## 2 Model of the Tilt Tri-Rotor UAV
### Nonlinear Equations of Motion  
In this part, we focus on deriving the 6-DOF nonlinear mathematical model of the tilt tri-rotor UAV. Assuming the UAV as a rigid body, the dynamic model can be derived based on Newton-Euler formulation. The schematic diagram of the tilt tri-rotor UAV coordinate system is shown in Figure 2. \(O_{c}X_{c}Y_{c}Z_{e}\) is the world frame, \(O_{b}X_{b}Y_{b}Z_{b}\) is the body frame, and \(O_{r}X_{r}Y_{r}Z_{r}\) is the rotor frame. It is noted that the right rotor, the left rotor, and the rear rotor are labeled by 1, 2, and 3, respectively. The transformation from the rotor frame to the body frame is done using the matrix \(R_{ri}^{b}\) as follows,  
\[R_{ri}^{b}= \begin{bmatrix}\cos\alpha_{i}&0&\sin\alpha_{i}\\ 0&1&0\\ \sin\alpha_{i}&0&\cos\alpha_{i}\end{bmatrix} i=1,2 \tag{1}\]  
Figure 1: Schematic diagram of the tilt tri-rotor unmanned aerial vehicle (UAV). \[R_{73}^{b}= \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}  \tag{2}\]  
where \(\alpha_{i}\) represents the tilting angle of rotor \(i\), and it is defined as \(-90^{\circ}\) if the UAV is in the hover mode. In the hover mode, the aerodynamic forces are small enough compared to the thrust and the gravity so that they can be ignored. The resultant force of the tilt tri-rotor UAV \(F^{b}\) includes the rotor thrust \(F_{r}^{b}\) and the gravity \(F_{g}^{b}\). \[F^{b}=F_{g}^{b}+F_{r}^{b} \tag{3}\]  
where  
\[F_{g}^{b}=R_{EBT} \begin{bmatrix}0\\ 0\\ mg\end{bmatrix}  \tag{4}\]  
\[F_{r}^{b}=-R_{r1}^{b} \begin{bmatrix}0\\ 0\\ k_{f}\omega_{1}^{2}\end{bmatrix} -R_{r2}^{b} \begin{bmatrix}0\\ 0\\ k_{f}\omega_{2}^{2}\end{bmatrix} -R_{r3}^{b} \begin{bmatrix}0\\ 0\\ k_{f}\omega_{3}^{2}\end{bmatrix}  \tag{5}\]  
\[R_{EBT}= \begin{bmatrix}\cos\theta\cos\psi&\cos\theta\sin\psi&-\sin\theta\\ \sin\phi\sin\theta\cos\psi-\cos\phi\sin\psi&\sin\phi\sin\theta\sin\psi+\cos \phi\cos\psi&\sin\phi\cos\theta\\ \cos\phi\sin\theta\cos\psi+\sin\phi\sin\psi&\cos\phi\sin\theta\sin\psi-\sin \phi\cos\psi&\cos\phi\cos\theta\end{bmatrix}  \tag{6}\]  
where \(R_{EBT}\) denotes the transformation matrix from the world frame to the body frame. The mass of the UAV is expressed as \(m\), \(g\) is the gravitational acceleration, and \(\omega_{i}\) represents the rotational angular velocity of rotor \(i\). The parameter \(k_{f}\) is pull coefficient of the rotor. The total moment \(\tau^{b}\) consists of the moment \(\tau^{b}_{d}\) created by the thrust of rotors, the reaction torque of rotors \(\tau^{b}_{F}\) and the moment \(\tau^{b}_{G}\) created by the gyroscopic effect of the propellers, and the body [30]:  
\[\tau^{b}=\tau^{b}_{F}+\tau^{b}_{d}+\tau^{b}_{G} \tag{7}\]  
Figure 2: Tilt tri-rotor UAV coordinate system. where  
\[\tau_{F}^{b}= \begin{bmatrix}r_{1x}\\ r_{1y}\\ r_{1z}\end{bmatrix} \times \begin{bmatrix}-k_{f}\omega_{1}^{2}\sin\alpha_{1}\\ 0\\ -k_{f}\omega_{1}^{2}\cos\alpha_{1}\end{bmatrix} + \begin{bmatrix}r_{2x}\\ r_{2y}\\ r_{2z}\end{bmatrix} \times \begin{bmatrix}-k_{f}\omega_{2}^{2}\sin\alpha_{2}\\ 0\\ -k_{f}\omega_{2}^{2}\cos\alpha_{2}\end{bmatrix} + \begin{bmatrix}r_{3x}\\ r_{3y}\\ r_{3z}\end{bmatrix} \times \begin{bmatrix}0\\ 0\\ -k_{f}\omega_{3}^{2}\end{bmatrix}  \tag{8}\]  
\[\tau_{d}^{b}=R_{r1}^{b} \begin{bmatrix}0\\ 0\\ k_{d}\omega_{1}^{2}\end{bmatrix} +R_{r2}^{b} \begin{bmatrix}0\\ 0\\ -k_{d}\omega_{2}^{2}\end{bmatrix} +R_{r3}^{b} \begin{bmatrix}0\\ 0\\ k_{d}\omega_{3}^{2}\end{bmatrix}  \tag{9}\]  
In Equations (7) and (8), \([r_{ix},r_{iy},r_{iz}]^{T}\) denotes the position of rotor \(i\) in the body frame, and \(k_{d}\) is torque coefficient of the rotor. The components of the gyroscopic moment are complicated due to the tilting motion of the rotors, so it is necessary to analyze the gyroscopic moment separately. The changes of the attitude influences the gyroscopic moment of the rear rotor. The gyroscopic moment of the front rotors is not only related to the change of the attitudes but also the tilting motion.