## 4 Simulation  
In order to successfully land on a convex surface, the quadrotor needs to descend vertically to the landing target to avoid rolling over. The proposed RBFNN method will control the quadrotor in order to maintain straight attitude during the landing, under the conditions of external disturbances. The soft landing gear will demonstrate its ability to absorb the remaining impact force and help the quadrotor to perch on a convex surface, as shown in Figure 6.  
To validate the real-time performance of the controller and assess the reliability and robustness of the proposed RBFNN-based backstepping control approach in the challenge of external disturbances, a Hardware-in-the-Loop (HITL) environment was developed for the UAV. The simulation platform was made up of two main parts: A hardware part and a software part. The hardware part was the Pixhawk autopilot unit, which was used in the field tests. The software part was the Matlab aerospace simulation environment. The hardware and software parts were connected by USB/UART to send and receive flight data. The RBFNN-based control approach was implemented by MATLAB 2018b and PX4 Autopilots Support from Embedded Coder. The quadrotor helicopter model was used, for simulation validation, with the physical parameters listed in Table 1. The disturbances were added to the model by \(\Xi=0.3\cos(0.2t)\) through an individual plugin. The RBFNN in the simulation used 2, 5, and 1 neuron(s) in the input, hidden, and output layers, respectively. The centre and width in Equation (17) were chosen as \(m=2\) and \(\sigma=6\), respectively. The coefficients \(\zeta_{1}\) and \(\zeta_{2}\) were chosen as 0.1 and 0.3, respectively. The parameters in Equation (25) were tuned by trial and error to achieve satisfying control performance, as shown in Table 2. \begin{table}
\begin{tabular}{c l c c} \hline \hline
**Symbol** & **Description** & **Value** & **Units** \\ \hline \(m\) & Mass & 2 & kg \\ \(l\) & Body length & 450 & mm \\ \(r\) & Rotor radius & 100 & mm \\ \(I_{x}\) & Moment of Inertia & 1.85 & \(10^{-3}\) kg \(\cdot\) m\({}^{2}\) \\ \(I_{y}\) & Moment of Inertia & 1.85 & \(10^{-3}\) kg \(\cdot\) m\({}^{2}\) \\ \(I_{z}\) & Moment of I inertia & 2.98 & \(10^{-3}\) kg \(\cdot\) m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the quadrotor helicopter. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Roll** & **Pitch** & **Yaw** \\ \hline \(\alpha\) & 11 & 14 & 12 \\ \(k\) & 0.4 & 0.4 & 0.3 \\ \(\gamma\) & 16 & 23 & 21 \\ \(h\) & 1.1 & 3.9 & 1.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the backstepping sliding-mode control. Figure 6: Quadrotor successfully landed on convex surfaces with the support of the proposed soft landing gear. Five simulation tests were carried out to investigate the accuracy and repeatability of the proposed method, and the simulation path can be seen in Figure 7. The simulation steps were as follows:  
1. The quadrotor started from an initial position \(A(0,0,0)\). 2. The quadrotor flew to the position \(B(2,2,6)\) and hovered there for attitude control tests; these results can be seen in Figure 8.
3. The quadrotor descended to the position \(C(4,2,2)\) with a speed between 5-8 m/s. 4. The quadrotor, then, received the landing position \(D(6,2,0)\). The UAV began to adjust its altitude, attitude, and velocity for the landing tests. 5. The quadrotor landed on the target without a high-speed impact. In these tests, the numerical simulation demonstrates that all the quadrotor attitude states had a small steady-state error, subject to the external disturbance. As shown in Figure 8 and Table 3, the steady-state error was thoroughly eliminated in a short time using the RBFNN-based backstepping control method. Table 4 demonstrates the Root Mean Square Error (RMSE) of the touchdown position, related to the target surface position.