## 3 Aircraft Flight Dynamics Model  
### Aircraft Dynamic Model  
The fixed-time spin recovery control law is founded on the nonlinear dynamic model of the F-18/HARV [12]. The aircraft dynamic model with twelve state variables \([u,v,w,p,q,r,\phi,\theta,\psi,H,X,Y]^{T}\) can be written as follows:  
\[\left\{\begin{array}{l}\dot{u}=rv-qw-g\sin\theta+\frac{\psi^{2}S(C_{1}(a, \theta,\delta_{c})\sin a-C_{2}(a,\theta,\delta_{c})\cos a)}{2m}+\frac{T_{m} \delta_{c}}{m}\\ \dot{v}=pw-ru+g\sin\phi\cos\theta+\frac{\psi^{2}S_{C}(\frac{1}{2}pr,\delta_{c },\delta_{c},\delta_{r})}{2m}\\ \dot{w}=qu-pv+g\cos\theta\cos\phi-\frac{\psi^{2}S(C_{1}(a,\theta,\delta_{c}) \cos a+C_{2}(a,\theta,\delta_{c})\sin a)}{2m}\end{array}\right. \tag{5}\]  
\[\left\{\begin{array}{l}\dot{p}=I_{1}pq-I_{2}qr+\frac{\psi^{2}S}{2m}(I_{3}C_{ 1}(\beta,p,r,\delta_{c},\delta_{a},\delta_{r})+I_{4}C_{n}(\beta,p,r,\delta_{c },\delta_{a},\delta_{r}))\\ \dot{q}=I_{5}pr-I_{6}\big{(}p^{2}-q^{2}\big{)}+\frac{\psi^{2}S^{2}}{2m}C_{m}( \alpha,q,\delta_{c})\\ \dot{r}=I_{7}pq-I_{1}qr+\frac{\psi^{2}S}{2m}\big{(}I_{4}C_{1}(\beta,p,r,\delta _{c},\delta_{a},\delta_{r})+I_{8}C_{n}(\beta,p,r,\delta_{c},\delta_{a},\delta _{r}))\end{array}\right. \tag{6}\]  
\[\left\{\begin{array}{l}\dot{\phi}=p+q\sin\phi\tan\theta+r\cos\phi\tan\theta \\ \dot{\theta}=q\cos\phi-r\sin\phi\\ \dot{\psi}=\sec\theta(r\cos\phi+q\sin\phi)\end{array}\right. \tag{7}\]  
\[\left\{\begin{array}{l}\dot{H}=u\sin\theta-v\sin\phi\cos\theta-w\cos\phi \cos\theta\\ \dot{X}=u\cos\theta\cos\psi+v(\sin\phi\sin\theta\cos\psi-\cos\phi\sin\psi)+w( \cos\phi\cos\psi\sin\theta+\sin\phi\sin\psi)\\ \dot{Y}=u\cos\theta\sin\psi+v(\sin\phi\sin\theta\sin\psi+\cos\phi\cos\psi)+w( \cos\phi\sin\psi\sin\theta-\sin\phi\cos\psi)\end{array}\right.\]  
where \(u,v,\) and \(w\) denote velocity components, respectively. \(p\), \(q\), and \(r\) are roll, pitch, and yaw angular velocities. \(\phi\), \(\theta\), and \(\psi\) represent roll, pitch, and heading angles. \(X\) and \(Y\) are the positions of the aircraft. \(H\) is aircraft altitude. \(I_{xx}\), \(I_{yy}\), \(I_{zz}\), and \(I_{xz}\) represent roll, pitch, yaw, and cross-product moment of inertia. Let \(I=I_{xx}I_{zz}-I_{xz}^{2}\), then the inertia parameters \(I_{i}\), \(i=1,2,\ldots 8\) in Equation (7) can be defined as \(I_{1}=I_{xz}\big{(}I_{xx}-I_{yy}-I_{xz}\big{)}/I\), \(I_{2}=\big{(}I_{zz}\big{(}I_{xx}-I_{yy}\big{)}+I_{zz}^{2}\big{)}/I\), \(I_{3}=I_{zz}/I\), \(I_{4}=I_{xz}/I\), \(I_{5}=(I_{zz}-I_{xx})/I_{yy}\), \(I_{6}=I_{xz}/I_{yy}\), \(I_{7}=\big{(}I_{xx}\big{(}I_{xx}-I_{yy}\big{)}+I_{xz}^{2}\big{)}/I\), and \(I_{8}=I_{xx}/I\). \(T_{\text{m}}\) denotes maximum available thrust.  
The aerodynamic model can be represented by the following nonlinear equations:  
\[\left\{\begin{array}{l}C_{i}(\alpha,q,\delta_{e})=C_{i}(\alpha)+C_{iq}\left( \frac{q^{2}}{2V}\right)+C_{i_{k}}\delta_{e},i=L,D,m\\ C_{j}(\beta,p,r,\delta_{e},\delta_{a},\delta_{r})=C_{j_{0}}+C_{j_{p}}\beta+C_{ j_{p}}\left(\frac{pb}{2V}\right)+C_{j_{r}}\left(\frac{rb}{2V}\right)+C_{j_{k}} \delta_{e}+C_{j_{k}}\delta_{a}+C_{j_{k}}\delta_{r},j=Y,l,n\end{array}\right. \tag{9}\]  
where \(C_{Y}\), \(C_{L}\), and \(C_{D}\) denote coefficients of lateral force, lift force, and drag force, respectively. \(C_{l}\), \(C_{m}\), and \(C_{m}\) represent coefficients of roll, pitch, and yaw moment. The nonlinear models of aerodynamical coefficients are the functions of variables \(\alpha\), \(\beta\), \(\delta_{e}\), \(\delta_{a}\), \(\delta\), \(p\), \(q\), and \(r\), which can be obtained from Refs. [12; 15].