## 1 Introduction
The adaptive law, however, may require a long time to converge to an equilibrium point because of the damping terms. Furthermore, these two approaches do not theoretically complete the proof of the system stability because they have not demonstrated their inherent stability criteria. To solve these problems, a new adaptive time-delayed control (ATDC) [21] scheme was developed, which was applied to robot manipulators. First, this approach theoretically proved the stability criteria for the TDC with adaptive control gains. This adaptive law is based on the sliding variable without damping terms, which is proportional to the magnitude of sliding variable. Additionally, the adaptive control gains of ATDC can be appropriately adjusted to obtain precise tracking performance because the control gains can increase significantly from small positive values. From these benefits, ATDC provides precise tracking performance with robustness. Although this approach is effective, its control gains create serious and inherent problems because they comprise a _discontinuous function_. In other words, as the control gains cause the chattering or fluctuation, the tracking performance and robustness may be degraded in robot manipulators. Furthermore, the adaptive law for the control gain has a time-invariant boundedness, which provides a point where the control gain increases or decreases. Setting the time-invariant boundedness is a time-consuming task and hence is difficult to achieve precise tracking performance without undesired side effects. In this regard, a new ATDC with effective and practical adaptive law should be developed. In this paper, we propose an ATDC based on the sliding-mode (ATDC-SM). The proposed ATDC-SM provides a new adaptive law with imposing time-varying boundedness for adjusting the control gains appropriately. This approach guarantees that the sliding variables enter a vicinity of the sliding manifold within a finite time. The vicinity of the sliding manifold is small, which is called time-varying boundedness (TVB) in this paper. It can be adjusted to achieve an appropriately adaptedspeed with the control gains, which is inversely proportional to the magnitude of the sliding variables. For example, when the magnitude of the sliding variables is large, there is room to increase the control gains because the TVB decreases. This is a result of fast response. Meanwhile, when the sliding variables stay near the sliding manifold, an increase of control gains may cause undesired side effects such as chattering or fluctuation. To remedy this problem, TVB will be arbitrarily increased, which creates a region where the control gains decrease to avoid undesired side effects. Therefore, TVB can be adjusted appropriately to ensure fast response without undesired side effects from a new adaptive law. Moreover, the control gains of the proposed ATDC-SM comprise a _continuous function_, producing stable control gains. Although the continuous function may exhibit a slow adaptation speed of control gain compared with the discontinuous function, the proposed one provides a room for improving the adaptation speed owing to the TVB. Also, the proposed ATDC-SM uses the TDE technique to cancel out complex nonlinear dynamics and unknown disturbances and provides a simply structured but effective method. It is shown that the tracking errors are uniformly ultimate boundedness (UUB) by using a Lyapunov approach [22]. The effectiveness of the proposed ATDC-SM is illustrated through a simulation with a one-link robot manipulator, which is compared with that of the conventional ATDC [21]. The reminder of this paper is organized as follows: In Section 2, the conventional ATDC scheme is briefly introduced and a presentation of the proposed ATDC-SM scheme for a robot manipulator is presented. In Section 3, the effectiveness of the proposed ATDC-SM scheme is explained through the simulations. In Section 4, the feasibility of the proposed ATDC-SM scheme is verified based on additional simulations. In Section 5, a summary of this paper is presented.