## 4 Simulation
### Analysis of Simulation Results
#### 4.2.3 Analysis of Curves under External Disturbance \(w_{3}\)  
Figures 8 and 9 show the variation curves of the displacement error and velocity error of the flexible manipulator when it is subjected to external interference \(w_{3}\) during operation. As shown in Figure 8, with the combination of three simple harmonic motions, the superposition and canceling effects of displacement fluctuations are further enhanced. Sometimes the amplitude is very large, and sometimes it is at the equilibrium point; the maximum frequency is 500 Hz, and the amplitude of the wave is 0.4 mm. The three control methods also suppress the fluctuation frequency well. In terms of final convergence, the RBF neural network backstepping control method finally converges at the equilibrium position where the displacement error is 0.1 mm. The RBF neural network boundary control method and RBF neural network fuzzy backstepping control method have a smaller error at the final convergence point, but RBF neural network fuzzy backstepping control method has a better inhibition effect and shorter convergence time. In the velocity error curve shown in Figure 9, the three simple harmonics are superimposed on each other, and both the fluctuation frequency and the velocity error of the unsuppressed curve are large. Not only the fluctuation frequency but also the velocity error fluctuation frequency is small. The fluctuation frequency is 300 Hz, and the maximum velocity error fluctuation frequency is 2.5 Hz. The maximum velocity error is 1.5 mm/s. The RBF neural network boundary control method and the inversion control method cannot suppress vibration well. The RBF neural network backstepping method has a large convergence equilibrium position error, and the RBF neural network boundary control method has a high vibration frequency at the beginning of operation. According to the image, the RBF neural network fuzzy backstepping control method has the best effect on vibration frequency and velocity error suppression. Table 7 provides a detailed analysis of the data.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Control Method**} & \multicolumn{2}{c}{**Displacement Error**} & \multicolumn{2}{c}{**Velocity Error**} \\ \cline{2-5}  & **Steady Displacement Error (mm)** & **Stable Frequency (Hz)** & **Steady Velocity Error (mm/s)** & **Stable Frequency (Hz)** \\ \hline RBF boundary control & 0.04536 & Approximately equal to 0 & 0.0867 & 1 \\ RBF backstepping control & 0.03059 & Approximately equal to 0 & 0.0721 & 0.714 \\ RBF fuzzy backstepping control & 0.00533 & 0 & 0.0133 & 0.333 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of the data when the manipulator is under the \(w_{2}\) interference.  
Figure 8: Displacement error of the manipulator under the \(w_{3}\) interference.  
Based on the analysis of the above simulation results, the RBF neural network fuzzy backstepping control method has better vibration suppression effect than the RBF neural network boundary control method and the RBF neural network backstepping control method, regardless of whether there is simple harmonic motion or multiple harmonic motion phase superposition. In most cases, the vibration frequency and vibration error can be quickly suppressed, and the final convergence equilibrium position error is small.