## 3 Preliminaries
### Model Derivation  
The force diagram of the flight vehicle is illustrated in Figure 2, in which \(MX_{b}Y_{b}Z_{b}\) is the body coordinate system. \(V\) is the velocity of the flight vehicle.  
The nonlinear attitude dynamics of the flight vehicle with uncertainties can be described by [28]  
\[\left\{\begin{array}{l}\dot{\gamma}_{v}=\cos\alpha\sec\beta\omega_{x}-\sin \alpha\sec\beta\omega_{y}+\\ \frac{Y(\tan\theta\sin\gamma_{v}+\tan\beta)+Z\tan\theta\cos\gamma_{v}-mg\cos \theta\cos\gamma_{v}\tan\beta}{\lambda}\\ \dot{\alpha}=-\cos\alpha\tan\beta\omega_{x}+\sin\alpha\tan\beta\omega_{y}+ \omega_{z}+\frac{mg\cos\theta\cos\gamma_{v}-Y}{mV\cos\beta}\\ \dot{\beta}=\sin\alpha\omega_{x}+\cos\alpha\omega_{y}+\frac{m\cos\theta\sin \gamma_{v}+Z}{mV}\\ \mathbf{I}\mathbf{\omega}+\mathbf{\widetilde{\omega}}\mathbf{I}\mathbf{\omega}=\mathbf{M}_{a}+\mathbf{d} _{\omega}\end{array}\right. \tag{1}\]  
where \(\gamma_{v},\ \alpha,\ \beta\) are the bank angle, angle of attack, and angle of sideslip, respectively. \(Y\) and \(Z\) are the lift and side forces. \(m\) is the mass of the flight vehicle and \(g\) is the gravity acceleration value. \(\theta\) is the flight path angle and \(\mathbf{\omega}=\left[\omega_{x}\quad\omega_{y}\quad\omega_{z}\right]^{T}\) is the angular rate vector. \(V\) is the velocity value of the flight vehicle. \(\mathbf{I}\in\mathbb{R}^{3\times 3}\) is the inertia matrix of the flight vehicle and \(\mathbf{M}_{a}=\begin{bmatrix}M_{x}&M_{y}&M_{z}\end{bmatrix}^{T}\) is the aerodynamic moment, which will be given later. \(\mathbf{d}_{\omega}\) denotes the uncertainties. Furthermore, the translational dynamics can be given by  
\[\left\{\begin{array}{l}\dot{x}=V\cos\theta\cos\psi_{v}\\ \dot{y}=V\sin\theta\\ \dot{z}=-V\cos\theta\sin\psi_{v}\\ m\dot{V}=P\cos\alpha\cos\beta-X-mg\sin\theta\\ mV\dot{\theta}=P(\sin\alpha\cos\gamma_{v}+\cos\alpha\sin\beta\sin\gamma_{v})+Y \cos\gamma_{v}-Z\sin\gamma_{v}-mg\cos\theta\\ -mV\cos\theta\dot{\psi}_{v}=P(\sin\alpha\sin\gamma_{v}-\cos\alpha\sin\beta\cos \gamma_{v})+Y\sin\gamma_{v}+Z\cos\gamma_{v}\end{array}\right. \tag{2}\]  
where \(x,y,z\) are the positions of the flight vehicle. \(\psi_{v}\) is the flight heading angle. \(X\) is the drag force and \(P\) is the thrust force. In our research, only the terminal guidance phase is considered. Therefore, \(P=0\) is assumed.  
The lift and side force, \(Y\) and \(Z\), are given by  
\[\left\{\begin{array}{l}Y=qSc_{y}^{a}\alpha+d_{Y}\\ Z=qSc_{z}^{b}\beta+d_{Z}\end{array}\right. \tag{3}\]  
in which \(q=\rho V^{2}/2\) is the dynamic pressure and \(\rho\) is the air density. \(S\) is the reference size. \(c_{y}^{a}\) and \(c_{z}^{b}\) are partial derivatives of the lift force coefficient with respect to \(\alpha\) and the side force coefficient to \(\beta\), respectively. \(d_{Y}\) and \(d_{Z}\) are the uncertainty terms caused by modeling  
Figure 2: Force diagram of the flight vehicle.  
errors and neglecting higher-order terms in approximation. Similarly, the aerodynamic moment can be simplified as  
\[\left\{\begin{array}{l}M_{x}=qSL\Big{(}m_{x}^{\alpha}\alpha+m_{x}^{\beta}\beta+m _{x}^{\delta_{x}}\delta_{x}\Big{)}+d_{Mx}\\ M_{y}=qSL\Big{(}m_{y}^{\beta}\beta+m_{y}^{\delta_{y}}\delta_{y}\Big{)}+d_{My}\\ M_{z}=qSL\Big{(}m_{x}^{\alpha}\alpha+m_{z}^{\beta_{z}}\delta_{z}\Big{)}+d_{Mz }\end{array}\right. \tag{4}\]  
where \(L\) represents the reference length. \(\delta_{x},~{}\delta_{y}\) and \(\delta_{z}\) denote the three axis fin deflections, respectively. \(m_{i}^{j}(i=x,y,z;j=\alpha,\beta,\delta_{i})\) represents the partial derivatives of the three axis moment coefficients with respect to \(\alpha\), \(\beta\), or fin deflections \(\delta_{i}\). \(d_{Mx}\), \(d_{My}\), and \(d_{Mz}\) are the uncertainty terms of the aerodynamic moment.