## 6 Numerical Simulation Results and Discussions
### Simulation Results and Discussions  
In the presence of both parameterized uncertainties and external disturbances our new controller is expected to perform similarly to the RA-BSC approach as long as the control input is not saturated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value** & **Unit** \\ \hline \(m\) & 2.0 & kg \\ \(J_{1}\) & 0.0121 & kg \(\cdot\)m\({}^{2}\) \\ \(J_{2}\) & 0.0119 & kg \(\cdot\)m\({}^{2}\) \\ \(J_{3}\) & 0.0223 & kg \(\cdot\)m\({}^{2}\) \\ \(l\) & 0.225 & m \\ \(c_{d}\) & 0.05 & \\ \(g\) & 9.81 & m/s\({}^{2}\) \\ \(\Lambda_{1}\) & diag\([0.01,0.01,0.01]\) & \\ \(\Lambda_{2}\) & diag\([0.012,0.012]\) & \\ \(u_{2}^{\max}\) & 30 & N \\ \(u_{3}^{\max}\) & 1.8 & Nm \\ \(u_{4}^{\max}\) & 3 & Nm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: System’s nominal parameters used in the simulation. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Value and Unit** & **Description** \\ \hline \(P_{d}\) & \([20\)sin(0.1 t), \(10\)sin(0.05 t), 0.1 t m & Trajectory reference \\ \(\psi_{d}\) & \(0^{\circ}\) & Desired heading angle \\ \(P_{0}\) & \([-5,3,0]\) m & Initial position \\ \(P_{0}\) & \([0,0,0]\) m/s & Initial velocity \\ \(\Theta_{0}\) & \([0,0,6]^{\circ}\) & Initial attitude \\ \(\Theta_{0}\) & \([0,0,0]^{\circ}/\)s & Initial angular velocity \\ \(\omega_{1},\omega_{2}\) & \(5,35\) & ESO’s bandwidth \\ \(k_{1}\) & diag\([0.8,0.8,0.27]\) & Controller gains \\ \(k_{2}\) & diag\([0.38,0.38,0.38]\) & Controller gains \\ \(k_{3}\) & diag\([1.05,1.05,1.75]\) & Controller gains \\ \(k_{4}\) & diag\([2.0,2.0,0.8]\) & Controller gains \\ \(k_{5}\) & diag\([1.0,1.0,0.8]\) & Controller gains \\ \(k_{6}\) & diag\([4.0,4.0,5.0]\) & Controller gains \\ \(c_{j}\) & 1 & Controller gains \\ \(\tau_{2},\tau_{5}\) & 0.01 & LPF’s parameters \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Trajectory reference, desired heading angle, initial conditions, and controller gains used in the simulation. Figure 2: Three-dimensional view of the trajectory tracking performance of the three controllers. If saturation does occur, our controller should exhibit a superior performance compared to both BSC and RA-BSC. To highlight the effectiveness of our proposed RAS-BSC, the simulation was carried out in two stages: trajectory tracking control in (i) the absence and (ii) the presence of lumped disturbances. For comparison, we also simulated the performance of the BSC and RA-BSC prescribing the same flight conditions. The first stage covers the time period from \(t=0\) to \(t=50\) s. At \(t=0\) s, the vehicle is initialized and then takes off from the initial position. All three controllers are generally able to drive the quadrotor to the trajectory reference (Figures 2-9) and stabilize the vehicle's attitude (Figures 10-12) within a few seconds after taking off. However, all control signals (Figures 13-16) reach different saturation values causing significant differences between the three controllers in their convergence times (Table 3). For the BSC, it is \(\sim\)9 s for \(x-\)position, 14 s for \(y\)-position, 23 s for \(z\)-position, and 6\({}^{\circ}\) for heading angle. For the RA-BSC, it is 15 s, 20 s, 35 s, and 9\({}^{\circ}\) for \(x\)-, \(y\)-, \(z\)-position, and heading angle, respectively. Whereas with our new algorithm, it is \(\sim\  
Here, we can make two main observations: (1) although the RA-BSC converges significantly slower than our new RAS-BSC controller in the presence of control input saturation (first flight stage), the convergence times become much more similar when the control input is not saturated (second flight stage), and (2) although the BSC converges quickly and delivers an acceptable performance in the absence of external disturbances (first flight stage), its tracking mission deteriorates considerably once the external disturbances are applied (second flight stage). During the second stage (from \(t=50\) s to \(t=150\) s), the lumped disturbances are applied to the simulation.