# A Novel Inverse Simulation Method of Helicopter Maneuvering Flight
## 3 Application to the Helicopter System  
According to the right-hand rule of the frame system, an earth-fixed inertial frame (\(\theta_{\rm E}x_{\rm E}y_{\rm E}z_{\rm E}\)) is defined at an arbitrary reference point (\(o_{\rm E}\)) on the ground, with the axes \(o_{\rm E}x_{\rm E}\), \(o_{\rm E}y_{\rm E}\), and \(o_{\rm E}z_{\rm E}\) toward the respective east, north, and vertical up. Following this, a flight-path frame (\(ox_{\rm p}y_{\rm E}z_{\rm p}\)) and a body frame (\(ox_{\rm B}y_{\rm B}z_{\rm B}\)) are then defined at the helicopter center of gravity (\(o\)), with the axis \(ox_{\rm P}\) toward the direction of flight velocity vector \(V_{\rm P}\), and the axis \(ox_{\rm B}\) toward the helicopter nose, and parallel to the cabin floor. Figure 1 shows these definitions of the frame system, where \(\mathbf{\Phi}_{\rm BP}=[\theta_{\rm BP},\ \theta_{\rm BP},\ \phi_{\rm BP}]^{\rm T}\) is the Euler angle between frames \(ox_{\rm B}y_{\rm B}z_{\rm B}\) and \(ox_{\rm P}y_{\rm P}z_{\rm P}\). \(\phi_{\rm BP}\) also indicates the sideslip angle of the helicopter, and \(\phi_{\rm PE}\) and \(\phi_{\rm PE}\) are the yaw angle and climbing angle of the flight path relative to frame \(o_{\rm E}x_{\rm E}y_{\rm E}z_{\rm E}\), respectively, which can be written as a vector form of \(\mathbf{\Phi}_{\rm PE}=[0,\ \theta_{\rm PE},\phi_{\rm PE}]^{\rm T}\) for convenience.  
The motion of a helicopter in the air can, in general, be described by a combination of rotations and translations. The rotations include pitch around axis \(oy_{\rm B}\), roll around axis \(ox_{\rm B}\), and yaw around axis \(oz_{\rm B}\). The translations include move forward or backward along axis \(ox_{\rm B}\), move right or left along axis \(oy_{\rm B}\), and move downward or upward along axis \(oz_{\rm B}\). If the helicopter is regarded as a rigid body, a conventional helicopter flight dynamics system can be written in the form:  
\[\left\{\begin{array}{l}m_{\rm B}\dot{V}_{\rm B}+\mathbf{\omega}_{\rm B}\times(m_ {\rm B}V_{\rm B})=\mathbf{F}_{\rm B}^{\rm AERO}+m_{\rm B}\cdot\mathcal{B}_{\rm B}\\ \mathbf{I}_{\rm B}\dot{\mathbf{\omega}}_{\rm B}+\mathbf{\omega}_{\rm B}\times(\mathbf{I}_{\rm B }\mathbf{\omega}_{\rm B})=M_{\rm B}^{\rm AERO}\\ T_{\it Euler}\cdot\mathbf{\Phi}_{\rm B}=\mathbf{\omega}_{\rm B}\end{array}\right. \tag{11}\]  
where \(\mathbf{V}_{\rm B}=[u_{\rm B},\ \upsilon_{\rm B},w_{\rm B}]^{\rm T}\in\mathcal{R}^{3}\) is the body translational velocity, \(\mathbf{\omega}_{\rm B}=[p_{\rm B},\ q_{\rm B},r_{\rm B}]^{\rm T}\in\mathcal{R}^{3}\) is the body angular velocity, \(\mathbf{\Phi}_{\rm B}=[\phi_{\rm B},\ \theta_{\rm B},\psi_{\rm B}]^{\rm T}\in\mathcal{R}^{3}\) contains the respective body roll,  
Figure 1: Definitions of the frame system.  
pitch, and yaw angles, \(m_{\rm B}\in\mathcal{R}^{1}\) is the mass, \(\mathbf{I}_{\rm B}\in\mathcal{R}^{3\times 3}\) is the inertia matrix, \(g_{\rm B}\in\mathcal{R}^{3}\) is the acceleration of gravity in the frame \(\alpha x_{\rm B}y_{\rm B}z_{\rm B}\), \(T_{\rm Enler}\in\mathcal{R}^{3\times 3}\) is the Euler transformation matrix, and \(\mathbf{F}_{\rm B}^{\rm AERO}\in\mathcal{R}^{3}\) and \(\mathbf{M}_{\rm B}^{\rm AERO}\in\mathcal{R}^{3}\) are the respective resultant aerodynamic force and resultant aerodynamic moment generated by the main rotor, tail rotor, fuselage, and horizontal and vertical tails for a single-rotor helicopter.  
Three processes (i.e., the construction of a transitional vector, the development of the functional forms of the state vector, and the degeneration of the first-order differential equations) need to be carried out when using the degeneration method to conduct inverse simulation.