# Trajectory Tracking with Adaptive Robust Control for Quadrotor
## 4 Adaptive Controller and Robust Mechanisms Design
### Trajectory Following
This is verified by the analysis results obtained with \(\mathcal{L}_{2}[u_{\Theta}]\) in Table 1.  
On the other hand, for pitch angle, the PD adaptive controller, which applied a bigger control response to achieve the control objective, is the adjustment mechanism based on the MIT rule. The smaller effort to achieve the control objective is MIT's adjustment mechanism with high order sliding mode. It is verified by the analysis results obtained with \(\mathcal{L}_{2}[u_{\Theta}]\) in Figure 3:0  
Thus, the adjustment mechanism based on the MIT rule of the control effort \(\mathcal{L}_{2}[u_{\Theta}]\) to achieve the control objective is bigger than the MIT-SM, MIT-Twisting, and MIT-HOSM. On the other hand, the adjustment mechanism based on the MIT-HOSM presented a smaller effort to achieve the desired roll angle than the based on MIT, MIT-SM, and MIT-Twisting. The results obtained are present in Table 2, and in Figure 9 the response of the roll angle is presented with the PD adaptive controller with the different adjustment mechanism proposed in this work. Analyzing the yaw angle with (30) and (31), the PD adaptive controller which presented a smaller error is the adjustment mechanism based on the MIT with sliding modes (MIT-SM) than the MIT, MIT-Twisting, and MIT-HOSM; see Figure 3:1 Furthermore, the adaptive control based on the adjustment mechanism for the yaw angle with a bigger error is the MIT with Twisting (MIT-Twisting). On the other hand, the adaptive controller's adjustment mechanism, which presented a greater control effort to achieve the control objective, is based on the MIT with high order sliding mode (MIT-HOSM). The MIT rule showed a smaller control effort than the MIT-SM, MIT-Twisting, and MIT-HOSM. In Figure 10 the response of the yaw angle is presented with the PD adaptive controller with the different adjustment mechanism proposed in this work. \begin{table}
\begin{tabular}{c c c} \hline
**Controller** & \multicolumn{2}{c}{**Pitch [deg]**} \\ \hline  & \(\mathcal{L}_{2}[e_{\phi}]\) & \(\mathcal{L}_{2}[u_{\theta}]\) \\ \hline MIT & 0.5452 & 0.1565 \\ \hline MIT-SM & 0.5333 & 0.1528 \\ \hline MIT-Twisting & 0.0051 & 0.0170 \\ \hline MIT-HOSM & 0.0026 & 0.0097 \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical results with \(\mathcal{L}_{2}\) of the adaptive robust controllers in pitch angle. \begin{table}
\begin{tabular}{c c c} \hline
**Controller** & \multicolumn{2}{c}{**Roll [deg]**} \\ \hline  & \(\mathcal{L}_{2}[e_{\phi}]\) & \(\mathcal{L}_{2}[u_{\phi}]\) \\ \hline MIT & 0.5191 & 1.15806 \\ \hline MIT-SM & 0.3571 & 1.12103 \\ \hline MIT-Twisting & 0.0029 & 0.0094 \\ \hline MIT-HOSM & 0.0011 & 0.0033 \\ \hline \end{tabular}
\end{table} 
Table 2: Numerical results with \(\mathcal{L}_{2}\) of the adaptive robust controllers in roll angle. Figure 9: Roll angle responses. Figure 10: Yaw angle responses. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Controller** & \multicolumn{2}{c}{**Yaw [deg]**} \\ \hline  & \(\mathcal{L}_{2}[e_{\psi_{z}}]\) & \(\mathcal{L}_{2}[u_{\psi}]\) \\ \hline MIT & 0.0249 & 0.0135 \\ \hline MIT-SM & 0.0134 & 0.0292 \\ \hline MIT-Twisting & 2.0546 & 4.8103 \\ \hline MIT-HOSM & 1.4752 & 6.6313 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical results with \(\mathcal{L}_{2}\) of the adaptive robust controllers in yaw angle. In Figure 11, the control response of the PD adaptive controller is presented with the adjustment mechanism based on the MIT rule, and Figure 12 shows the minimization of the cost function with the adaptive control law and the adjustment mechanism based on the MIT rule. Figures 13 and 14 present the convergence of the adaptive gains with the adjustment mechanism based on the MIT rule. Figure 11: Controller response with MIT rule. Figure 12: Minimization of \(\int_{\theta,\psi,\phi}\) with MIT rule. Figure 13: Convergence \(\hat{k}_{p}\) response with MIT rule. Figure 15 shows the control response of the adaptive controller based on the MIT rule with sliding mode (MIT-SM).