## 4 Simulations and Experimental Test
### Performance against Disturbances and Parameters Perturbation  
In this part, the proposed control system is assessed regarding robustness and capabilities. On the one hand, the controller is examined for its ability to reject low-frequency disturbances. On the other hand, the robustness against model parameter variations is also tested. The simulations against the external disturbances are divided into two aspects: the position control and the attitude control. In the simulations of the position control, the process is designed as follows. At 0 s, the UAV takes off and hovers at a given point. At 5 s and 10 s, the position step signal with the amplitude of 1m is introduced in the Y and X direction. Then, a low-frequency sinusoidal disturbance is applied to the three channels of the position subsystem at 8-10 s, 12-14 s, and 16-18 s, respectively. The magnitude of the low-frequency sinusoidal disturbance is 5 N, and can be given as  
\[d_{x}(t)= \begin{cases}5 sin(\pi(t-16)),&16\leq t\leq 18\\ 0,&t<16,t>18\end{cases}  \tag{63}\]  
\[d_{y}(t)= \begin{cases}5 sin(\pi(t-12)),&12\leq t\leq 14\\ 0,&t<12,t>14\end{cases}  \tag{64}\]  
\[d_{z}(t)= \begin{cases}5 sin(\pi(t-8)),&8\leq t\leq 10\\ 0,&t<8,t>10\end{cases}  \tag{65}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Magnitude** \\ \hline \(m\) & mass & 5.6 kg \\ \(r_{1x}\) & Right rotor distance along \(x\) axis & 0.22 m \\ \(r_{1y}\) & Right rotor distance along \(y\) axis & 0.2635 m \\ \(r_{1z}\) & Right rotor distance along \(z\) axis & 0 m \\ \(r_{3x}\) & rear rotor distance along \(x\) axis & \(-\)0.42 m \\ \(r_{3y}\) & rear rotor distance along \(y\) axis & 0 m \\ \(r_{3z}\) & rear rotor distance along \(z\) axis & 0 m \\ \(l_{xx}\) & Moment of inertia around \(x\) axis & 0.3556 kg \(\cdot\) m\({}^{2}\) \\ \(l_{yy}\) & Moment of inertia around \(y\) axis & 0.3553 kg \(\cdot\) m\({}^{2}\) \\ \(l_{zz}\) & Moment of inertia around \(z\) axis & 0.6084 kg \(\cdot\) m\({}^{2}\) \\ \(k_{f}\) & pull coefficient of the rotor & \(4.531\times 10^{-5}\) \\ \(k_{d}\) & Moment coefficient of the rotor & \(9.409\times 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters. Figure 4 shows the tracking curves of the position control subsystem. Moreover, the disturbances are scaled to one-tenth to be shown in the figure, so that the influence of the external disturbance can be demonstrated clearly. After the step signal is applied to the UAV in the \(X\) and \(Y\) directions, the position controller can make the errors of two channels converge to zero in \(\sim\)3 s. When the sinusoidal disturbances are introduced into the three directions, the position errors begin to increase. It is noteworthy that the curve of position \(X\) almost coincides with \(Y\), and the position error of the two curves caused by the disturbances is less than 0.1 m. However, the curve of \(Z\) almost has no change. The simulation results are consistent with the proposed control strategy. The altitude is controlled by the thrust directly. Meanwhile, the position control in \(X\)-axis and \(Y\)-axis are achieved by the adjustment of the attitudes. As for the simulations of the attitude tracking and external disturbances suppression of the attitude controller, we make the UAV hover at a set point and set a sinusoidal reference signal in the three directions at 0-4 s, 6-10 s, and 12-16 s, respectively. Besides, a sinusoidal disturbance torque with amplitude of 3 N\(\cdot\)m is added to the three directions at 8-10 s, 12-14 s, and 16-18 s, respectively. The sinusoidal disturbance torque is given as  
\[d_{\phi}(t)= \begin{cases}3 sin(\pi(t-8)),&8\leq t\leq 10\\ 0,&t<8,t>10\end{cases}  \tag{66}\]  
\[d_{\theta}(t)= \begin{cases}3 sin(\pi(t-12)),&12\leq t\leq 14\\ 0,&t<12,t>14\end{cases}  \tag{67}\]  
\[d_{\psi}(t)= \begin{cases}3 sin(\pi(t-16)),&16\leq t\leq 18\\ 0,&t<16,t>18\end{cases}  \tag{68}\]  
Figure 5 shows the tracking curves of the attitude control subsystem, and the disturbances are also scaled to one-tenth shown in the figure.