## 3 Nonlinear Identification Method Based on Broad Learning System and Particle Swarm Optimization
### Hybrid Optimization-Based Parameter Identification Framework  
In dynamic parameter identification, traditional minimal parameter set regression models can provide a clear structural mapping from a physical modeling perspective. However, they often fail to ensure accuracy in output prediction and consistency in control response when faced with unstructured disturbances in real-world systems. Therefore, achieving high-precision torque estimation and effective subsequent vibration suppression control depends critically on the efficient integration of structural and nonstructural modeling approaches. In the standard linear regression model based on the minimal parameter set, all dynamic characteristics of the system can be explained by structural terms. However, in practical systems, robotic manipulators are affected by unstructured disturbances that lack a unified modeling form. To enhance the model's fitting capability and improve the accuracy of feedforward control compensation, nonstructural disturbance terms are introduced into the original structural model. The overall model can then be expressed as:  
\[\tau(t)=Y(x_{t})P+\tau_{f}(t)+e(t) \tag{40}\]  
By combining the nonlinear modeling capability of the BLS with the global optimization ability of PSO, a unified framework is established for jointly modeling both the structural dynamic components and unstructured disturbances in robotic manipulators. In this mechanism, the minimal parameter set model serves as the foundation for capturing the primary structural dynamics, while the BLS is used to model residual terms arising from factors such as friction. PSO is employed to optimize the architecture and hyperparameters of the BLS network. Together, this forms a structure-driven hybrid identification framework that supports high-frequency vibration compensation and high-precision feedforward torque prediction. The structural model terms are estimated using the least squares method:  
\[\hat{\tau}_{\text{struct}}\left(t\right)=Y(x_{t})P \tag{41}\]  
The residual is defined as:  
\[r(t)=\tau(t)-\left(\hat{\tau}_{\text{struct}}\left(t\right)+K(\theta-q)+B(\hat {\theta}-\dot{q})\right) \tag{42}\]  
This definition characterizes the residual as the main source of unstructured disturbances. To prevent it from propagating through the control system and exciting structural vibrations, it must be accurately modeled and suppressed. A disturbance modeling function is constructed using the BLS:  
\[\hat{r}_{t}=\mathcal{F}_{\text{BLS}}(x_{t};W,\beta) \tag{43}\]  
The final system identification model can be expressed as:  
\[\hat{r}(t)=\hat{r}_{\text{struct}}(t)+\hat{r}_{t}=Y(x_{t})P+\mathcal{F}_{\text{ BLS}}(x_{t}) \tag{44}\]  
By integrating structural modeling with a disturbance compensation mechanism, the proposed model not only enhances the resolution of vibration source modeling but also significantly improves system prediction performance, reduces the amplitude of compensation excitation, and exhibits strong real-time update capability, making it suitable for feedforward control scenarios. Figure 4 illustrates the control framework for compensating unstructured joint torque in robotic arms using a BLS-based friction modeling approach enhanced by PSO optimization. The system receives joint position, velocity, and acceleration as inputs, which are fed into both the nominal dynamic model and the friction modeling network. The dynamic model estimates the structured torque under ideal rigid-body assumptions, while the BLS friction network captures nonlinear residual components such as joint friction. These two outputs are combined to form the total estimated torque, which is then compared against the actual measured torque to calculate an error signal. This error serves as the fitness input for the PSO optimizer, which continuously adjusts the structure and parameters of the BLS network. The optimized results are passed through a weight updating module and fed back into the BLS network, enabling real-time adaptation.