## 3 Simulation
### Simulation Results  
Figure 3 shows TVB generated by the proposed ATDC-SM that allows the adaptive law used in Equation (11) to provide remarkable results. The change in adaptive law directly affects the control gain. In other words, the TVB is decreased to obtain fast convergence rate, and thus control gain is increased  
Figure 1: A one-link robot manipulator. Figure 2: Trajectory of the reference angle. rapidly. Contrastingly, the TVB is increased to avoid undesired side effects such as fluctuations or chattering caused by too large control gain, and it is a result that the control gain is rapidly decreased. The detailed results of the control gain are shown in Figure 4.  
Figure 4 shows the control gains generated by the conventional ATDC and the proposed ATDC-SM. The discontinuous control gain of the conventional ATDC moves instantaneously to its minimum value, causing chattering and fluctuations as shown in Figure 5. These effects may result in a severe degradation in the tracking performance of robot manipulators. The proposed ATDC-SM with TVB, however, provides a fast adaptation speed, and its control gain is decreased to avoid the undesired side effects while maintaining a continuous control gain. As shown in the next figures, the proposed ATDC-SM is effective in reducing undesired side effects while achieving better tracking performance. Figure 5 shows the control torques generated by the conventional ATDC and the proposed ATDC-SM. As shown in Figure 4, the control gain is directly related to the control torque. In fact, the conventional ATDC causes undesired side effects, such as chattering, because it employs the control gain with a discontinuous function. By contrast, the control gain of the proposed ATDC-SM is based on continuous functions, which does not cause undesired side effects such as chattering. However, the chattering generated in the transient region, as shown in Figure 5, is caused by the sensor noise term used in Equation (12) of the revised paper. In other words, the proposed ATDC-SM results in reducing undesired side effects such as chattering and producing a stable control torque compared with the conventional ATDC. Figure 4: Control gains computed by (**a**) the conventional ATDC and (**b**) the proposed adaptive time-delayed control based on the sliding-mode (ATDC-SM). Figure 3: Time-varying boundedness computed by the proposed ATDC-SM. Figure 6 shows the sliding variable generated by the proposed ATDC-SM. The sliding variable represents irregular values that provide instantaneous large values. These points appear at approximately 2 s, 5 s, 7 s, 8 s, 10 s, 11 s, and 14 s, as shown in Figure 2. Therefore, the undesired side effects are caused by the non-differentiable point and the friction forces [27] when the direction of operation of the motor installed in the robot manipulator changes. However, as shown in Figure 6, the proposed ATDC-SM helps the sliding variables converge to the sliding manifold, and subsequently they stay in the vicinity of the sliding manifold. Figure 7 shows the tracking errors of four control approaches, i.e., MNTSMC [16], ASMC [18], the conventional ATDC [21], and the proposed ATDC-SM. The effect of change on control gain generated by the adaptive law with a time-varying boundedness in the proposed ATDC-SM is significant, and it can be observed that the fast convergence speed is achieved at approximately 2 s, 5 s, 7 s, 8 s, 10 s, 11 s, and 14 s. In addition, the proposed ATDC-SM provides more stable results without undesired side effects such as chattering compared with the existing control approaches, including MNTSMC, ASMC, and the conventional ATDC, even at the point where no derivative is possible at approximately 7 s, as shown in Figure 2. Contrastingly, it can be observed that the existing control approaches produces degraded tracking performance owing to chattering or restricted convergence speed.