## 2 UAV Motion Model
### Quadrotor Motion Principle  
The motion principles of the quadrotor are illustrated in Figure 1, in which opposite propellers (1, 3) and (2, 4) are paired and rotate in opposite clockwise and counterclockwisedirections to cancel out rotational torques. By controlling the speeds of rotors, motions in various directions can be achieved [47], as shown in Figure 1b, in which (from left to right) (1) increasing/decreasing four propellers with the same speed results in a vertical motion; (2) the difference in the speeds of the propeller pair (1, 3) results in changes in the pitch angle (around the y-axis) and thus movement along the \(x\)-axis; (3) the difference in the speeds of the propeller pair (2, 4) results in changes in the roll angle (around the \(x\)-axis) and thus movement along the y-axis; and (4) the difference in the speeds of propeller pair (1, 3) with pair (2, 4) results in counter-torque between pairs and thus changes in the yaw angle (around the \(z\)-axis), leading to a rotation around the \(z\)-axis.  
Figure 1: Quadrotor motion principle and kinematic-dynamic modeling: (**a**) quadrotor motion in the Earth reference frame (\(O\)) and its rigid body reference frame (\(O_{B}\)); (**b**) rotating propellers 1 to 4 create forces \(F_{1}\) to \(F_{4}\), and the resultant force of propellers with various speeds results in quadrotor motion in various directions.  
where \(\xi=\begin{bmatrix}x&y&z\end{bmatrix}^{\top}\in\mathbb{R}^{3}\) denotes the UAV 3D transition in the Earth reference frame; \(v=\begin{bmatrix}v_{x}&v_{y}&v_{z}\end{bmatrix}^{\top}\in\mathbb{R}^{3}\) denotes the UAV linear velocity in the Earth reference frame; \(\overrightarrow{z}=\begin{bmatrix}0&0&1\end{bmatrix}^{\top}\) is the unit vector of the \(z\)-axis; \(m\) [kg] is the vehicle mass and \(\mathbf{g}\left[\frac{Nm^{2}}{kg^{2}}\right]\) is the gravitational constant; \(\hat{\omega}\) is the skew-symmetric matrix such that:  
\[\hat{\omega}v=\omega\times v,\text{ and }\hat{\omega}^{\top}=-\hat{\omega}.\] \[\hat{\omega}=\begin{bmatrix}0&-\omega_{2}&\omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{bmatrix}, \tag{6}\]  
and \(\mathcal{R}\) denotes the rotation matrix based on \(\mathcal{E}=\begin{bmatrix}\varphi&\theta&\psi\end{bmatrix}^{\top}\in\mathbb{ R}^{3}\) denoting [PERSON]'s roll (\(\varphi\)), pitch (\(\theta\)), and yaw (\(\psi\)) angles with respect to the \(x-y-z\) axes, respectively, as:  
\[\mathcal{R}=\begin{bmatrix}c\theta c\psi&s\varphi s\theta c\psi-c\varphi s\psi &s\varphi s\psi+c\varphi s\theta c\psi\\ c\theta s\psi&s\varphi s\theta s\psi+c\varphi c\psi&c\varphi s\theta s\psi-s \varphi c\psi\\ -s\theta&s\varphi c\theta&c\varphi c\theta\end{bmatrix}; \tag{7}\]  
where \(\mid\) and s denote shorthand forms of the cosine and sine functions, respectively. The state space model of the quadrotor can be developed by plugging \(\hat{\omega}\) and \(\mathcal{R}\) into \(\mathcal{DM}\), [47; 49]. Various dynamics-aware UAV trajectory optimization algorithms have been proposed in the literature [35; 37; 50]. However, as mentioned earlier, it is not computationally efficient to use complicated models for trajectory design when the sampling time is larger than the UAV motion dynamics. Therefore, it is preferable to use linear kinodynamic models.