## 4 Simulation
### Analysis of Simulation Results  
#### 4.2.1 Analysis of Curves under External Disturbance \(w_{1}\)  
Figures 4 and 5 show the displacement error and velocity error curves of each system when the manipulator is subjected to external interference \(w_{1}\), in which the red curve is the error curve when the vibration suppression of the system is not applied, the black curve is the error curve using the RBF neural network boundary control method, and the blue curve is the error curve using the RBF neural network inversion control method. The green curve is the error curve of the RBF neural network fuzzy backstepping control method. As can be seen from the displacement error of the manipulator in Figure 4, there is an unusual displacement error of the manipulator at the start of the motor, which is mainly due to the sudden change of force and the change of motion state at the start of the motor. Then it fluctuates within a smaller range. Under the interference of simple harmonic vibration,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Meaning** & **Unit and Size** \\ \hline \(EI_{k}\) & Bottom stiffness of the flexible manipulator & 1.61 N-m \\ \(I_{k}\) & Moment of inertia at bottom of the flexible manipulator & 0.0021 kg\(\cdot\)m \\ \(EI_{m}\) & Stiffness coefficient of the motor & 70 N-m \\ \(EI_{g}\) & Stiffness coefficient at the top of the flexible manipulator & 0.00767 N-m \\ \(I_{L}\) & Moment of inertia at the top of the flexible manipulator & 0.0059 kg\(\cdot\)m \\ \(R_{m}\) & Damping coefficient of the motor & 2.6 N-s/m \\ \hline \hline \end{tabular}
\end{table}
Table 4: Nonlinear state space equation parameters of the flexible manipulator.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Meaning** & **Unit and Size** \\ \hline \(m_{m}\) & Mass of the motor shaft & 1.5 kg \\ \(L_{m}\) & Length of the motor shaft & 10 mm \\ \(I_{m}\) & Moment of inertia of the motor & 0.4 kg\(\cdot\)m \\ \(m_{L}\) & Mass of the flexible manipulator & 500 g \\ \(L_{L}\) & Moment of inertia of the flexible & 0.4 kg\(\cdot\)m \\  & manipulator & 9.8 m/s\({}^{2}\) \\ \(g\) & Acceleration of gravity & 9.8 m/s\({}^{2}\) \\ \(EI\) & Stiffness coefficient & 0.8 N\(\cdot\)m \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters of the linear state space equations for the flexible manipulator.  
the error fluctuation is not obvious. However, the small fluctuation frequency in the local range is very high. It is about 1000 Hz. Long-term and high-frequency vibration affects the working accuracy and service life of the manipulator, so vibration suppression is needed. Under the vibration suppression of the RBF neural network boundary control method and the RBF neural network backstepping control method, the vibration frequency of the flexible arm can be well suppressed, and the vibration displacement error is suppressed to about 0.05 mm. The RBF neural network fuzzy backstepping control method not only restrains the vibration frequency well, but also restrains the displacement error of the manipulator, so that the final convergence result is close to 0. The velocity error curve is similar to the displacement error curve when vibration is not suppressed. After the three control methods start vibration suppression, there is a certain fluctuation frequency in the initial operation, which is the largest for the RBF neural network backstepping control method, the second-largest for the RBF neural network boundary control method, and the least for the RBF neural network fuzzy backstepping control method. According to the final convergence results, RBF neural network fuzzy inversion control method is still the best, and the velocity error and vibration frequency error are almost eliminated. Table 5 provides a detailed analysis of the data.  
Figure 4: Displacement error of manipulator under the \(w_{1}\) interference.