# Practical Adaptive Sliding-Mode Control Approach for Precise Tracking of Robot Manipulators
###### Abstract
As such, even though various control approaches have been performed for the ASMC approach, there are still some limitations with regard to the adaptation rate of the switching gain. In other words, given that the existing ASMC approaches only provide a fast or a stable adaptation rate, it may not be easy for practicing engineers to achieve the favorable tracking performance without the undesirable side effects. Therefore, it is necessary to develop a more simple and practical control approach that can provide the switching gains with fast and stable adaptation rate. In this paper, we propose a practical ASMC approach, namely, the strong and stable adaptive sliding-mode control (SS-ASMC) approach, that offers strong and stable switching gains in the large and small magnitudes of the sliding variable, respectively. There is a significant effort towards addressing the technical challenges associated with the new time-varying switching gains having two adaptive laws which is called parent and child adaptive laws. A parent adaptive law is designed to be proportional to the magnitude of the sliding variable. Therefore, the proposed SS-ASMC approach can provide strong switching gains through fast adaptation rate when causing inappropriately large sliding variables by the large negative effects, e.g., abrupt disturbances. Then, the switching gains employ the time-varying update parameters with the child adaptive law that are designed to stay on its low bound while away from the sliding manifold. These adaptive laws of the proposed ASMC guarantee that within a finite time, all of the sliding variables enter an arbitrarily small vicinity of the sliding manifold and then stay around it. On the contrary, as mentioned previously, given that the parent adaptive law is designed to be proportional to the magnitude of the sliding variable, it may provide a slow adaptation rate in the vicinity of the sliding manifold. To achieve a fast and stable adaptation rate, the time-varying update parameters with the child adaptive law are activated, which has an effect of adjusting the parameters to the switching gains of the proposed SS-ASMC approach. This is a reason why the switching gains can be quickly adjusted without making them unstable. Therefore, even though the switching gains become somewhat large to achieve precise tracking performance while guaranteeing the robustness, they are not sufficiently affected by the chattering because the adaptation rate is fast and stable owing to the child adaptive law. Further, all control approaches based on sliding-mode control are in need of an equivalent control approach and thus require the complex and exact system dynamics. However, the proposed SS-ASMC approach employs a time-delayed estimation (TDE) to solve the negative demands generating in an equivalent control approach, which can compensate uncertainties, nonlinear dynamics, and unknown disturbances in the robot manipulators. Moreover, the pole-placement (PPT) is also employed to obtain the error dynamics with the dominant pole, which helps to converge to the equilibrium point. For these reasons, the proposed SS-ASMC approach can be taken without the knowledge of the system information in the robot manipulators, which makes it unnecessary to have a rather complete system dynamics. From these benefits, the proposed method works well for simulations accompanying undesirable side effects such as the parameter variations and the abrupt disturbances. It is shown that the proposed SS-ASMC approach guarantees the bounded-input bounded-output stability through Lyapunov function, meaning the tracking errors are uniformly ultimately bounded. The effectiveness of the new time-varying switching gains in the proposed SS-ASMC approach is illustrated in simulations with robot manipulators. It turns out the proposed one has strong and stable tracking performance without the chattering, and then compared with the existing control approaches. The remainder of this paper is organized as follows.