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This paper presents a new widely and stably adaptive sliding-mode control (WS-ASMC) with nonsingular terminal sliding variable to enhance the performance in reaching and sliding phases. The proposed WS-ASMC has developed new widely and stably adaptive switching gains (W-ASG and S-ASG). The W-ASG is based on an adaptive law with a fast adaptation rate to appropriately suppress the errors generated by using state information from previous time. Therefore, it provides a fast convergence rate in the reaching phase. The S-ASG is directly designed to be related to the negative magnitude of sliding variables. It helps to enhance the system stability while providing a fast convergence rate from the W-ASG. A nonsingular terminal sliding variable is applied to the proposed WS-ASMC. It has strong attractivity and hence improves the convergence rate in the sliding phase. Additionally, as it exhibits a synergistic effect with the S-ASG, it is a solution to mitigate the problem of the insufficient control torque generated by applying this sliding variable. Based on these benefits, the proposed WS-ASMC can provide precise tracking performance with robustness owing to the synergistic effects of the enhanced reaching and sliding phases. It is shown that the tracking errors are uniformly ultimately bounded. The effectiveness of the proposed control is clearly demonstrated through the simulation of robot manipulators and is compared with that of existing control approaches. INDEX TERMSAdaptive sliding-mode control, time-delayed control, nonsingular terminal sliding variable, fast adaptation, tracking control, robot manipulator. I. INTRODUCTION He was a Postdoctoral Researcher with the Department of Creative IT Engineering, POSTECH. He is currently a Researcher with the Electronics and Telecommunications Research Institute (ETRI), Daegu, South Korea. His research interests include control and automation, multiobjective control, robot manipulator, and deep learning. CHANGMOOK KANG (Member, IEEE) received the B.S. and Ph.D. degrees in electrical engineer-

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