## 1 Introduction  
In today's complex work environment, where human-robot interaction has become increasingly frequent, a diverse range of robots has emerged to cater to distinct implementation scenarios. Among these robots, FJRs have garnered significant attention due to their inherent compliance, which enhances the safety of human interactions [1]. Typically, FJRs comprise motors with load links, connected via low-stiffness flexible couplings. This joint elasticity allows FJRs to be modeled as a two-mass system. To effectively control these systems, numerous control methods have been proposed in the academic literature. For instance, [PERSON] et al. [2] combined command filtering with adaptive fuzzy control methods to control FJRs with time-varying full-state constraints. [PERSON] et al. [3] elaborated on how to integrate event-triggered control with adaptive neural networks to design a control strategy, aiming to achieve precise tracking control of FJRs. [PERSON] et al. [4] combined a PID-type control strategy with motor position measurement information to achieve precise control of FJRs. In practical scenarios, it is crucial to confine the tracking error within a predefined range. To this end, PPC has garnered significant attention from researchers [5; 6; 7]. [PERSON], for instance, introduced an adaptive fuzzy control approach for single-link flexible-joint robotic manipulators, leveraging PPC to ensure transient performance guarantees [8]. Furthermore, FJRs can achieve desired impedance against external contact forces and superior position tracking in free motion, thanks to robust prescribed performance position tracking controllers [9]. However, it is worth noting that the aforementioned control methods often rely on backstepping or dynamic surface control techniques, which are renowned for their intricate and repetitive design procedures. Consequently, their application in real high-order FJRs remains limited. In FJRs, the position and velocity of links exhibit slower time-scale variations, whereas the position and velocity of motors undergo faster dynamics. This characteristic allows the overall system dynamic equations to be recast into a standard singularly perturbed form, facilitating the utilization of SPT to simplify controller design. [PERSON] et al. [10], for instance, applied SPT to decompose the higher-order model of a serial elastic actuator into two lower-order models. Subsequently, they developed a second-order sliding mode control law for each lower-order model, ensuring semi-global exponential stability. Another study by [PERSON] et al. [11] introduced a nonlinear hybrid control scheme for a redundant parallel robot system, leveraging SPT and Tikhonov's theorem to achieve both trajectory tracking and vibration suppression. [PERSON] et al. proposed a prescribed performance controller for FJRs with exogenous disturbances, incorporating SPT and a time-varying tangent barrier Lyapunov function to expand the initial error constraint boundary [12]. However, these methods fail to identify and compensate for unknown dynamics in FJRs, resulting in relatively large control errors. To address the challenges posed by unknown nonlinear dynamics in practical systems, NNs have been extensively employed due to their remarkable universal approximation capabilities [13; 14; 15; 16; 17]. In the development of NN-based adaptive control methods, the algorithm for identifying NN weights plays a crucial role, as it profoundly affects both the tracking performance and the robustness of the closed-loop system [18]. NN-based adaptive control methods can be categorized into two main branches: direct adaptive control methods [19; 20; 21] and indirect adaptive control methods [22]. Direct adaptive control primarily relies on gradient-descent-based updating laws for online identification of unknown controller parameters. However, these methods may not yield satisfactory identification results [23].