# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 3 Application to the Helicopter System
### Degeneration of the First-Order Differential Equations  
To degenerate the first-order differential equations, functional forms of some other variables, i.e., \(F_{\text{B}}^{\text{AERO}}\) and \(M_{\text{B}}^{\text{AERO}}\), as well as \(\mathbf{g_{\text{B}}}\) in Equation (11), need to be developed.  
Generally, collective (\(\delta_{\text{C}}\)), longitudinal cyclic (\(\delta_{\text{B}}\)), lateral cyclic (\(\delta_{\text{S}}\)), and pedal (\(\delta_{\text{R}}\)) controls are used to manipulate \(F_{\text{B}}^{\text{AERO}}\) and \(M_{\text{B}}^{\text{AERO}}\), to fly the helicopter. In detail, collective, longitudinal cyclic, and lateral cyclic controls are used to control the main rotor, and pedal control is used to control the tail rotor. Note that \(F_{\text{B}}^{\text{AERO}}\) and \(M_{\text{B}}^{\text{AERO}}\) are also dependent on the state vector of the helicopter system. Thus, these can be deduced in the forms:  
\[\left\{\begin{array}{l}\mathbf{F_{\text{B}}^{\text{AERO}}=f_{\text{F}}(V_{\text {B}},\mathbf{\omega_{\text{B}}},\;\mathbf{\Phi_{\text{B}}},\;\mathbf{U_{4\times 1}}\;)}\\ \mathbf{M_{\text{B}}^{\text{AERO}}=f_{\text{M}}(V_{\text{B}},\mathbf{\omega_{\text{B}}},\;\mathbf{\Phi_{\text{B}}},\;\mathbf{U_{4\times 1}}\;)}\end{array}\right. \tag{21}\]  
where the control vector \(\mathbf{U_{4\times 1}}=[\delta_{\text{C}},\delta_{\text{B}},\delta_{\text{S}}, \delta_{\text{R}}]^{\text{T}}\in\mathcal{R}^{4}\).  
\(g_{\text{B}}\) can be written in the form:  
\[g_{\text{B}}=\mathbf{T}_{x}(\phi_{\text{BP}})\mathbf{T}_{y}(\theta_{\text{BP}})\mathbf{T}_{ z}(\psi_{\text{BP}})\mathbf{T}_{y}(\theta_{\text{PE}})\begin{bmatrix}0\\ 0\\ g\end{bmatrix} \tag{22}\]  
where \(g\in\mathcal{R}^{1}\) is the acceleration of gravity.  
By combining with the prescribed vector \(\mathbf{\eta}(t)\), the functional forms of the state vector \(\mathbf{x}\), and Equations (21) and (22), the degenerated functional form of Equation (11) can be written in the form:  
\[\mathcal{T}_{6\times 6}\Big{(}V_{\text{P}},\dot{V}_{\text{P}},\;\mathbf{\Phi}_{ \text{PE}},\;\dot{\mathbf{\Phi}}_{\text{PE}},\;\ddot{\mathbf{\Phi}}_{\text{PE}},\;\mathbf{ \Phi}_{\text{BP}},\;\dot{\mathbf{\Phi}}_{\text{BP}},\;\ddot{\mathbf{\Phi}}_{\text{BP}}, \;\mathbf{U_{4\times 1}}\Big{)}=\mathbf{0}_{6\times 1} \tag{23}\]The time derivatives of \(\phi_{\text{BP}}\) and \(\theta_{\text{BP}}\) are developed based on the one differentiation-based approach [29]:  
\[\begin{cases}\left(\phi_{\text{BP}}^{(r+1)}\right)_{m}=\frac{\left(\phi_{\text{ BP}}^{(r)}\right)_{m}-\left(\phi_{\text{BP}}^{(r)}\right)_{m-1}}{\Delta t}\\ \left(\theta_{\text{BP}}^{(r+1)}\right)_{m}=\frac{\left(\theta_{\text{BP}}^{(r )}\right)_{m}-\left(\theta_{\text{BP}}^{(r)}\right)_{m-1}}{\Delta t}\end{cases} \tag{24}\]  
where \(m\) represents the \(m\)th simulation iteration.  
Thus, Equation (23) can be further written in the form:  
\[\mathcal{T}_{6\times 6}\left[\ \boldsymbol{\eta}_{m}^{(n)}(t),\ \left(\phi_{\text{BP}} \right)_{m-2},\ \left(\theta_{\text{BP}}\right)_{m-3},\left(\phi_{\text{BP}} \right)_{m-1},\ \left(\theta_{\text{BP}}\right)_{m-1},\ \left(\phi_{\text{BP}}\right)_{m},\ \left(\theta_{\text{BP}}\right)_{m^{\prime}} \left(\boldsymbol{U}_{4\times 1}\right)_{m}\right]=\boldsymbol{0}_{6\times 1} \tag{25}\]  
where \(n\) equals to 0, 1 or 2. Because \(\boldsymbol{\eta}_{m}^{(n)}(t)\) is known due to the prescribed flight path, there are six unknown variables, i.e., \(\left(\phi_{\text{BP}}\right)_{m}\), \(\left(\theta_{\text{BP}}\right)_{m}\), and \(\left(\boldsymbol{U}_{4\times 1}\right)_{m}\) in the \(m\)th iteration. Thus, this nonlinear equation can be enclosed in each iteration, and degeneration can be realized.