# RBF Neural Network Based Backstepping Control for an Electrohydraulic Elastic Manipulator
## 1 Introduction  
Nowadays, many researchers have focused on developing high-performance machines with capabilities comparable to humans, especially with respect to motion, safety, as well as energy efficiency. From the analysis of human and animal behaviors, it was found that the adaptable compliance and variable stiffness play important roles. Recently, actuators and systems have been developed with the ability to adjust the stiffness, called a variable stiffness series elastic system (VSSES). It is a kind of variable stiffness system, which consists of two actuators where one regulates the equilibrium position of the VSSES, and the other adjusts its stiffness. On an aspect of safety of the robot manipulator, a series elastic manipulator (SEM) contains a variable stiffness actuator (VSA) providing the ability to decouple the inertia of the actuator proper from the inertia of the last link. That makes robots safe and improves their adaptability in the human-robot cooperation field [1]. In literature, many applications of the variable stiffness system (VSS) [2] such as shock absorbance [3; 4; 5], stiffness variability with a constant load [6; 7], and cyclic movement [8], the safety robot [9] proved the effectiveness of the variable stiffness system although this research focused on tuning the system stiffness optimally. Normally, primary torque or force of these VSSs is generated by electric motors [3; 4; 5; 8; 9], or pneumatic actuators [6; 7; 9]. Therefore, it is difficult to use these VSSs for applications which require high power and compact size. To enhance the VSS working range with the above abilities at high stiffness states, an electrohydraulic actuator is a good choice to develop a new VSS because of its advantages of small size-to-power ratios and large force/torque output [10]. However, high nonlinearity and uncertainties in its dynamicscan increase the complexity of the system dynamics, which are great challenges associated with the control VSS. To control a nonlinear system under the presence of uncertain environments and external disturbances, sliding mode control (SMC), a nonlinear robust control, is utilized [11]. The SMC strategies [12; 13; 14] have been effectively developed in many systems with matched uncertainties which mean the uncertainties and the control input act on exactly the same channel [15]. However, its main drawback is a chattering effect, which can affect the control performance [16]. In order to solve this issue and enhance the control performance, the SMC has been combined with some adaptive strategies, such as an integral adaptive switching gain, adaptive parametric uncertainties, and adaptive neural network. The integral adaptive laws [17] did not require any knowledge of the uncertainties. The robust gains increase until they bound the maximum uncertainties, then the robustness and stability are guaranteed. However, when the magnitude of the uncertainties is significantly smaller than the robust gain, the chattering issue can still occur. The adaptive laws [18; 19] were derived to estimate the parametric uncertainties in a hydraulic system. The quality of these adaptive laws depends on the known dynamical model. Sliding mode controls [20; 21; 22; 23] with radial basis function neural networks (RBFNNs) were used to approximate the unknown nonlinearities and the upper bound of the estimated disturbances of a three-phase shunt active power filter by simulations. Although these approximations did not require any knowledge of the uncertainties, their adaptive laws are sensitive to external disturbances and measurement noise [24]. Motivated by the previous works about the variable stiffness system and SMC, an electrohydraulic elastic manipulator (EEM) which includes an adjustment based stiffness mechanism (ABSM) and an electrohydraulic system (EHS) has been investigated as a VSSES. This paper proposes an adaptive robust control for controlling the equilibrium position of the EEM with the presence of the variant stiffness and the uncertainties.