the feasibility of the proposed ATDC-SM in multi-input-multi-output (MIMO) system, we have performed simulations through the system model of two-link robot manipulator. In this simulation, reference trajectories with non-differentiable point (Figure 9) was designed to be of both low and high frequencies. Two control approaches, i.e., the conventional ATDC [21] and the proposed ATDC-SM, have been applied to a two-link manipulator as an independent control method. \begin{table}
\begin{tabular}{c c} \hline \hline
**Control Strategies** & **Joint Angle (deg)** \\ \hline MNTSMC [16] & \(1.32\times 10^{-2}\) \\ ASMC [18] & \(0.98\times 10^{-2}\) \\ Conventional ATDC [21] & \(0.97\times 10^{-2}\) \\ Proposed ATDC-SM (9) & \(0.59\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Root-mean-square (RMS) tracking errors of the conventional ATDC and the proposed ATDC-SM. Figure 8: Comparison of the tracking of reference trajectory (solid-line) and the proposed ATDC-SM (dotted-line). Figure 7: Comparison of the tracking errors of MNTSMC [16] (dashed-line), ASMC [18] (dotted-dashed-line), the conventional ATDC [21] (solid-line), and the proposed ATDC-SM (dotted-line). Subsequently, all of their parameters are tuned in a low frequency trajectory. The system model of two-link robot manipulator [28] can be represented as follows:  
\[\mathbf{M}(\mathbf{q}_{t})= \begin{bmatrix}l_{2}^{2}m_{2}+2l_{1}l_{2}m_{2}c_{2}+l_ {1}^{2}(m_{1}+m_{2})&l_{2}^{2}m_{2}+l_{1}l_{2}m_{2}c_{2}\\ l_{2}^{2}m_{2}+l_{1}l_{2}m_{2}c_{2}&l_{2}^{2}m_{2}\end{bmatrix} \]  
\[\mathbf{C}(\mathbf{q}_{t},\dot{\mathbf{q}}_{t})\dot{\mathbf{q}}_{t}= \begin{bmatrix} -m_{2}l_{1}l_{2}s_{2}q_{2,t}^{2}-2m_{2}l_{1}l_{2}s_{2}q_{1,t}q_{2,t}\\ m_{2}l_{1}l_{2}s_{2}q_{2,t}^{2}\end{bmatrix} \]  
\[\mathbf{G}(\mathbf{q}_{t})= \begin{bmatrix}m_{2}l_{2}gc_{12}+(m_{1}+m_{2})l_{1 }gc_{1}\\ m_{2}l_{2}gc_{12}\end{bmatrix} \]  
\[\mathbf{F}(\dot{\mathbf{q}}_{t})= \begin{bmatrix}\kappa_{1}\text{sgn}(\dot{q}_ {1,t})\\ \kappa_{2}\text{sgn}(\dot{q}_{2,t})\end{bmatrix}  \tag{13}\]  
where \(q_{i}\) is the \(i\)-th joint angle, and \(s_{i}\), \(c_{i}\), \(c_{ij}\) are defined as \(\sin(q_{i,t})\), \(\cos(q_{i,t})\), and \(\cos(q_{i,t}+q_{j,t})\), respectively. The lengths of the links are chosen to be \(l_{1}=20\) (cm) and \(l_{2}=10\) (cm). Gravity acceleration is set to be \(g=9.81\) (m/s\({}^{2}\)), the friction coefficients are set to be \(\kappa_{1}=\kappa_{2}=50\). The loads of the joints 1 and 2 are given as \(m_{1}=10\) (kg) and \(m_{2}=5\) (kg), respectively. To the illustrate the effectiveness of the proposed ATDC, the conventional ATDC [21] is employed for comparison. Additional simulation trajectories with non-differentiable point (Figure 9) are designed to be both low and high frequencies. All parameters of two control approaches are tuned in a low frequency trajectory. The simulation results are depicted in Figures 10-12. Figure 10 shows TVB computed by the proposed ATDC-SM. The TVB is inversely proportional to the magnitude of the sliding variables, and accordingly, the adaptive law \(\dot{\omega}_{t}\) in Equation (11) of the revised paper yields remarkable results. Actually, the TVB decreases if sliding variable moves away from sliding manifold, and hence, high control gains are achieved rapidly. Contrastingly, if the sliding variable is closer to the sliding manifold, the TVB should be increased to reduce the magnitude of the control gains rapidly. The effect of change on the TVB is shown in Figure 11.  
Figure 9: Trajectories of the reference angles: (**a**) Joint 1; (**b**) Joint 2. Figure 11 shows the control gains generated by the proposed ATDC-SM. The control gains with TVB result in both high and low values with a fast adaptation speed. In particular, high control gains provide a fast convergence rate which results in precise tracking performance.