## 4 Simulation
### Analysis of Simulation Results
#### 4.2.2 Analysis of Curves under the External Disturbance \(w_{2}\)  
Figures 6 and 7 show the variation curves of the displacement error and velocity error of the flexible manipulator when it is subjected to external interference \(w_{1}\) during operation. As shown in Figure 6, in the displacement error curve, due to the superposition of two simple harmonic motions, the overall displacement error fluctuates more, and so does the velocity error. However, in the wave frequency, due to the combination of two simple harmonic motions, when the two simple harmonic motions have the same direction, is a superposition. When the two harmonic motion directions are opposite, the frequency and amplitude cancel each other. In this simulation, the two sets of simple harmonic motion cancel each other, making the maximum frequency about 400 Hz. However, in terms of fluctuation error, if the instantaneous error at start-up is not considered, the maximum fluctuation error reaches 0.1448 mm, so vibration suppression is required. Both the RBF neural network boundary control method and the RBF neural network backstepping control method can restrain the vibration frequency well and limit the displacement error to less than 0.1 mm. The RBF neural network fuzzy inversion control method can eliminate the vibration frequency and displacement error well under external interference \(w_{2}\), and almost eliminate the vibration error and displacement error. Although the velocity error is not large in the velocity error curve in Figure 7, the fluctuation frequency is 500 Hz. Although the RBF neural network boundary control method and RBF neural network backstepping control method have a certain fluctuation frequency at the initial stage of operation, they both suppress the fluctuation frequency well in the end, and the final convergence equilibrium position is \(-0.1\) mm/s. The RBF neural network fuzzy backstepping control method also  
Figure 5: Velocity error of manipulator under the \(w_{1}\) interference.  
\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 & 0.0578 & Approximately equal to 0 & 0.0162 & Approximately equal to 0 \\ RBF backstepping control & 0.0574 & Approximately equal to 0 & 0.02 & Approximately equal to 0 \\ RBF fuzzy backstepping control & 0.0071 & 0 & 0.0049 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of the data when the manipulator is under the \(w_{1}\) interference.  
has certain fluctuation frequency and velocity errors when the system starts running, but in the final phase of operation, the fluctuation frequency and velocity error balance point is better than for the first two control methods, and the speed error is better suppressed. Table 6 provides a detailed analysis of the data.  
Figure 6: Displacement error of the manipulator under the \(w_{2}\) interference.  
Figure 7: Velocity error of the manipulator under the \(w_{2}\) interference.