# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 3 Application to the Helicopter System
### Development of the Functional Forms of the State Vector  
Here, we use the prescribed time histories \(\mathbf{\eta}(t)\) to develop the functional forms of the state vector \(\mathbf{x}\). The state vector \(\mathbf{x}\) of the helicopter system includes the elements \(\mathbf{V}_{\rm B}\), \(\mathbf{\omega}_{\rm B}\), and \(\mathbf{\Phi}_{\rm B}\):  
\[\mathbf{x}=[\mathbf{V}_{\rm B},\ \mathbf{\omega}_{\rm B},\ \mathbf{\Phi}_{\rm B}]^{\sf T} \tag{14}\]  
Through coordinate system transformation, \(\mathbf{V}_{\rm B}\) is deduced in the form:  
\[\mathbf{V}_{\rm B}^{(r)}=\mathbf{T}_{x}(\phi_{\rm BP})\mathbf{T}_{y}(\theta_{\rm BP})\mathbf{T} _{z}(\phi_{\rm BP})\mathbf{V}_{\rm P}^{(r)} \tag{15}\]  
According to the angular velocity composition theorem of rigid body motion [28], \(\mathbf{\omega}_{\rm B}\) can be deduced in the form:  
\[\begin{split}\mathbf{\omega}_{\rm B}^{(r)}=\mathbf{T}_{x}(\phi_{\rm BP}) \mathbf{T}_{y}(\theta_{\rm BP})\mathbf{T}_{z}(\psi_{\rm BP})\big{[}\mathbf{e}_{\mathbf{x}},\ \mathbf{e}_{\mathbf{y}},\ \mathbf{T}_{y}(\theta_{\rm PE})\mathbf{T}_{x}(180^{\circ})\mathbf{e}_{\mathbf{z}}\big{]}\mathbf{ \Phi}_{\rm PE}^{(r+1)}\\ +\big{[}\mathbf{e}_{\mathbf{x}},\ \mathbf{T}_{x}(\phi_{\rm BP})\mathbf{e}_{\mathbf{y}},\ \mathbf{T}_{x}(\phi_{\rm BP})\mathbf{T}_{y}(\theta_{\rm BP})\mathbf{e}_{\mathbf{z}}\big{]}\mathbf{ \Phi}_{\rm BP}^{(r+1)}\end{split} \tag{16}\]where \(\mathbf{e_{x}}\), \(\mathbf{e_{y}}\), and \(\mathbf{e_{z}}\) are the unit vectors along the respective x-, y-, and z-axes, and \(\mathbf{T}_{x}(\cdot)\) is the Euler transformation matrices relative to the \(x\)-axis.  
The spatial position of the axes \(ox_{\text{B}}\) and \(oy_{\text{B}}\) in frame \(o_{\text{E}}x_{\text{E}}y_{\text{E}}z_{\text{E}}\) can indicate the quantitative value of the elements of \(\mathbf{\Phi}_{\text{B}}\)  
\[\theta_{\text{B}}=\arcsin\big{(}\epsilon_{x,\text{E}}^{z}\big{)} \tag{17}\]  
\[\phi_{\text{B}}=\left\{\begin{array}{ccc}0&,&e_{y,\text{E}}^{Z}=0\\ &-\arccos\Big{(}\frac{\mathbf{e_{1}}\cdot\mathbf{e_{y,\text{E}}}}{|\mathbf{e_{1}}|}\Big{)} &,&e_{y,\text{E}}^{Z}>0\\ &\arccos\Big{(}\frac{\mathbf{e_{1}}\cdot\mathbf{e_{y,\text{E}}}}{|\mathbf{e_{1}}|}\Big{)} &,&e_{y,\text{E}}^{Z}<0\end{array}\right. \tag{18}\]  
\[\psi_{\text{B}}=\text{atan}\Big{(}\frac{e_{x,\text{E}}^{Y}}{e_{x,\text{E}}^{X }}\Big{)} \tag{19}\]  
where:  
\[\mathbf{e_{i,\text{E}}}=\begin{bmatrix}e_{i,\text{E}}^{X}\\ e_{i,\text{E}}^{Y}\\ e_{i,\text{E}}^{Z}\end{bmatrix}=\mathbf{T}_{x}(180^{\circ})\mathbf{T}_{z}(-\psi_{\text {PE}})\mathbf{T}_{y}(-\theta_{\text{PE}})\mathbf{T}_{z}(-\psi_{\text{BP}})\mathbf{T}_{y}(- \theta_{\text{BP}})\mathbf{T}_{x}(-\phi_{\text{BP}})\mathbf{e}_{i,\text{ B}} \tag{20}\]  
where the subscript \(i\) denotes \(x\) or \(y\), \(\mathbf{e}_{i,\text{ B}}\) denotes the unit vectors \(\mathbf{e}_{x}\) or \(\mathbf{e}_{y}\) in frame \(o_{\text{E}}y_{\text{B}}z_{\text{B}}\), \(\mathbf{e}_{i,\text{E}}\) denotes the unit vectors \(\mathbf{e}_{x}\) or \(\mathbf{e}_{y}\) in frame \(o_{\text{E}}x_{\text{E}}y_{\text{E}}z_{\text{E}}\), and \(\mathbf{e}_{1}=\begin{bmatrix}e_{x,\text{ E}}^{Y}-e_{x,\text{ E}}^{X},\end{bmatrix}^{\text{T}}\in\mathcal{R}^{3}\).