## 1 Introduction  
The hydraulic-driven flexible robotic manipulator system (HDFRMS) has wide implementation potentials in a large number of engineering fields human can hardly deal with directly, such as forest exploitation, heavy payload motion control, construction work, mobile equipment applications and other industrial areas [1, 2, 3], since their energy consumption and weight are lower, but the speed is higher. Unlike rigid manipulators, the dynamics of flexible manipulators is the nonlinear rigid-flexible coupling infinite-order model [4, 5]. Indeed, many control methods have been presented for flexible multibody dynamics system, in particular flexible link manipulators and flexible joint manipulators, involving PID control [6, 7], neural network control [8], sliding mode control [9], adaptive control [10], optimal control [11], etc. Nevertheless, ill-conditioned numerical issues and inaccurate or unreliable control will occur, because these dynamic model-based control methods are directly designed ignoring multiple timescales in dynamics. Singular perturbation theory (SPT) is used to deal with the problem of control under different timescales. The two-order dynamic model of flexible systems can be decoupled into the slow subsystem (SS) in slow timescale, which is the traditional timescale, and the fast subsystem (FS) in new timescale, namely fast timescale based on SPT. In [4, 12] the flexible link manipulator system is decomposed through the perturbation parameter that the small-time constant is defined as, and two sub-controllers are designed. The composite control consisting of fuzzy sliding mode control in SS and linear quadratic regulator (LQR) optimal control in FS are proposed in [13]. Additionally, in [14], the slow controller is substituted by the learning controller and the disturbance observer to improve the trajectory tracking precision. An adaptive non-saturated control is preferentially considered to handle actuator saturation involving dynamics parameters in the flexible joint manipulator system, where there are any parametric uncertainties [4,15]. Additionally, the adaptive neural backstepping control in SS is presented as the uncertain nonlinearity compensator [16]. Furthermore, the fuzzy control [17], the robust fuzzy sliding mode control [18] and the adaptive high-gain Kalman filter [19] are investigated to improve robustness. Compared with the electrical-driven manipulator, hydraulic-driven manipulators have higher power to handle heavier payloads whose weights are greater than 100 kg, and they have lower stiffness and response time. However, the order of their dynamic equations that are more complicated is raised from the second order to the third order. It is so difficult to control them because of electro-hydraulic coupling and inherent elastic vibration. A hybrid controller including a feedforward control and feedback control [20], and model-based adaptive energy shaping controller [21] are proposed in the hydraulic manipulator system. Many observers, such as the adaptive observer and variable structure disturbance observer, are also designed to handle uncertainties and vibrations [22, 23, 24]. However, the description of dynamics is not complete yet. That is, flexible vibration or hydraulic drive has always been ignored in past controller designs. Therefore, the control method which can track desired trajectories precisely and suppress flexible vibrations based on the complete electro-hydraulic coupling model needs be presented. In the work, we discuss an ADP-based trajectory tracking control law and singular perturbation decoupling for the HDFRMS with uncertainties and varying payloads. The main contributions include the following three aspects. (1) The whole third-order dynamic model including three link rigid-flexible manipulator, asymmetric four-way valve controlled hydraulic cylinder and hydraulic actuator is derived. It is more beneficial and complete than the actual engineering system, since both the flexible vibration and hydraulic drive are considered in the dynamic model.