# Short Landing for Flying-Wing Unmanned Aircraft with Thrust Vector
## 2 Aircraft Dynamics Model
### Thrust Vector Model  
Table 3 displays the second-order dynamic system model used to represent the thrust response to throttle input, accounting for position and rate limitations. It should be noted that positive rotation of the 3 BSD nozzle generates negative moments and that the nozzle is aligned with the \(x\)-axis of the body frame. The thrust vector changes with the rotation of the nozzle. To avoid a singularity at 90 degrees of pitch, the nozzle is first rotated around its pitch axis by an angle, \(\delta_{qr}\), and then the new frame is rotated around its yaw axis by an angle, \(\delta_{q\eta}\). The rotation matrix from the nozzle frame to the body frame is \(M_{bT}\). To prevent a singularity from occurring at 90 degrees of pitch, the thrust vector is adjusted through the rotation of the nozzle. Specifically, the nozzle is initially rotated around its pitch axis by an angle of \(\delta_{q\eta}\). Following this, the new frame is then rotated around its yaw axis by an angle of \(\delta_{qr}\) to arrive at the desired orientation. The corresponding rotation matrix from the nozzle frame to the body frame is denoted as \(M_{bT}\), and it can be expressed as follows:  
\[M_{bT}= \begin{bmatrix}\cos\delta_{qr}&-\sin\delta_{qr}&0\\ \sin\delta_{qr}&\cos\delta_{qr}&0\\ 0&0&1\\ \end{bmatrix}\begin{bmatrix}\cos\delta_{q\eta}&0&\sin\delta_{q\eta}\\ 0&1&0\\ -\sin\delta_{q\eta}&0&\cos\delta_{q\eta}\\ \end{bmatrix}  \tag{6}\]  
The resulting expression for the thrust vector in the body frame is as follows:  
\[ \begin{bmatrix}T_{vx}\\ T_{vy}\\ T_{vy}\\ \end{bmatrix} =M_{bT} \begin{bmatrix}-\delta_{qr}T_{v\max}\\ 0\\ 0\\ \end{bmatrix} = \begin{bmatrix}\delta_{qr}T_{v\max}\cos\delta_{q\eta}\cos\delta_{qr }\\ \delta_{qr}T_{v\max}\cos\delta_{q\eta}\sin\delta_{qr}\\ -\delta_{qr}T_{v\max}\sin\delta_{q\eta}\\ \end{bmatrix}  \tag{7}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Effector** & **Explanation** & **Damping** & **Natural** & **Position** & **Rate Limit** \\ \hline \(\delta_{r}\) & Elevator angle & 0.8 & 30 rad/s & \([-30^{\circ},+30^{\circ}]\) & \(\pm 110^{\circ}/s\) \\ \(\delta_{q}\) & Ailéron angle & 0.8 & 30 rad/s & \([-30^{\circ},+30]\) & \(\pm 110^{\circ}/s\) \\ \(\delta_{LSSD}\) & Left SSD deflection angle & 0.8 & 30 rad/s & \([0^{\circ},60^{\circ}]\) & \(\pm 110^{\circ}/s\) \\ \(\delta_{RSSD}\) & Right SSD deflection angle & 0.8 & 30 rad/s & \([0^{\circ},60^{\circ}]\) & \(\pm 110^{\circ}/s\) \\ \(\delta_{q\eta}\) & 3 BSD nozzle pitch angle & 1 & 10 rad/s & \([0^{\circ},90^{\circ}]\) & \(\pm 30^{\circ}/s\) \\ \(\delta_{qr}\) & 3 BSD nozzle yaw angle & 1 & 10 rad/s & \([-15^{\circ},+15^{\circ}]\) & \(\pm 30^{\circ}/s\) \\ \(\delta_{qrT}\) & Turbojet engine throttle & 1 & 1 rad/s & \([0,1]\) & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effector characteristics of the UAV. To analyze the moments generated by the thrust vector, it is necessary to take the cross product of the thrust vector and its corresponding lever arm. This yields the following equation, which represents the moments generated by the thrust vector:  
\[ \begin{bmatrix}M_{vx}\\ M_{vy}\\ M_{v2}\end{bmatrix} = \begin{bmatrix}0&0&0\\ 0&0&-R_{v}\\ 0&R_{v}&0\end{bmatrix}\begin{bmatrix}T_{vx}\\ T_{vy}\\ T_{v2}\end{bmatrix} = \begin{bmatrix}0\\ -R_{v}\delta_{vT}T_{\text{rm max}}\sin\delta_{vq}\\ R_{v}\delta_{vT}T_{\text{rm max}}\cos\delta_{vq}\sin\delta_{vr}\end{bmatrix}  \tag{8}\]  
where \(R_{v}\) represents the distance from the nozzle to the center of gravity along the body's \(x\)-axis, as shown in Figure 4. To balance the pitch moment generated by the thrust vector, the force that the lift fans should provide is calculated as follows:  
\[T_{f}=\frac{R_{v}\delta_{vT}T_{\text{rm max}}\sin\delta_{vq}}{nR_{f}} \tag{9}\]  
where \(n\) is the number of lift fans (\(n=2\)), and \(R_{f}\) is the distance between the lift fan center and CG along the body \(x\)-axis.