## 3 Model Description of Helicopter and Wind Gust
### Helicopter Modeling  
Based on the assumptions and simplifications mentioned above, the helicopter attitude dynamic model described in the fuselage-fixed reference frame is given in (1). \[\left\{ \begin{array}{l}\dot{\Theta}=S^{-1}(\Theta)\omega\\ \dot{I}\omega=-\omega\times J\omega+Q\\ Q=Q_{A}\tau+Q_{B}\\ Q_{A}=\left[\begin{array}{ccc}h_{t}&Q_{m}&T_{m}h_{m}\\ 0&T_{m}h_{m}&-Q_{m}\\ l_{t}&0&-T_{m}l_{m}\\ \end{array}\right],Q_{B}=\left[\begin{array}{c}0\\ T_{m}l_{m}\\ -Q_{m}\\ \end{array}\right]\end{array} \right. \tag{1}\]  
where \(\Theta=[\phi,\theta,\psi]^{T}\) are Euler angles and \(\omega=[p,q,r]^{T}\) is angular velocity vector, respectively. \(S^{-1}(\Theta)\) is the transformation matrix from the angular velocities to Euler angles derivatives. \(J=diag(I_{xx},I_{yy},I_{zz})\) is a diagonal matrix of moments of inertia in fuselage-fixed reference frame with the inertia coupling terms neglected, and \(Q=[L,M,N]^{T}\) are the lumped moments exerted on the helicopter. \(T_{m}\), \(Q_{m}\) denote the main rotor thrust and counteractive torque; \(\tau=[T_{t},\beta_{1c},\beta_{1s}]^{T}\) are the system inputs of the model, with \(T_{t}\) being the tail rotor thrust and \(\beta_{1c}/\beta_{1s}\) being longitudinal/lateral Tip-Path-Plane flapping angles of the main rotor. \(h_{m}\), \(h_{t}\), \(l_{m}\) and \(l_{t}\) represent the horizontal and vertical distances from the center of the rotor hub to the center of gravity. As the main rotor flapping dynamics is the main source of nonlinear forces, the flapping dynamics are introduced in the helicopter model. The most complete and accurate analytical equation of rotor flapping dynamics is expressed as below in (2) [35]. \[\left[ \begin{array}{c}\ddot{\beta}_{1 cw}\\ \dot{\beta}_{1 sw}\end{array} \right] =-\Omega\!\left[ \begin{array}{cc}\frac{7}{8}&2\\ -2&\frac{7}{8}\end{array} \right]\left[ \begin{array}{c}\dot{\beta}_{1 cw}\\ \dot{\beta}_{1 sw}\end{array} \right]-\Omega^{2}\!\left[ \begin{array}{cc}P^{2} -1+\frac{7K_{1}\mu^{2}}{16}&\frac{7}{8}(1+\frac{\mu^{2}}{2})\\ -\frac{7}{8}(1+\frac{\mu^{2}}{2})&P^{2}-1+\frac{3\gamma K_{1}\mu^{2}}{16} \end{array} \right]\left[ \begin{array}{c}\beta_{1 cw}\\ \beta_{1 sw}\end{array} \right] \tag{2}\] \[+\Omega^{2}\!\left[ \begin{array}{cc}0&0&\frac{7}{8}(1+\frac{ \mu^{2}}{2})&0\\ -\frac{7\mu}{3}&-\frac{7\mu}{4}&0&-\frac{7}{8}(1+\frac{3\mu^{2}}{2})\end{array}  \right]\left[ \begin{array}{c}\theta_{0}\\ \theta_{t}\\ \theta_{1 cw}\\ \theta_{1 sw}\end{array} \right]\] \[+\Omega\!\left[ \begin{array}{cc}-2&-\frac{7}{8}&0&-\frac{1}{ \Omega}\\ -\frac{7}{8}&2&-\frac{1}{\Omega}&0\end{array} \right]\left[ \begin{array}{c}p _{w}\\ q_{w}\\ \dot{p}_{w}\\ \dot{q}_{w}\end{array} \right]+\Omega^{2}\!\left[ \begin{array}{c}0\\ -\frac{7\mu}{4}\end{array} \right]\lambda\]  
A more specific explanation of this formula can be found in [35]. In order to simplify the controller design and model complexity, only the first-order of this formula is used here in (3) [36]. \[\ddot{B}_{1w}=K_{\omega}\omega_{12w}+K_{B}B_{1w}+K_{\theta}\theta_{1w} \tag{3}\]  
where \(B_{1w}=[\beta_{1 cw},\beta_{1 sw}]^{T}\), \(\omega_{12w}=[p_{w},q_{w}]^{T}\), \(\theta_{1w}=[\theta_{1 cw},\theta_{1 sw}]^{T}\). \(\theta_{1 cw}/\theta_{1 sw},\theta_{0}\) are the longitudinal/lateral cyclic pitch and collective pitch of the swash plate, which means the actual system inputs. The subscript_w denotes the states in the wind-hub frame [35]. A detailed explanation of the parameter matrices \(K_{\omega}\), \(K_{B}\) and \(K_{\theta}\) is provided in [36]. Due to the fact that the flapping angles are much smaller than the collective pitch, the effect of wind gust disturbances on the flapping dynamics could be simplified as \(\zeta=[\gamma\Omega\mu/6,0]^{T}\), where \(\mu\) denotes the advance ratio. Then the main rotor flapping dynamics described above could be rewritten in (4).