## 2 Modeling
### Kinematics  
#### 2.2.1 Kinematic Model  
The body coordinate frame (BCF) of Tilt-X, consisting of \(o_{b}x_{b}y_{b}z_{b}\), and the earth coordinate frame (ECF), consisting of \(o_{e}x_{e}y_{e}z_{e}\), are shown in Figure 2a; the origin coincides with the COG. For the attitude angles, roll is the rotation of the body around the \(x\)-axis, denoted by \(\phi\), pitch is the rotation of the body around the \(y\)-axis, denoted by \(\theta\), and yaw is the rotation of the body around the \(z\)-axis, denoted by \(\psi\). The distance from each propeller to \(z_{b}\) is \(d\). Figure 2b shows the situation where all four propellers tilt at the same angle toward the negative \(x\)-axis, i.e., the pitch-axis, of the BCF. The tilt angles are denoted as \(\beta_{i}^{b}\) (\(i=1\), \(2\), \(3\), \(4\)). Servo motors \(1\) and \(4\), which tilt propellers \(1\) and \(4\), and servo motors \(2\) and \(3\), which \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(e\), \(b\), \(i\) & The superscripts representing the earth coordinate frame (ECF), the body coordinate frame (BCF), and the \(i\)th propeller, respectively \\ \(o\), \(p\), \(v\), \(\omega\) & The origin, position, velocity, and angular velocity, respectively \\ \(x\), \(y\), \(z\) & The subscripts representing the vectors in Cartesian coordinate system \\ \(\Theta\) & The Euler angle \\ \(\phi\), \(\theta\), \(\psi\) & The roll angle, pitch angle, and yaw angle, respectively \\ \(\beta\) & The propeller tilt angle \\ \(s\), \(c\), \(t\) & The abbreviations for sine, cosine, and tangent, respectively \\ \(R\), \(A\) & The rotation matrix and the relationship between attitudes 1 [FOOTNOTE:1]Footnote 1: Consistent with the meanings of the relationship of Euler angles. 2 \(\epsilon_{1}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix}\), \(\epsilon_{2}=\begin{bmatrix}0\\ 1\\ 0\end{bmatrix}\), \(\sigma=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix}\). \\ \(a\), \(r\) & The propeller blade area and propeller radius, respectively \\ \(d\) & The distance between the body center and any rotor \\ \(m\) & The mass \\ \(g\) & The gravitational acceleration \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of all symbols. Figure 2: **(a)** Coordinate systems BCF and ECF, and the attitude angles \(\phi\), \(\theta\), and \(\psi\). Two ways of tilting propellers to maximize (**b**) the horizontal pulling force \(F_{x}^{e}\) by tilting propellers \(1\) to \(4\) in the same direction, and (**c**) the yaw moment \(-M_{z}^{e}\) by tilting propellers \(1\) and \(4\), and propellers \(2\) and \(3\) in opposite directions, respectively. tilt propellers 2 and 3, are symmetrically installed at the ends of the cantilever arms of Tilt-X. The tilt angles of the former are assigned to positive signs and those of the latter to negative signs. Propellers 1 and 2 are counterclockwise (CCW) propellers with angular velocities denoted as \(\omega_{1}^{b}\) and \(\omega_{2}^{b}\), respectively, while propellers 3 and 4 are clockwise (CW) propellers with angular velocities denoted as \(-\omega_{3}^{b}\) and \(-\omega_{4}^{b}\). Since the thrust direction of the propellers is along the negative \(z\)-axis, their signs are negative. Tilting all the propellers in this manner results in a horizontal pulling force, \(F_{x}^{e}\), acting on the environment. Figure 2c shows the situation where propellers 1 and 4 tilt at the same angle toward the negative \(x\)-axis of the BCF, while propellers 2 and 3 tilt at the same angle toward the positive \(x\)-axis of the BCF. Each propeller generates a moment in the direction opposite to its angular velocity, denoted as \(M_{i}^{b}\). M-UAVs achieve yaw motion in the CCW and CW directions through the difference in the magnitudes of these moments. Tilting all the propellers in this manner results in a yaw moment, \(-M_{x}^{e}\), acting on the environment.