# Short Landing for Flying-Wing Unmanned Aircraft with Thrust Vector
## 3 Short-Landing Performance Analysis
### Attainable Linear and Pitch Angular Acceleration  
To ensure flight safety during short landings, it is important to select an appropriate angle of attack and velocity for the aircraft. The conventional flying quality requirements based on maximum angular acceleration and attitude change in 1 s under an abrupt (0.3-s) ramp-control input [30] are not suitable for the flying-wing UAV, due to its unique size and inertial properties. Instead, a methodology can be used to identify the safest angle of attack for a given velocity by evaluating the maximum linear and angular acceleration generated by control effectors. This methodology involves selecting a series of equilibrium points [31] for the aircraft during a short landing. The longitudinal dynamic model of the UAV can be derived from Figure 4.  
\[ \begin{split}&\dot{V}_{gx}=-\frac{1}{m}\Big{(}D\cos\gamma+L\sin \gamma-T\cos\big{(}\delta_{ny}+\theta\big{)}+T_{v}\frac{R_{x}}{R_{f}}\sin \delta_{vq}\sin\theta\Big{)}\\ &\dot{V}_{gz}=-\frac{1}{m}\Big{(}-D\sin\gamma+L\cos\gamma+T\sin \big{(}\delta_{nyq}+\theta\big{)}-gm+T_{v}\frac{R_{x}}{R_{f}}\sin\delta_{vq} \cos\theta\Big{)}\end{split}  \tag{11}\]  
Equation (11) describes the longitudinal dynamic model of the UAV during a short landing, where \(V_{gx}\) and \(V_{gz}\) are the velocity components in the earth coordinate frame; \(L\) and \(D\) are the lift and drag forces in the wind coordinate frame; \(\gamma\) and \(\theta\) are the flight path angle and pitch attitude angle, respectively; and \(g\) is the gravitational acceleration. To simplify the analysis, Equation (11) is expanded using a Taylor series, resulting in Equation (12), which considers only the first-order term of the expansion:  
\[\dot{x}\approx\dot{x}_{0}+\underbrace{\frac{\partial f(x,u)}{\partial x}}_{F( \cdot)}\underbrace{ \begin{array}{c}x=x_{0}\\ u=u_{0}\end{array} }_{F(\cdot)}\underbrace{ \begin{array}{c}(x-x_{0})+\\ \underbrace{\begin{array}{c}\frac{\partial f(x,u)}{\partial u}\end{array}}_{ \begin{array}{c}x=x_{0}\\ u=u_{0}\end{array}}_{\begin{array}{c}x=x_{0}\\ u=u_{0}\end{array}}\end{array} }_{G(\cdot)}(u-u_{0}) \tag{12}\]  
where the state vector is \(x=[V_{gx}\ V_{gz}]^{\mathrm{T}}\), and the control input vector is \(u=[\delta_{nyq}\ T_{v}]^{\mathrm{T}}\); \(x_{0}\) and \(u_{0}\) are the trimmed values for state and control; and \(F(\cdot)\)\(\in\)R\({}^{2\times 2}\) and \(G(\cdot)\)\(\in\)R\({}^{2\times 2}\) are vectors of nonlinear algebraic equations. The control input matrix, \(G(\cdot)\), represents the acceleration response to the control inputs, as shown in the following equation:  
\[G(\cdot)= \begin{bmatrix}\frac{\partial\dot{V}_{gx}}{\partial\delta_{nyq}}& \frac{\partial\dot{V}_{gx}}{\partial T_{v}}\\ \frac{\partial\dot{V}_{gz}}{\partial\delta_{nyq}}&\frac{\partial\dot{V}_{gz}}{ \partial T_{v}}\end{bmatrix}  \tag{13}\]  
Equation (13) can be expanded using partial derivatives, as demonstrated in Equation (14). \[ \begin{split}&\frac{\partial\dot{V}_{gx}}{\partial\delta_{nyq}}=- \frac{T_{v}}{m}\Big{(}\sin\big{(}\delta_{nyq}+\theta\big{)}+\frac{R_{x}}{R_{f }}\cos\delta_{vq}\sin\theta\Big{)}&\frac{\partial\dot{V}_{gx}}{\partial T_{v }}=\frac{1}{m}\Big{(}\cos\big{(}\delta_{nyq}+\theta\big{)}-\frac{R_{x}}{R_{f} }\sin\delta_{vq}\sin\theta\Big{)}\\ &\frac{\partial\dot{V}_{gz}}{\partial\delta_{nyq}}=-\frac{T_{v}}{m}\Big{(} \cos\big{(}\delta_{vq}+\theta\big{)}+\frac{R_{x}}{R_{f}}\cos\delta_{vq}\cos \theta\Big{)}&\frac{\partial\dot{V}_{gz}}{\partial T_{v}}=-\frac{1}{m}\Big{(} \sin\big{(}\delta_{nyq}+\theta\big{)}+\frac{R_{x}}{R_{f}}\sin\delta_{vq}\cos \theta\Big{)}\end{split}  \tag{14}\]  
Figure 10 displays the matrix \(G(\cdot)\) calculated at a trimmed angle of attack of 4 degrees and various velocities. The gray area indicates a continuous error bar for the changing sink rate. In Figure 10a, tangential acceleration shows a negative correlation with the nozzle pitch angle, indicating that the UAV gains more deceleration capability as velocity decreases due to the larger trimmed nozzle pitch angle.