## 3 Nonlinear Identification Method Based on Broad Learning System and Particle Swarm Optimization
### BLS Network Optimized by PSO  
In the dynamic parameter identification of robotic manipulators under vibration suppression, the identification process is often affected by friction components and low-frequency structural vibrations. These factors may lead to output drift or distorted responses as excitation conditions change, particularly in the presence of nonlinear disturbances. To address this issue, PSO is introduced to optimize the structural parameters of the BLS network, significantly enhancing its local fitting capability for nonlinear disturbances and improving the noise tolerance of the overall regression structure. PSO is a swarm intelligence evolutionary algorithm proposed by [PERSON] and [PERSON] in 1995. It is inspired by the cooperative foraging behavior of bird flocks in multimodal environments. The algorithm dynamically adjusts the search path of each particle in a high-dimensional solution space through a dual-memory mechanism: individual best experience and global best performance. Compared with conventional optimization methods, PSO exhibits superior convergence stability and global search capability in high-dimensional nonlinear systems. It is especially effective in frequency response modeling and controller parameter tuning, where it offers strong overshoot suppression and response speed control [21]. The performance of the BLS model is influenced by multiple factors, including the number of feature mappings, the type of activation functions, and the regularization coefficient. Traditional heuristic methods struggle to find an optimal configuration in such  
Figure 3: The simplified structure of BLS for nonlinear system control. a multidimensional parameter space, limiting the applicability of BLS in complex nonlinear disturbance scenarios. The introduction of PSO transforms the network training process into a parameter optimization problem: each particle represents a specific configuration of the BLS network and is evaluated through a fitness function (e.g., torque prediction error). The parameter update is guided by feedback from both individual and global optima. This approach not only improves the BLS network's ability to locally fit nonlinear friction and flexible coupling disturbances but also enhances the robustness and generalization ability of the overall regression model. Assuming a swarm of \(N\) particles, the position update of each particle is determined by tracking its personal best and the global best positions, as defined in Equations (31) and (32):  
\[v_{i}^{k+1}=wv_{i}^{k}+c_{1}r_{1}\Big{(}p_{i}-x_{i}^{k}\Big{)}+c_{2}r_{2}\Big{(} g-x_{i}^{k}\Big{)} \tag{31}\]  
\[x_{i}^{k+1}=x_{i}^{k}+v_{i}^{k+1} \tag{32}\]  
To further enhance the search effectiveness of the particle swarm and mitigate premature convergence, a constriction factor model \(\chi\) is introduced. This model controls the dynamic convergence rate of the step size and improves particle stability in complex parameter spaces. Its specific formulation is given in Equation (33):  
\[\chi=\frac{2}{\left|2-\varphi-\sqrt{\varphi^{2}-4\varphi}\right|} \tag{33}\]  
The final expression is given in Equation (34):  
\[v_{i}^{k+1}=\chi\Big{[}wv_{i}^{k}+c_{1}r_{1}\Big{(}p_{i}-x_{i}^{k}\Big{)}+c_{ 2}r_{2}\Big{(}g-x_{i}^{k}\Big{)}\Big{]} \tag{34}\]  
Here, \(\varphi\) represents the sum of \(c_{1}\) and \(c_{2}\); \(x_{i}\) is the current position vector of the \(i\)-th particle; \(v_{i}\) is the velocity vector; \(p_{i}\) denotes the personal best position of the particle; \(g\) is the global best position; and \(w\) is the inertia weight. To improve the boundary control capability during the particle swarm optimization process, the particles are initialized as shown in Equation (35):  
\[x_{i}(0)\sim\mathcal{U}(x_{\text{min}},x_{\text{max}}),v_{i}(0)\sim\mathcal{U} (v_{\text{min}},v_{\text{max}}) \tag{35}\]  
The initial positions and velocities of the particles are uniformly sampled within the upper and lower bounds to ensure global coverage of the search space and to prevent divergence due to boundary violations.