## 1 Introduction  
Due to advantages such as high load efficiency, small size-to-power ratio, and fast response, hydraulic actuators have been widely investigated in construction [1; 2], aerospace [3], motion simulator [4], as well as robotic area [5; 6; 7]. Boston Dynamics' hydraulic robots such as BigDog [5], Atlas [6], and SARCOS's robot exoskeletons [7] are some examples of advanced hydraulic robot system. One of the crucial challenges in control of the hydraulic manipulator is undesired behavior due to extremely nonlinear behaviors of the system and actuator dynamics, uncertainties of the system, and external disturbance. In previous works [8; 9], the actuator dynamics were usually excluded from the manipulator dynamics to simplify the control procedure. However, the uncertainties in actuator dynamics affects the control performance, as well as the stability of the whole system [10]. Consequently, the actuator dynamics have been considered in robotic control design in recent researches [10; 11; 12; 13; 14; 15; 16; 17; 18]. The system dynamics arises both unmatched and matched uncertainties in manipulator dynamics and actuator dynamics, respectively. Many studies have been provided to deal with these problems, which can be divided into two categories. Firstly, some advanced controllers have been investigated on system dynamics which were expressed by taking derivative of the acceleration variable of the manipulator to including the actuator dynamics [10; 11; 12; 13]. Their results proved the effectiveness of this approach. However, they meet the observable problems due to the noise measurement. On the other hand, some advanced controllers have been employed on the system dynamics, which includes the manipulator dynamics and actuator dynamics, independently [14; 15; 16; 17]. So, they take advantages of the sensors which had already been equipped in the system. Additionally, these controllers have been developed based on the backstepping technique because it is well-known as a good technique for handling the matched and unmatched uncertainties [19]. The main property of backstepping design, is that it stabilizes the system states through a step-by-step recursive process [20]. However, classical backstepping design mainly supposes that the uncertainties and the disturbance are constant or slowly altering. To enhance the compensation capability of the backstepping control, some adaptive laws [21] have been used to cope with the uncertainties. But, the derivatives of the model uncertainty and the disturbance cannot reach to zero, the backstepping with adaptive laws is no longer relevant. The sliding mode control (SMC) has been extensively used to control uncertain nonlinear systems with high dynamic uncertainties because of its robustness [22]. The fundamental idea of the SMC is to use a discontinuous control term for driving the controlled system's error state variables toward zero. Chattering effect may be activated in cases of large control gains used [23]. So, the adaptive mechanisms [24; 25; 26; 27; 28] have been provided to eliminate this effect and to improve the ability of the SMC, which is named an adaptive SMC (ASMC). However, the SMC is not good at dealing with the unmatched uncertainties in the nonlinear system. Considering the characteristics of ASMC and backstepping, it is possible to combine these methods together for reserving their advantages and reducing their limitation at the same time. In previous studies [14; 15; 17], the adaptive backstepping sliding mode control was applied to a manipulator including electric actuator dynamics for position control problem. The adaptive mechanism were developed based on linear regressor method [14; 17], and least square-support vector machine [15]. Although their results proved that these controllers dealt well with the uncertainties and disturbances. Their structure and initial values are usually selected by designer's experiences, so it is challenging to employ them in practice.