# Adaptive Robust Fuzzy Impedance Control of an Electro-Hydraulic Actuator
## 1 Introduction  
Electro-hydraulic systems play an important role in construction vehicles, industrial applications, and underwater operations. Accurate velocity control is necessary, e.g., hydraulic elevators [1, 2]. Accurate force control is also required, e.g., load simulators [3, 4]. Both velocity and force control are required in robot manipulators [5, 6, 7, 8], underwater manipulators [9, 10] and tunnel boring machines [11, 12]. It is necessary to keep an expected dynamic balance between the motion of the actuator and the contact force with the environment or workpieces. [PERSON] [13, 14, 15] developed impedance control to deal with the dynamics between the actuator and the environment. Impedance control adjusts the dynamic relationship between the position and force of the end effector as a mass-damper-spring system. After this, many researchers paid attention to impedance control. [PERSON] et al. [16] developed an impedance control strategy for a lightweight intrinsically compliant parallel ankle rehabilitation robot powered by pneumatic muscle actuators. [PERSON] et al. [17] implemented proprioceptive impedance control in the MIT Cheetah. [PERSON] et al. [18] proposed an adaptive impedance control for underwater manipulator teleoperation. [PERSON] et al. [19] presented adaptive impedance control for robotic teleoperation of contact tasks. [PERSON] and [PERSON] [20] proposed a control strategy combining an adaptive variable impedance tracking controller for the slabstone force/position control. [PERSON] et al. [21] proposed learning algorithms to find the optimal impedance control of human-robot interactions. Because of the strong parameter uncertainties and uncertain nonlinearities in electro-hydraulic systems, many control algorithms have been employed to obtain high-accuracy motion control. Feedback linearization [22] cancels the known nonlinear terms but cannotdeal with system uncertainties. Adaptive control (AC) [23; 24] improves the control system's dynamic performance and static precision. Parameter drift and even instability are the disadvantages of AC when encountering disturbances or measurement noise. Sliding mode control (SMC) [25; 26] can compensate system uncertainties and external disturbances, but high frequency modes may be excited and degrade system performance. [PERSON] put forward a control strategy called adaptive robust control (ARC) [27; 28; 29; 30; 31], which synthesized AC and SMC effectively. Parameter estimation with smooth limitation can be adopted in ARC to handle parameter drift and system instability, and the chattering problem can be solved in ARC through employing continuous processing of the control input. A disturbance observer is another way to achieve better motion tracking performance. Since disturbance is always impossible to measure directly in practice, a disturbance observer is an effective strategy to estimate disturbance and cancel its influence. [PERSON] [32] developed a nonlinear disturbance observer to attenuate the disturbance generated by an exogenous system. [PERSON] et al. [33], [PERSON] et al. [34], [PERSON] et al. [35], [PERSON] et al. [36], [PERSON] et al. [37], and [PERSON] et al. [38] developed an extended disturbance observer to estimate both external disturbances and uncertain parameters. State estimation errors and tacking errors are both employed to drive the extended disturbance observer, whose effectiveness has been verified in different kinds of electro-hydraulic systems. Fuzzy control [39; 40; 41] is commonly used and does not require a precise system model. [PERSON] et al. [42] used fuzzy logic-based impedance control to realize force control. [PERSON] and [PERSON] [43] integrated fuzzy logic and neural networks into impedance control to improve performance of the controller. Thus, fuzzy logic could be employed to regulate parameters of impedance rules, which helps to obtain a better performance in different environmental stiffnesses [44]. The current control strategies for hydraulic systems only focus on impedance control or motion control.