## 3 Nonlinear Identification Method Based on Broad Learning System and Particle Swarm Optimization
### Excitation Function Design and Sampling  
Dynamic parameter identification relies on sufficient excitation of the system's state space through input trajectories. In vibration-suppression-oriented parameter identification, the excitation trajectory must cover the full motion range of the joints while also activating subtle friction disturbances and flexible coupling modes, enabling the model to comprehensively identify unstructured sources that induce vibration. Insufficient excitation leads to incomplete coverage, resulting in degradation of the identification matrix and making it difficult to model errors, which may cause improper compensation in the control response [20].  
Figure 4: Overall system control block diagram.  
To meet the requirements of identifiability, excitation smoothness, and periodic closure, a fifth-order Fourier excitation trajectory with zero boundary conditions is designed. Its basic form is given in the following equation:  
\[q_{d,i}(t)=\sum_{k=1}^{5}\left(\frac{a_{i,k}}{\omega_{k}}sin(\omega_{k}t)-\frac {b_{i,k}}{\omega_{k}}cos(\omega_{k}t)\right) \tag{45}\]  
Here, \(q_{d,i}(t)\) represents the excitation displacement trajectory of the iii-th joint; \(\omega_{k}\) denotes the base frequency of the \(k\)-th sine/cosine component; and \(a_{i,k}\) and \(b_{i,k}\) are the corresponding Fourier coefficients for the joint. All joint trajectories constructed using the zero-boundary Fourier series satisfy the constraint conditions defined in the following equation, which ensures smooth excitation and prevents non-physical vibration caused by discontinuities.  
\[q_{d,i}(0)=q_{d,i}(T)=0, \tag{46}\]  
\[\dot{q}_{d,i}(0)=\dot{q}_{d,i}(T)=0, \tag{47}\]  
\[\ddot{q}_{d,i}(0)=\ddot{q}_{d,i}(T)=0. \tag{48}\]  
To ensure that the sampled data can effectively excite the full state of the system, the excitation trajectory is sampled at 500 Hz, satisfying the Nyquist criterion. The trajectory period is set to 10 s to cover the complete cycles of all five Fourier orders, preventing truncation effects. The trajectory constraints ensure that both joint angles and velocities are zero at the start and end points, guaranteeing continuity and differentiability at the boundaries and avoiding false vibration responses caused by abrupt changes.  
Figure 5 shows the zero-boundary fifth-order Fourier excitation trajectories for the robotic manipulator, representing the prescribed input signals at the motor-side. Here, J\({}_{1}\) denotes the motor-side joint input angle for joint \(i\), used to avoid ambiguity with other rotational variables. These trajectories are specifically designed as ideal reference signals to excite the robotic system sufficiently for dynamic parameter identification. It should be noted that these input trajectories do not depict the actual joint output responses, which would reflect additional vibrations due to joint flexibility and other nonlinear effects.  
Figure 5: Zero-boundary Fourier excitation input trajectories of joint angles J\({}_{j}\) (motor-side), i = 1, 2, \(\ldots\), 6.  
Table 1 lists the excitation trajectory parameters, which include the fifth-order Fourier coefficients used for all six joints. These coefficients correspond to the amplitude scaling factors for the sine and cosine components, defining the relative contribution of each frequency order within the excitation signal. They determine the spectral structure and time-domain energy distribution of the trajectory. These coefficients are directly used to generate the drive signals, which are applied in both simulation and physical experimental systems to collect modeling sample data, excite system dynamic responses, and ultimately support the modeling of unstructured disturbances and the identification of vibration characteristics.