## 1 Introduction
[PERSON] et al. proposed a friction compensation approach by embedding experimentally validated LuGre friction models into a state feedback controller for motor-mechanical systems. [PERSON] et al. further extended this approach by incorporating nonlinear least squares into a sliding-mode control structure to enable time-varying friction identification, thereby achieving high-precision trajectory tracking for multi-DOF robotic [6]. In robotic manipulator systems characterized by flexible joints, harmonic drives, and strong structural coupling, the system response contains a substantial amount of nonlinear and dynamic disturbance components. Traditional structural models struggle to capture the complex frictional effects and load coupling characteristics inherent in such systems. To address these challenges, increasing attention has been directed toward unstructured modeling approaches. [PERSON] and [PERSON] employed a BackproPagation (BP) neural network architecture to learn the mapping between inputs and outputs autonomously, treating network weights as parameters to accomplish inertial parameter identification tasks [7]. [PERSON] et al. compared the performance of Auto-Regressive with eXogenous (ARX) and Nonlinear Auto-Regressive with eXogenous (NARX) models in continuum robotic systems, and they found that the NARX model provided a more accurate description of nonlinear dynamic characteristics [8]. [PERSON] et al. utilized a BackPropagation Multilayer Perceptron (BPMLP) to perform nonlinear dynamic modeling of a flexible beam structure with fixed-free boundary conditions. In comparison with the traditional RLS method, the neural network exhibited superior accuracy in both time and frequency domains. The effectiveness of the model was validated through correlation tests and prediction error analysis, offering a solid foundation for the subsequent design of vibration suppression controllers [9]. However, these methods often suffer from high training complexity, difficulty in online parameter updating, and limited generalization capability. To establish a nonlinear dynamic model under vibration control, recent research has begun to incorporate intelligent optimization algorithms to assist in parameter identification. [PERSON] et al. proposed a method combining Haar wavelet transformation with PSO, which enhanced nonlinear modeling capability through signal compression and global optimization. This approach successfully controlled the identification error of a multi-DOF robotic manipulator within 2% [10]. [PERSON] et al. applied the identified parameters in a Nonlinear Model Predictive Control (NMPC) framework, constructing feedforward compensation for friction, thereby improving trajectory tracking performance in six-axis robotic systems [11]. The BLS, first introduced by [PERSON] et al. in 2017 [12], constructs a width-expandable network using a combination of feature mapping and enhancement nodes. By employing pseudoinverse or regularized least squares to solve output weights, BLS avoids the iterative gradient-based training process typically required in conventional neural networks [12]. [PERSON] et al. applied BLS to nonlinear system modeling and control by developing a fuzzy incremental BLS architecture, enabling online learning and control of complex systems [13]. [PERSON] et al. embedded BLS into the control of a micromanipulation platform to perform small-sample nonlinear identification, successfully modeling microscale friction disturbances and achieving robust control [14]. Due to its fast training speed and support for incremental parameter updates, BLS is particularly suitable for friction modeling in industrial robots where fast response and dynamic adjustment are essential. In robotic manipulator systems, complex disturbances such as nonlinear friction, flexible joint connections, and inertial mismatches are key contributors to high-frequency micro-vibrations and low-frequency residual oscillations.