## 2 UAV Dynamic Model  
Aerodynamic modeling of the UAV adopts the airframe axis coordinate system and the earth axis system (see in Figure 2). The origin of the axis system is at the center of gravity of the UAV. The \(x\)-axis points forward along the plane of symmetry of the fuselage. The \(y\)-axis is perpendicular to the \(x\)-axis. The \(z\)-axis is determined by the right-hand rule. The \(x\)-axis and \(z\)-axis constitute the symmetry of the fuselage. Modeling of the lift fan and duct is based on the self-body axis system, which is an independent system of the lift fan and duct. The process of UAV take-off and transition to level flight is shown in Figure 3. During vertical take-off and landing, the duct deflects to provide upward thrust, and the lift fan in front also generates upward thrust. In the transition process, the forward-flying thrust is obtained by deflecting the duct inclination, and the transition from vertical take-off to horizontal flight is completed under certain control strategies. In level flight, the duct is deflected to the horizontal position, and the flight is controlled by the deflection of the control surface. Figure 1: Conceptual graphs of vertical take-off and landing fixed-wing UAV, which includes a lift fan and ducted lift-increasing system. Figure 3: Duct deflection in three flight states. Figure 2: Bod \(y\)-axis coordinate system and earth axis coordinate system. In the process of modeling the vertical take-off and landing UAV, the coordinate system and angle need to be defined. [PERSON] [17] studied the coordinate system and the conversion relationship, and the Euler transformation relationship between the ground coordinate system and the airframe coordinate system (see in Figure 4) is used in this article. Yaw angle \(\psi\): the angle between the projection of the body axis \(Ox_{b}\) on the horizontal plane \(Ox_{g}y_{g}\) and the \(Ox_{g}\) axis. The angle is positive when the aircraft yaws right. Pitch angle \(\theta\): the angle between the body axis \(Ox_{b}\) and the horizontal plane \(Ox_{g}y_{g}\). When the head of the aircraft is raised, the angle is positive. Roll angle \(\phi\): the angle between the plane of symmetry of the aircraft and the vertical plane containing the \(Ox_{b}\) axis. The angle is positive when the aircraft rolls right. According to the general law of coordinate transformation, the transformation matrix \(R_{bg}\) from the earth coordinate system to the body coordinate system can be obtained as  
\[R_{bg}=\left[ \begin{array}{ccc}cos\theta cos\psi&cos\theta sin\psi&-sin\theta \\ sin\theta sin\theta cos\psi-cos\phi sin\psi&sin\theta sin\theta sin\psi+cos\phi cos \psi&sin\phi cos\theta\\ sin\theta cos\phi cos\psi+sin\theta sin\phi&sin\theta cos\phi sin\psi-sin\phi cos \phi&cos\phi cos\theta\\ \end{array} \right]\]  
We project the aerodynamic forces and aerodynamic moments of the lift fan and ducted lift-increasing system, and combine the control equations to obtain the nonlinear flight dynamics mathematical equations of the vertical take-off and landing fixed-wing UAV. The mathematical model of nonlinear flight dynamics can generally be written in the form of a first-order differential equation:  
\[\dot{y}=f(y,u;t) \tag{1}\]  
where \(y\) represents all of the state quantities of the vertical take-off and landing fixed-wing UAV flight dynamics equation, and u represents the control quantities of the vertical take-off and landing fixed-wing UAV, and they are defined as  
\[y=\left[u,v,w,p,q,r,\theta,\phi,\psi\right]^{T} \tag{2}\]  
\[u=\left[p_{f},p_{d},\delta p_{r},\delta p_{p},\delta_{fu},\delta_{fd},\delta_{ di},\delta_{e},\delta_{a}\right]^{T} \tag{3}\]  
The force and moment of the body expressed by the state quantity of the dynamic equation, that is, the reference motion parameter \(y\) used in the equation, are as follows:  
\[\left.