## 4 Discussion
### The Feasibility of the Proposed ATDC-SM in Two-Link Robot Manipulator
However, the control gains may result in the excessive values that may cause undesired side effects, e.g., fluctuations and chattering. To avoid these problems, the control gains decrease rapidly owing to the increase in TVB, as shown in Figure 10. In other words, the TVB helps to reduce the control gains to maintain the system stability. The next figure shows that the proposed ATDC-SM is effective in achieving better tracking performance as compared with the conventional ATDC. Figure 12 shows the tracking errors of two control approaches, including the conventional ATDC and the proposed ATDC-SM. The effect on the control gains generated by the adaptive law of Equation (11) with the TVB (seen in Figure 10) in the proposed ATDC-SM is significant. It can be observed that  
Figure 11: Control gains computed by the proposed ATDC-SM: (**a**) Joint 1; (**b**) Joint 2. Figure 10: Time-varying boundedness computed by the proposed ATDC-SM: (**a**) Joint 1; (**b**) Joint 2.  
the fast convergence speed is provided around in 2 s, 5 s, 7 s, 8 s, 10 s, 11 s, and 14 s. Moreover, the proposed ATDC-SM provides more stable results without undesired side effects, such as chattering, compared with the existing control approach even at the point where no derivative is possible, i.e., at approximately 7 s.  
Although the proposed ATDC-SM shows better tracking performance when compared with the conventional ATDC in two-link robot manipulator, applying the proposed one to the MIMO system with independent control may not guarantee global system stability. The proposed ATDC-SM needs to be designed to ensure the system stability in the MIMO system through a future work, and then it would be very meaningful to develop the proposed control-based approach to apply to the MIMO system. The Performance of the Proposed ATDC-SM in One-Link Robot Manipulator With Irregular Reference Trajectory  
The results of the following figures are compared with those of tracking errors in the _Simulation Results_ of this paper. The simulation mathematical model is the same as one-link robot manipulator model seen in Equation (12). Figure 13 shows reference trajectory of the low-to-high frequency with both non-differentiable point and variable amplitude, unlike Figure 2.  
Figure 14 shows the tracking errors of four control approaches, i.e., MNTSMC [16], ASMC [18], the conventional ATDC [21], and the proposed ATDC-SM. As shown in Figures 2 and 13, in reference trajectory between 0-7 s, the tracking errors of all control approaches in Figure 14 are also the same  
Figure 12: Comparison of the tracking errors of the conventional ATDC [21] (dotted-line) and the proposed ATDC-SM (solid-line): (**a**) Joint 1; (**b**) Joint 2. Figure 13: Trajectory of the reference angle. as those in Figure 7 because the reference trajectories of Figures 2 and 13 are the same. However, the tracking errors shown in Figure 14 differ from those shown in Figure 7 around 7 s. As one of the differences, the tracking performance of the existing control approaches is significantly degraded in reference trajectory with non-differentiable point. They cause larger chattering and fluctuation around 7 s. On the other hand, the proposed ATDC-SM has good tracking performance without undesired side effects such as chattering. It implies that the proposed one shows better robust tracking performance than the existing control approaches. As another difference, since the reference trajectory provides large amplitude with high frequency in 7-15 s, the existing control approaches have slow convergence to the equilibrium point, and then the peak point for tracking errors is larger. On the other hand, the proposed ATDC-SM not only maintains fast convergence speed but also shows little change in peak point for tracking errors. RMS errors of all control approaches are shown in Table 2.  
Figure 15 shows both tracking and reference trajectories generated by the proposed ATDC-SM.