## 6 Simulation Results
### Conducted Simulation Results of the Quadrotor USV System under Nominal Mass  
The parameters of the controller and observer were selected appropriately using the trial-and-error method by checking the output performance. The parameter values are listed in Tables 2-4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Quantity** & **Value** \\ \hline \(m_{0}\) & nominal mass of the body & 2 kg \\ \(l\) & length of arm & 0.2 m \\ \(l_{x}\), \(l_{y}\), \(l_{z}\) & moment of inertia at each axis & 1.25, 1.25, 2.2 Ns\({}^{2}\)/rad \\ \(l_{r}\) & moment of inertia at propeller & 1 Ns\({}^{2}\)/rad \\ \(b\), \(d\) & constant & 5 Ns\({}^{2}\), 5 Nms\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter values of the quadrotor UAV.  
The fractional orders were selected as \(\gamma_{1}=0.8\) and \(\gamma_{2}=0.1\). The positioning command were taken as follows:  
\[x_{d}(t)=\left\{\begin{array}{l}2m,0<t\leq 40\\ 5m,40<t\leq 60\\ 1m,60<t\leq 100\end{array}\right.,y_{d}(t)=\left\{\begin{array}{l}1m,0<t\leq 3 0\\ 2m,30<t\leq 80\\ 0.5m,80<t\leq 100\end{array}\right.,z_{d}(t)=\left\{\begin{array}{l}5m,0<t\leq 78 \\ 2m,78<t\leq 100\end{array}\right.,\] \[\psi_{d}(t)=\left\{\begin{array}{l}0 rad,\qquad 0<t\leq 28\\ 0.25 rad,\qquad 28<t\leq 78\\ 0.125 rad,\qquad 78<t\leq 100\end{array}\right.,\phi_{d}(t)=0 rad,\theta_{d}(t)=0 rad.\]  
The other perturbations are as follows:  
\[f_{dx}=(-\Delta m\ddot{x}-d_{x}\dot{x}+1.5\sin t)/m_{0}.f_{dy}=(-\Delta m \ddot{y}-D_{y}\dot{y})/m_{0},f_{dz}=(-\Delta m\ddot{x}-D_{z}\dot{z})/m_{0}, \tau_{d\phi}=\]  
\[-d_{\phi}|\dot{\phi}/J_{x},\tau_{d\theta}=-d_{\theta}|\dot{\theta}/J_{y},\tau_ {d\phi}=-d_{\phi}|\dot{\psi}/J_{z},d_{x}=0.1\,\mathrm{N\ sec/m},d_{y}=0.01\, \mathrm{N\ sec/m},\]  
\(d_{z}=0.1\,\mathrm{N\ sec/m},d_{\phi}=d_{\theta}=d_{\psi}=0.012\,\mathrm{N\ sec/rad}\).  
For the nominal UAV system, Figure 4 shows the simulation results. Figure 4 shows the position tracking result in 3-D space. To express clearly the tracking performance of each control system, the position tracking errors of the three translation axes and the heading angle tracking error are presented in Figure 4b-e, respectively. As shown in Figure 4b,c, the size of the total tracking errors of the LBSC is smaller than that of the BSC system, but the LBSC system has a larger overshoot in the direction change range.  
The positioning performance of the ZBSC system is improved than those of the BSC and LBSC systems. However, the proposed system has better tracking performance than the ZBSC system. The positioning results in the z- and \(\psi\)-axes of the proposed system outperforms that of the other three systems as illustrated in Figure 4h-k. The control inputs generated from the altitude and heading controllers are presented in Figure 4f,g, respectively.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Axis** & **Value** \\ \hline Altitude & \(\Lambda_{1}=7,\ c_{11}=2,\ c_{12}=2,\ r_{11}=20,\ r_{12}=20,\ \kappa_{1}=1.1\) \\ Longitudinal and Lateral & \(\Lambda_{3}=50,\ c_{31}=2,\ c_{32}=2,\ r_{31}=20,\ r_{32}=20,\ \kappa_{3}=1.1\) \\ \(\Lambda_{5}=50,\ c_{51}=2,\ c_{52}=2,\ r_{51}=10,\ r_{52}=10,\ \kappa_{5}=1.1\) \\ Heading & \(\Lambda_{7}=20,\ c_{71}=2,\ c_{72}=2,\ r_{71}=20,\ r_{72}=20,\ \kappa_{6}=1.1\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameter values of the compensating signal.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Axis** & **Value** \\ \hline Altitude & \(\Lambda_{1}=7,\ c_{11}=2,\ c_{12}=2,\ r_{11}=20,\ r_{12}=20,\ \kappa_{1}=1.1\) \\ Longitudinal and Lateral & \(\Lambda_{3}=50,\ c_{31}=2,\ c_{32}=2,\ r_{31}=20,\ r_{32}=20,\ \kappa_{3}=1.1\) \\ \(\Lambda_{5}=50,\ c_{51}=2,\ c_{52}=2,\ r_{51}=10,\ r_{52}=10,\ \kappa_{5}=1.1\) \\ Heading & \(\Lambda_{7}=20,\ c_{71}=2,\ c_{72}=2,\ r_{71}=20,\ r_{72}=20,\ \kappa_{6}=1.1\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameter values of the tracking differentiator.  
Figure 4: _Cont._