## 1 Introduction  
In a number of industrial fields, robot manipulators have been used to perform complex and repetitive tasks such as assembling [1], drilling [2], and assisting [3] works. Such robot manipulators are constantly updated to perform more sophisticated tasks for various applications in biotechnology [4] and nanotechnology [5]. These industries require precise tracking performance with robustness. However, as the dynamics of robot manipulators typically contains high non-linearity, coupling dynamics, uncertainties, and unknown disturbances, it is challenging to obtain exceptional tracking performance while operating robot manipulators. To implement the precise tracking of robot manipulators, various control approaches have been developed over the past few decades [1; 6; 7; 8; 9; 10]. In [1; 6], computed torque approaches have been designed for compensating uncertain and unknown dynamics while using a system model. In [7; 8], sliding-mode control approaches have been presented for enhancing robustness while suppressing unknown disturbances. In [9; 10], adaptive control approaches have been developed for adjusting appropriate gains, which are robust control approaches. However, since these control approaches are complicated and depended on the information of a system model, they cannot be used to easily implement the robot manipulators. To address the problem, the time-delayed control (TDC) scheme has received considerable attention owing to its properties. As a key property of the TDC, TDC uses time-delayed estimation (TDE) which is a one-sample delayed information to cancel uncertainties and unknown disturbances on robot manipulators. Furthermore, since the structure of the TDC is simple, it is an inexpensive solution. For this reason, TDC has been known to be a practical and effective control approach, which has been successfully applied in various applications, including robot manipulators [11], unmanned aerial vehicles [12], DC-DC converters [13], and missile guidance systems [14]. However, studies regarding TDC have not fully matured yet primarily owing to the difference between TDE and real information (called TDE error) which is significantly affected by abrupt disturbances such as Coulomb friction. Some researchers have attempted to solve these problems by combining TDC with auxiliary control approaches, including gradient estimators [15], terminal sliding-mode controls [16], fuzzy controls [17], and adaptive sliding-mode controls [18]. As the above-mentioned approaches effectively suppress TDE errors by using the auxiliary controls, they provide better tracking performance than TDC without auxiliary controls. However, when abrupt disturbances (e.g., Coulomb friction) occur, the excessive gains of auxiliary controls may be required, thereby resulting in chattering or fluctuation. Furthermore, the TDC scheme employs fixed control gains whose tuning may be time-consuming, and then the tracking performance of TDC with inappropriate control gains can still be improved. Another commonly implemented solution is the pairing of two TDCs with adaptive control gains [19; 20]. These control approaches have been developed to provide more fundamental remedies, which have time-varying control gains instead of fixed control gains in the TDC, thereby yielding appropriate control gains in the TDC results. In [19], this approach employed a modified Nussbaum function-based scheme that is applied to the time-varying control gains of the TDC for estimating control gains appropriately. However, since the adaptive law used to adjust the control gains is based on angular acceleration using numerical differentiation, undesired side effects such as noise may be generated, and hence it may not be easy to obtain precise tracking performance in real robot manipulators. In [20], the adaptive law of control gains is based on sliding variables with damping terms derived from sliding-mode control. Owing to the control gains in the damping term, this technique provided more stable results than time-varying control gains in [19].