# Short Landing for Flying-Wing Unmanned Aircraft with Thrust Vector
## 2 Aircraft Dynamics Model
### Lift Fan Model  
During short-landing operations, the lift fans play a crucial role in controlling the UAV's sink rate and counteracting the negative pitch moment caused by the 3 BSD nozzle's pitching. To maintain the UAV's attitude, a combination of lift fan control and elevator input is utilized. The lift fan system comprises electric motors, as detailed in Table 2. To obtain a high-fidelity model, the force and torque produced by the lift fans are measured during bench testing. These measurements are denoted as \(T_{f}\) and \(Q_{f}\), respectively, and are calculated as outlined in [26]:  
\[\left\{ \begin{aligned} & T_{f}=\rho C_{\text{f}}\omega_{f}{}^{2}D_{f}^{4}\\ & Q_{f}=\rho C_{q}\omega_{f}{}^{2}D_{f}^{5}\end{aligned} \right. \tag{1}\]  
where \(\omega_{f}\) is the rotation speed of the culvert fan, \(\rho\) is the air density, and \(C_{\text{f}}\) and \(C_{q}\) are the lift fan's force and torque coefficient, respectively. The least-squares estimation method is employed to estimate the thrust and torque coefficients, using the bench-test data, as illustrated in Figure 3a,b. These coefficients are subsequently utilized in the controller design. While the static bench-test data provide valuable information, they are insufficient to fully characterize the lift fan's dynamic response. Therefore, the dynamic torque of the lift fan is computed by incorporating instantaneous electric power: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Explanation** & **Value** \\ \hline \(m\) & Mass & 62 kg \\ S & Wing area & 1.8 m\({}^{2}\) \\ \(b\) & Wingspan & 3.2 m \\ \(T_{0}\) max & Maximum thrust of the turbojet engine & 270 N \\ \(T_{f}\) max & Maximum force per lift fan & 75 N \\ \(I_{xx}\) & Moment of inertia around body \(x\)-axis & 72.1 kg\(\cdot\)m\({}^{2}\) \\ \(I_{yy}\) & Moment of inertia around body \(y\)-axis & 63.7 kg\(\cdot\)m\({}^{2}\) \\ \(I_{zz}\) & Moment of inertia around body \(z\)-axis & 112.3 kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the target aircraft. Figure 2: Top view of the flying-wing UAV. \[Q_{fp}=\frac{UI}{\omega_{f}} \tag{2}\]  
where \(U\) and \(I\) are the current and voltage, respectively, and \(Q_{fp}\) is the calculated torque. As part of the dynamic modeling of the lift fan system, the derivative of the fan's rotation speed can be represented by the following model:  
\[\dot{\omega}_{f}(t)=\frac{\min\Bigl{(}\max\Bigl{(}K_{f}\Bigl{(}\omega_{fc}(t- \tau_{d})-\omega_{f}(t)\Bigr{)}I_{rf},-Q_{f}\Bigr{)},Q_{fp}+Q_{f}\Bigr{)}}{I_{rf}} \tag{3}\]  
where \(K_{f}\) is the control bandwidth of the electronic speed controller (ECS, \(K_{f}=30\) rad/s), \(\tau_{d}\) is the command delay for the rotation speed (\(\tau_{d}=10\) ms), and the rate of rotation speed \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Explanation** & **Value** \\ \hline \(m_{f}\) & Mass of blade culvert fan & 0.092 kg \\ \(D_{f}\) & Diameter of fan & 0.15 m \\ \(I_{rf}\) & Inertia moment of blade culvert fan & 0.0004 kg-m\({}^{2}\) \\ \(N\) & Pole pairs of the electric motor & 6 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical parameters of the lift fan. Figure 3: Bench test results for lift fans. (**a**) Lift fan static force test results and fitting curve; the estimated results are \(C_{t}=4.2278\). (**b**) Lift fan static torque test results and fitting curve; the estimated results are \(C_{q}=0.2303\). (**c**) Recorded dynamic electric power change with rotation speed. (**d**) Calculated results of the dynamic lift-fan torque. is limited by \(Q_{fp}\) and \(Q_{f}\). Finally, the moment of the lift fan, \(M_{f}\), is expressed in the body axis as follows:  
\[\mathbf{M_{f}}=l_{rf}\omega_{f}\times\mathbf{\Omega}+\mathbf{Q_{f}}+(\mathbf{CF-CG+BIAS})\times \mathbf{T_{f}} \tag{4}\]  
In Equation (4), we have the following:  
\[