## 1 Introduction
To reduce the effect of dynamic uncertainties, adaptive-based controllers are the most commonly used methods [16; 17; 18; 19; 20; 21; 22; 23; 24]. For example, in [17], an adaptive robust controller with ESO (ARCESO) was synthesized for high-accuracy motion control of a DC motor, in which the adaptive control was presented to deal with for the parametric uncertainty. To suppress the parametric uncertainty, a neural network learning adaptive robust controller was synthesized for an industrial linear motor stage to achieve good tracking performance and excellent disturbance rejection ability, where the parametric variations were handled by the adaptation part of the controller [18]. Although adaptive based controllers can achieve satisfactory performance in many physical systems, the parameter variation range should be known in advance [17; 18; 23], which is usually difficult to acquire in advance of practical application, thus leading to the limitation of wide application. To overcome the effect of unknown disturbances, disturbance observers are intuitive and effective methods [17; 18; 25; 26; 27; 28; 29; 30]. In [16], an ESO was constructed to estimate the unstructured uncertainties (including nonlinear friction, external disturbances, and unmodeled dynamics) for designing an ARCESO high-accuracy motion controller of a DC motor. In [25], a generalized proportional integral observer was designed for dealing with load torque disturbance and time-varying parameter uncertainties for the finite control set predictive current control of induction motor systems. Due to the universal approximation feature, neural networks (NNs) and fuzzy logic systems (FLSs) are also widely used to deal with the unknown disturbances. To approximate the unknown disturbances, a NNs learning algorithm was employed to design the adaptive robust controller for an industrial linear motor stage to achieve good tracking performance and excellent disturbance rejection ability [18]. FLSs were utilized to approximate the unknown disturbances of a bionic mechanical leg for the adaptive fuzzy robust controller design [28] in the previous work of the authors in this paper. In addition, robust control has also been a choice to some researchers to attenuate disturbances in physical systems, such as active disturbance rejection control [31; 32], sliding-mode control (SMC)-based methods [33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44], etc. Among them, the nonsingular fast terminal sliding-mode control (NFTSMC) [30; 38; 39; 40; 41; 42; 43; 44], as a new typical robust controller, has been widely used in controlling uncertain systems because of its attractive properties such as fast dynamic response, robustness against uncertainties, chattering phenomenon elimination, finite time convergence, and its simple design procedure [30; 38]. Motivated by the above observations, to provide a high-performance motion controller with capabilities of unmeasured system states self-estimating, dynamic uncertainties and unknown disturbances rejection for a deep-sea electric oil-filled joint actuator, a novel extended-state observer-based prescribed performance non-singular fast-terminal sliding-mode control (PP-NFTSMC-ESO) strategy was proposed in this paper. The ESO was utilized to estimate the unmeasured system states and the lumped uncertainties to make the precise model-based compensation, while the residual parts including estimation errors and stable tracking errors were discharged by the simple robust term and stable feedback term, respectively. In order to improve the transient and steady-state position responses, an error constraint transformation was developed to guarantee the prescribed time-varying performance.