## 6 Numerical Simulation Results and Discussions
### Simulation Results and Discussions
As a result, and despite having shown a faster initial response (in the first flight stage) than both RA-BSC and RAS-BSC, the BSC tracking performance deteriorates dramatically once the disturbances influence the system (Figures 2-12) and the position tracking error performance exhibits large peak-to-peak amplitudes of oscillation (Table 4). Furthermore, the BSC altitude oscillation remains at a high frequency and never converges to zero (Figure 9). During this stage, the control signals \(u_{1}\), \(u_{2}\), and \(u_{3}\) are not saturated (Figures 13-15), and, as expected, the position tracking error performance is nearly identical for the RA-BSC and our RAS-BSC. For the RA-BSC, it is, respectively, 0.6 m, 0.7 m, and 0.05 m for the \(x\)-, \(y\)-, and \(z\)-position. Meanwhile, for our new controller, it is 0.6 m, 0.55 m, and 0.05 m for the \(x\)-, \(y\)-, and \(z\)-position, respectively. Furthermore, during this second stage, the control signal \(u_{4}\) becomes saturated (Figure 16), resulting in a considerable degradation of the RA-BSC heading angle tracking performance (Figure 12 and Table 4) with a peak-to-peak oscillation amplitude of \(2.4^{\circ}\), whereas it is only \(0.8^{\circ}\) for our controller. The numerical simulation results also show that the saturation amounts corresponding to the virtual control signals \(\mathbf{F}\) and \(\mathbf{U_{2}}\) are all bounded  
Figure 5: Comparing the trajectory tracking performance of the three controllers along the \(y\)-direction. Figure 6: Comparing the trajectory tracking performance of the three controllers along the \(z\)-direction. Figure 11: Comparison of pitch attitude performance between our new controller (RAS-BSC) and existing controllers. Figure 12: Comparison of yaw attitude performance between our new controller (RAS-BSC) and existing controllers. Figure 10: Comparison of roll attitude performance between our new controller (RAS-BSC) and existing controllers. Figure 14: Comparison of control signal \(u_{2}\) between our new controller (RAS-BSC) and existing controllers. Figure 13: Comparison of control signal \(u_{1}\) between our new controller (RAS-BSC) and existing controllers. Figure 15: Comparison of control signal \(u_{3}\) between our new controller (RAS-BSC) and existing controllers. In summary, our new controller clearly outperforms the other backstepping-based controllers, despite either parameterized uncertainty in combination with external disturbances or control input saturation. The way our new controller exhibits the performance promises a robust and rapid response, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Controller** & \(x\)**-Position** & \(y\)**-Position** & \(z\)**-Position** & **Heading Angle** \\ \hline BSC & 2.9 m & 4.5 m & 0.7 m & 26\({}^{\circ}\) \\ RA-BSC & 0.6 m & 0.7 m & 0.05 m & 2.4\({}^{\circ}\) \\ RAS-BSC & 0.6 m & 0.55 m & 0.05 m & 0.8\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of peak-to-peak oscillation amplitude of control performance during the second flight stage (in the presence of external disturbances). Figure 16: Comparison of control signal \(u_{4}\) between our new controller (RAS-BSC) and existing controllers. Figure 17: The magnitude of the saturation amounts, \(\parallel\Delta\mathbf{F}\parallel\) and \(\parallel\Delta\mathbf{U}_{2}\parallel\), are all bounded, which verifies the reasonability of the saturation assumptions \(\parallel\Delta\mathbf{F}\parallel\leq v_{1}\) and \(\parallel\Delta\mathbf{U}_{2}\parallel\leq v_{2}\). even when the vehicle is flying in disturbance-rich environments and in the presence of control input saturation and sensor faults. This significantly extends the range of applications and environments in which UAVs can be deployed, especially if these applications require high stability, rapid response times, and high accuracy, e.g., for moving obstacle avoidance, precision landing on moving platforms, autonomous delivery to difficult (e.g., narrowed) areas, etc.