###### Abstract  
This paper is aimed at addressing the tracking control issue for an n-DOF manipulator regardless of unknown friction and unknown control direction. In order to handle the above issues, an adaptive sliding mode control (ASMC) is developed with a Nussbaum function. The sliding mode control (SMC) in the proposed control guarantees the tracking problem and fast responses for the manipulator. Additionally, there are adaptive laws for the robust gain in the SMC to deal with the unknown external disturbance and reduce the chattering effect in the system. In practice, the mistakes in the connection between actuators and drivers, named unknown control direction, cause serious damage to the manipulator. To overcome this issue, the Nussbaum function is multiplied by the ASMC law. A Lyapunov approach is investigated to analyze the stability and robustness of the whole system. Finally, several simulations are implemented on a 3-DOF manipulator and their results are compared with those of the existing controllers to validate the advantages of the proposed method. sliding mode control; manipulator; unknown control direction; Nussbaum function; Lyapunov approach Article  
Adaptive Sliding Mode Control for a Robotic Manipulator with Unknown Friction and Unknown Control Direction  
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D.-T. Adaptive Sliding Mode Control for a Robotic Manipulator with Unknown Friction and Unknown Control Direction  
gain regarding the uncertainties and disturbances without knowledge of them. In recent years, the unknown control direction [20; 21; 22], which is the wrong connection in the actuator, has been interesting to many researchers. The Nussbaum gain technique is well known as a control gain estimator for handling the unknown control directions and it was first presented in 1983 by [PERSON] in _Systems and Control Letters_. In [23], a fault-tolerant control conducted based on feedback linearization is combined with a Nussbaum gain function and an online parameter estimator for a hypersonic vehicle. In [24], a robust adaptive and fault-tolerant control scheme was developed from a novel PID-based fault-tolerant controller with a Nussbaum-type function for a wind turbine. In [25], a command filter-based adaptive tracking controller was designed based on FLS, a command filter, and Nussbaum functions for a class of nonlinear systems with unknown control directions. In [26], [PERSON] et al. developed an adaptive consensus tracking control based on some Nussbaum functions to manage the unknown control directions and the time-varying actuator faults simultaneously. In [27], [PERSON], etc. provided an adaptive fuzzy prescribed performance tracking control for a nonlinear switched stochastic system with unknown control directions. In this study, the adaptive controller also used a Nussbaum-type function to handle the control direction problem. From this literature, there has not been any study integrating the unknown control direction issue into the SMC design for a manipulator. Therefore, this is one of the motivations for our study. This paper presents a novel adaptive SMC for an n-degree of freedom (DOF) robotic manipulator regardless of the existence of the unknown friction and unknown control direction. The proposed control is designed based on the SMC, adaptive switching gain laws, and Nussbaum function. The adaptive switching gain laws are investigated to regulate the robust gain of the SMC online. This approach helps in not only reducing the chattering effect but also deal with the time-varying external disturbance in the system. Moreover, a Nussbaum function is integrated into the adaptive SMC to manage the wrong control connection. Additionally, the stability and robustness of the whole system with the proposed control are analyzed by a Lyapunov theorem. Finally, some simulations are implemented on a 3-DOF manipulator and their results are compared with those of previous methods to demonstrate the superiority of the proposed control.