## 1 Introduction  
With the widespread application of industrial robots in high-precision machining, aerospace manufacturing, and medical automation, the requirements for control accuracy and dynamic performance of robotic manipulators have become increasingly demanding. However, structural vibration has emerged as a major bottleneck hindering further performance improvement. In practical operations, due to structural flexibility, varying payloads, drive backlash, and nonlinear factors such as friction, robotic manipulators are prone to residual vibration, flexible vibration, and self-excited vibration during high-speed movements and frequent start-stop operations. These vibrations result in decreased positioning accuracy, prolonged execution time, and compromised system stability [1]. Recent studies have focused extensively on identifying the sources of vibration. [PERSON] et al. proposed a torque compensation method based on a tracking filter, demonstrating that constructing a feedforward term by estimating acceleration can effectively suppress the peak response of residual joint vibrations in robotic manipulators [1]. [PERSON] et al. further highlighted that abrupt variations in trajectory jerk (i.e., the derivative of acceleration) are also key contributors to vibration. By optimizing spline trajectory parameters using a Quasi-Genetic Algorithm (QGA), they significantly reduced jerk variance, thereby suppressing vibrations at the source [2]. To meet the dual demands of vibration suppression and high-precision control, [PERSON] et al. introduced a composite control strategy combining Zero-Vibration (ZV) input shaping and a Pole-placement Predictive Function Controller (PPFC). By employing gravity-compensated quasi-linear joint dynamics, the system was decoupled into four parallel first-order subsystems. On a 3-DOF manipulator, this method achieved a 45% reduction in tracking error, a 78% suppression of residual vibration, and a 32% reduction in motor torque fluctuations [3], thereby validating the effectiveness of the feedforward-feedback composite architecture in flexible joint control. [PERSON] et al. proposed a feedforward control method based on Time-Varying Filtered B-splines (TVFBS), which successfully suppressed residual vibrations in a 6-DOF robotic manipulator while maintaining operation speed. Compared to conventional Time-Varying Input Shaping (TVIP) methods, this approach improved trajectory tracking accuracy and established a modeling framework for the frequency response function applicable to multi-joint robotic systems [4]. However, the aforementioned methods typically rely on accurate modeling of the system's frequency response, which limits their applicability when dealing with unstructured disturbances such as frictional hysteresis and environmental perturbations. These approaches are often based on idealized models or are restricted to trajectory-level regulation, lacking precise nonlinear dynamic identification and modeling support for the system itself. Therefore, under complex nonlinear disturbances, constructing a high-precision dynamic model remains a major challenge for implementing active vibration control. Although traditional mechanism-based modeling methods can analytically estimate inertial parameters, they fall short in accurately characterizing unstructured disturbances such as harmonic drive flexibility and time-varying friction hysteresis. As a result, the deviation between the model-predicted torque and the measured torque often exceeds 18.5% [5]. This underscores the necessity for a novel identification method that operates without the assumption of frictionless modeling--one that can autonomously learn the dynamic behavior of friction and support online parameter optimization. Conventional identification methods are primarily based on Newton-Euler or Lagrangian dynamics and are widely used in rigid manipulator systems. In these frameworks, inertial and center-of-mass parameters are typically estimated using linear Least Squares (LS) or Recursive Least Squares (RLS) algorithms.