## 6 Simulation Results
### Conducted Simulation Results of the Unknown System with Disturbance Observer  
Next, a simulation was conducted to assess the robustness with respect to the load perturbation for the case that the load mass of the quadrotor increases a 200%. Additionally, a wind gust of \(1.5\sin(t)\) m/ sec\({}^{2}\) was disturbed to the x-axis. A simulation of the unknown system with the controller equipped with the disturbance observers of (90) (proposed), (91) (Levant [29,30]), and (92) (Zong [27,28]), was conducted to evaluate the estimation performance (Table 5). In Figure 5a,b, the simulation results of the 3-D positioning and z-axis positioning error are shown, respectively. The consequences of these figures imply that the positioning performance of the control system equipped with the proposed observer is superior to those of the other control system equipped with other observers. Figure 4: Conducted simulation results of the quadrotor system under nominal mass: (**a**) 3–D positioning result; (**b**) positioning error of x–axis; (**c**) positioning error of y–axis; (**d**) positioning error of z–axis; (**e**) angular positioning error of \(\psi\)–axis; (**f**) control input of the altitude controller; and (**g**) control input of heading controller. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Observer** & \multicolumn{2}{c}{**Value**} \\ \hline Altitude axis & \(\Lambda_{2}=8\), \(\eta_{\rho 21}=0.2\), \(\eta_{\rho 2}=0.2\), \(c_{\rho 21}=2.5\), \(c_{\rho 22}=2.5\), \(x_{z}=1.1\) \\ Longitudinal axis & \(\Lambda_{x}=10\), \(\eta_{\rho x1}=0.25\), \(\eta_{\rho 22}=0.2\), \(c_{\rho x1}=2.5\), \(c_{\rho x2}=2.5\), \(\kappa_{x}=1.1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter values of the disturbance observer. Figure 5: Cont. The estimated results for the observer states and disturbances of the proposed observer shown in Figure 5c,d, respectively, indicate that the proposed observer outperforms the Levant and Zong observers. This can be confirmed from the root mean square (RMS) values of the positioning error and estimation errors for the observer state and uncertainty shown in Table 6, where the RMS value of the estimation error for the proposed system decreases to a maximum 5% that for the Levant's system. As the second disturbance case, when the quadrotor mass including the load mass increases to a 300%, a simulation under the same other conditions was conducted to evaluate the estimation performance for the three disturbance observers of (87) (proposed), (88) (Levant [29; 30]), and (89) (Zong [27; 28]). The 8-shape positioning command trajectory is selected as  
\[\mathbf{P}_{d}(t)= \begin{bmatrix}x_{d}&y_{d}&z_{d}\end{bmatrix} ^{T}= \begin{bmatrix}0&0&-4\big{(}1-e^{-0.3t}\big{)}\end{bmatrix} ^{T}(m),t\leq 4s,\] \[(98)\mathbf{P}_{d}(t)= \begin{bmatrix}x_{d}\\ y_{d}\\ z_{d}\end{bmatrix} = \begin{bmatrix}8-8\cos\left(\frac{2\pi(t-4)}{12}\right)\\ 4\sin\left(\frac{4\pi(t-4)}{12}\right)\\ -4\big{(}1-e^{-0.3t}\big{)}\end{bmatrix} ,t\geq 4s\]  
where other rotation commands are set to be zero. In the first flight step, the quadrotor UAV moved vertically for 4 s for take-off flight. As the second step, an 8-shaped path flight was conducted while continuing to lift. In the second flight step, three flight performances for the lift, sideslip, and forward flights of the quadrotor UAV were continuously evaluated. Figure 5 shows the simulation results for these flights, where Figure 6a presents the 3-D positioning result, and the positioning errors on the x-, y-, z-axes are presented in Figure 6b-d, respectively. These results imply that the positioning performance of the proposed control system equipped with the proposed observer outperformed than those of the traditional controllers with other observers like the previous cases. The RMS values and the mean absolute error (MAE) values of the