# Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone
## 5 Simulation  
In this section, the effectiveness of the method is demonstrated through simulations on a flexible robotic arm. The parameters of the flexible robotic arm were selected as shown in Table 1.  
Selection of Fuzzy Membership Functions:  
\[\mu_{F_{2}^{2}}\Big{(}\xi_{1},\xi_{2},\xi_{3,f}\Big{)}=exp\bigg{[}-\frac{( \xi_{1}-l)^{2}}{2}\bigg{]}\times exp\bigg{[}-\frac{(\xi_{2}-l)^{2}}{2}\bigg{]} \times exp\bigg{[}-\frac{(\xi_{3,f}-l)^{2}}{2}\bigg{]}\] \[\mu_{F_{4}^{1}}\Big{(}\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5,f} \Big{)}=exp\bigg{[}-\frac{(\xi_{1}-l)^{2}}{2}\bigg{]}\times exp\bigg{[}- \frac{(\xi_{2}-l)^{2}}{2}\bigg{]}\times exp\bigg{[}-\frac{(\xi_{3}-l)^{2}}{2} \bigg{]}\] \[\times exp\bigg{[}-\frac{(\xi_{4}-l)^{2}}{2}\bigg{]}\times exp \bigg{[}-\frac{(\xi_{5,f}-l)^{2}}{2}\bigg{]}\] \[\mu_{F_{5}^{1}}\Big{(}\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5,u} \Big{)}=exp\bigg{[}-\frac{(\xi_{1}-l)^{2}}{2}\bigg{]}\times exp\bigg{[}-\frac {(\xi_{2}-l)^{2}}{2}\bigg{]}\times exp\bigg{[}-\frac{(\xi_{3}-l)^{2}}{2}\bigg{]}\] \[\times exp\bigg{[}-\frac{(\xi_{4}-l)^{2}}{2}\bigg{]}\times exp \bigg{[}-\frac{(\xi_{5}-l)^{2}}{2}\bigg{]}\times exp\bigg{[}-\frac{(u_{f}-l)^{ 2}}{2}\bigg{]}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline \(J_{1v},h\) & 50 Kgm\({}^{2}\) \\ \(K_{l}\) & 10 Nm/A \\ \(K_{b}\) & 0.976 Nm/A \\ \(m\) & 0.1 Ke \\ \(g\) & 9.8 N/Kg \\ \(F_{1g}F_{2}\) & 0.01 Nms/rad \\ \(R\) & 2.71 \\ \(\frac{K}{L}\) & 20 \\ \(\frac{L}{N}\) & 100 H \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the flexible robotic arm.  
Figure 8: The trajectories of \(\xi_{4}\) and \(\dot{\xi}_{4}\).  
Figure 6: The trajectories of \(\xi_{2}\) and \(\dot{\xi}_{2}\).  
Figure 11 represents the trajectories of \(e\) with different values of \(c_{i}\), where \(e=y-y_{r}\). The parameter e1 is chosen as follows: \(c_{1}=60\), \(c_{2}=15\), \(c_{3}=60\), \(c_{4}=10\), \(c_{5}=10\). The parameter e2 is chosen as follows: \(c_{1}=5\), \(c_{2}=5\), \(c_{3}=5\), \(c_{4}=5\), \(c_{5}=5\). The parameter e3 is chosen as follows: \(c_{1}=15\), \(c_{2}=15\), \(c_{3}=15\), \(c_{4}=15\), \(c_{5}=15\). The parameter e4 is chosen as follows: \(c_{1}=40\), \(c_{2}=40\), \(c_{3}=40\), \(c_{4}=40\), \(c_{5}=40\). As can be seen from Figure 9, by increasing the values of \(c_{1}\), \(c_{2}\), \(c_{3}\), \(c_{4}\), \(c_{5}\), the trajectory tracking error e decreases accordingly. The experimental results correspond to the conclusion mentioned in Section 4 which states that \(\lim\limits_{t\to\infty}\lvert z_{1}(t)\rvert\leq\sqrt{2D/c}\).  
Figure 12 represents the trajectories of \(v\) with different values of \(c_{i}\), where v1 is parameterized with \(c_{1}=5\), \(c_{2}=5\), \(c_{3}=5\), \(c_{4}=5\), \(c_{5}=5\). v2 is parameterized with \(c_{1}=15\), \(c_{2}=15\), \(c_{3}=15\), \(c_{4}=15\), \(c_{5}=15\). v3 is parameterized with \(c_{1}=40\), \(c_{2}=40\), \(c_{3}=40\), \(c_{4}=40\), \(c_{5}=40\). As \(c_{i}\) increases, better control performance can be achieved.  
The simulation results indicate that the observer designed in this study has a good estimation performance and that the control method employed exhibits strong robustness.  
Figure 10: The trajectories of \(v\).  
Figure 9: The trajectories of \(\xi_{5}\) and \(\xi_{5}\).