## 3 Nonlinear Identification Method Based on Broad Learning System and Particle Swarm Optimization  
### Construction of a Nonlinear Friction BLS Network  
In the context of vibration control, unstructured disturbances such as frictional interference and structural coupling residuals are major contributors to system micro-vibrations and end-effector drift. To enhance the system's ability to perceive such dynamic disturbances, a BLS with nonlinear fitting capability is introduced as the core of the vibration modeling module. The BLS learns the mapping between input state vectors and friction torque, capturing friction dynamics directly and enabling feedforward compensation in subsequent vibration controllers. BLS is a shallow neural network and an efficient alternative to deep learning, utilizing a width-expansion strategy. It consists of a feature mapping layer, enhancement layer, and output layer, and replaces traditional deep architectures with lateral expansion. BLS exhibits strong adaptability in handling nonlinear systems. By applying multiple random mapping functions, the input data are projected into a high-dimensional feature space, and the target output is approximated using least squares estimation [12]. Figure 2 illustrates the structure of a BLS network. Figure 2: The original structure of a Broad Learning System (BLS). The feature mapping layer of BLS transforms the input into a high-level feature space to approximate nonlinear functions. To better simulate velocity-dependent friction disturbances observed during robotic manipulator motion, the LuGre friction model is introduced as a physical reference form for unstructured disturbances. Its dynamic expression is given in Equation (20):  
\[\dot{z}=\dot{q}-\frac{\left|\dot{q}\right|}{\sigma_{0}g(\dot{q})}z,\tau_{f}= \sigma_{0}z+\sigma_{1}\dot{z}+\sigma_{2}\dot{q} \tag{20}\]  
Here, z denotes the elastic displacement state of microscopic surface contact, \(\sigma_{0}\) represents the stiffness coefficient of the model, \(\sigma_{1}\) is the damping coefficient, and \(\sigma_{2}\) is the viscous friction coefficient. Term \(g(\dot{q})\) is defined as the velocity-based Stribeck function, used to simulate the nonlinear stick-slip decay behavior. The specific form of \(g(\dot{q})\) is given in Equation (21):  
\[g(\dot{q})=\tau_{c}+(\tau_{s}-\tau_{c})e^{-\left(\frac{\dot{q}}{\tau_{0}} \right)^{2}} \tag{21}\]  
This structure characterizes the dynamic response of the friction torque to changes in joint velocity. Next, input x is constructed, and the joint state variables are defined as shown in Equation (22):  
\[x=\left[q,\dot{q},\ddot{q}\right]\in R^{3} \tag{22}\]  
Here, \(q,\dot{q},\ddot{q}\) represents the joint angle, velocity, and acceleration, which are used as the input to the BLS network to effectively provide information about the friction disturbance variation under the current joint state. Subsequently, the feature mapping layer is constructed, where the input undergoes feature transformation. Each mapping channel applies a random weight and bias followed by an activation function, as expressed in Equation (23):  
\[Z^{(i)}=\phi^{(i)}\Big{(}xW_{z}^{(i)}+b_{z}^{(i)}\Big{)}\in R^{1\times k},i=1, 2,\ldots,m \tag{23}\]  
Here, \(W_{z}^{(i)}\) denotes the weights applied to the input for each mapping node; \(\phi^{(i)}\) is the activation function responsible for introducing nonlinear characteristics; and each \(Z^{(i)}\) provides a set of corresponding high-dimensional features.