# Helicopter Safe Landing Trajectory after Main Rotor Actuator Failures
## 3 Helicopter Dynamic Model Formulation  
After reconfiguration of the original control mixer, a helicopter dynamic model, which takes into account the effect of an actuator failure, can be formulated. As is demonstrated in [13; 14], the attitude is controllable if the vertical channel is given up. It can also be concluded from Equations (4)-(9) that the lateral control is still independent from the longitudinal control after one of the actuators stuck. Thus, for numerical considerations, the helicopter dynamic model in this study is described by a nonlinear three-degree-of-freedom (DOF) rigid-body helicopter dynamic model, with longitudinal and vertical dynamic descriptions. The three-DOF helicopter dynamic model is widely used among helicopter autorotation problems [21], and is verified via flight tests [22], which is similar to the landing procedures in this study. States variables of the three-DOF dynamic model are chosen as  
\[\mathbf{s}=(x,z,v_{x},v_{z},\theta,q)\]  
which include the horizontal displacement \(x\), vertical displacement \(z\), horizontal velocity \(v_{x}\), rate of descent \(v_{z}\), pitch angle \(\theta\) and pitch rate \(q\). The control variables are the collective pitch \(\theta_{0}\) and longitudinal cyclic pitch \(\theta_{\rm 1s}\), which satisfy the constraints described in Equations (4)-(9). Descriptions of the helicopter dynamic equations are formulated as  
\[\left\{ \begin{array}{l}\dot{x}=v_{x}\\ \dot{z}=v_{z}\\ \dot{v}_{x}=-\frac{1}{m}(T_{\rm B}\sin\theta+H_{\rm B}\cos\theta+D_{\rm F}\cos \alpha_{\rm F})\\ \dot{v}_{z}=-\frac{1}{m}(T_{\rm B}\cos\theta-H_{\rm B}\sin\theta+D_{\rm F}\sin \alpha_{\rm F})+g\\ \dot{\theta}=q\\ \dot{q}=\frac{1}{I_{\rm P}}(M_{\rm Y}-T_{\rm B}\cdot l_{\rm R}+H_{\rm B}\cdot h _{\rm R}-L_{\rm H}\cos(\theta-\eta)\cdot l_{\rm H})\end{array} \right. \tag{10}\]  
where \(T_{\rm B},H_{\rm B},M_{\rm Y}\) are rotor thrust and drag forces resolved in the body axes and rotor pitch moment, respectively. \(D_{\rm F},L_{\rm H}\) are the fuselage drag and the horizontal tail lift.\(\theta,\alpha_{\rm F},\eta\) are pitch angle, angle of attack of the fuselage and the flight-path angle. \(l_{\rm R},h_{\rm R}\) are horizontal and vertical distance from rotor hub to the center of gravity. \(l_{\rm H}\) is the horizontal distance of the horizon tail from the center of gravity. \(g\) is the gravitational acceleration. Formulations of fuselage and horizontal tail forces are from [23]. \(T_{\rm B},H_{\rm B}\) are calculated by  
\[\left[ \begin{array}{c}T_{\rm B}\\ H_{\rm B}\end{array} \right]=\left[ \begin{array}{cc}\cos i_{s}&\sin i_{s}\\ -\sin i_{s}&\cos i_{s}\end{array} \right]\left[ \begin{array}{c}T\\ H\end{array} \right] \tag{11}\]  
where \(i_{s}\) is the inclination of rotor shaft, \(T,H\) are the rotor forces. \(T,H,M_{\rm Y}\) are calculated by normalized forms \(C_{T},C_{H},C_{MY}\), which are the rotor thrust coefficient, rotor drag force coefficient, and rotor pitch moment coefficient, respectively. The normalization factors can be found in [24]. The rotor thrust coefficient \(C_{T}\) is calculated by [24]:  
\[C_{T}=\frac{1}{2}a\sigma\Big{[}(\frac{1}{3}+\frac{1}{2}\overline{v}_{xb}{}^{2 })(\theta_{0}-\frac{3}{4}\theta_{tw})B^{3}+\frac{1}{4}(1+\overline{v}_{xb}{} ^{2})B^{4}-\frac{1}{2}\lambda(1+\frac{1}{2}\overline{v}_{xb}{}^{2})B^{2}\Big{]} \tag{12}\]  
where \(B\) is the blade tip-loss factor, \(a\) is the blade's mean lift-curve slope, \(\theta_{\rm tw}\) is the blade twist, and \(\sigma\) is the rotor solidity. The rotor inflow ratio \(\lambda\) is given by  
\[\lambda=\overline{v}_{xb}\sin\theta-\overline{v}_{xb}\cos\theta+\
u \tag{13}\]  
where the induced velocity coefficient \(\
u\), described as [24]:  
\[\
u=\kappa v_{\rm h}f_{\rm I} \tag{14}\]  
where \(\kappa\) is the nonuniform inflow empirical correction factor, and \(v_{\rm h}\) is the induced velocity at hover. \(f_{\rm I}\) is the induced velocity factor, which has a correction of the vortex-ring state, given by [24]:  
\[f_{\rm I}=\left\{