## 2 Dynamical Model  
### Quadrotor Model  
To simplify control design while ensuring model accuracy, we make the following assumptions for the quadrotor UAV. (1) The UAV is a rigid body with constant mass, moment of inertia, and center of gravity during flight. (2) The UAV structure is symmetrical. (3) Earth's rotation and revolution effects are negligible. (4) The only external forces acting on the UAV are gravity, propeller-generated lift and torque, and external disturbances. Using the Newton-Euler equations, the kinematic and dynamic models of the quadrotor unmanned aerial vehicle (UAV) are formulated in two coordinate reference frames: the inertial frame and the body frame ([PERSON], 2018). In this study, the inertial frame is denoted as frame \(\hat{F}_{E}=\left\{\hat{f}_{oe},\ \hat{f}_{ex},\ \hat{f}_{ey},\ \hat{f}_{ez}\ \right\}\), while the body frame is denoted as frame \(\hat{F}_{B}=\left\{\hat{f}_{bo},\ \hat{f}_{bx},\ \hat{f}_{by},\ \hat{f}_{bz}\ \right\}\). Here, \(\hat{f}_{oo}\) represents the origin of the inertial coordinate system, and \(\hat{f}_{bo}\) represents the center of mass of the quadrotor UAV. These definitions are illustrated in Figure 1. The Euler angles of the quadrotor, denoted as \(\Theta^{E}=\left[ \begin{matrix}\phi&\theta&\psi\end{matrix} \right]^{T}\in \mathbb{R}^{3\times 1}\), represent its roll angle (\(\phi\)), pitch angle (\(\theta\)), and yaw angle (\(\psi\)). These angles satisfy the conditions \(|\phi|<\frac{\pi}{2},|\theta|<\frac{\pi}{2},|\psi|<\pi\).The angular velocities of the drone, denoted as \(\Omega^{B}=[pqr]^{T}\in\mathbb{R}^{3\times 1}\), correspond to its rotations along the X, Y, and Z axes of the body coordinate system, respectively. The position of the quadrotor in the world coordinatesystem \(\hat{F}_{E}\) is defined as \(P^{E}=\left[p_{x}^{e}\quad p_{y}^{e}\quad p_{z}^{e}\right]^{T}\in\mathbb{R}^{3\times 1}\), while the velocity of the quadrotor's movement in the inertial coordinate system \(\hat{F}_{E}\) is defined as \(V^{E}=\left[v_{x}^{e}\quad v_{y}^{e}\quad v_{z}^{e}\right]^{T}\in\mathbb{R}^{3 \times 1}\). Consequently, the kinematic and dynamic model of the quadrotor's displacement and rotation can be expressed as follows:  
\[\dot{P}^{E}=V^{E}\]  
\[\dot{V}^{E}=m^{-1}\left[R_{B}^{E}T_{B}+F_{d}+dv\right]\]  
\[\dot{\Theta}^{E}=R_{\Theta}\cdot\Omega^{B}\]  
\[\dot{\Omega}^{B}=I^{-1}\left[\tau_{b}+G_{a}-\Omega^{B}\times\left(I\cdot\Omega ^{B}\right)+M_{d}\right] \tag{1}\]  
where \(m\) represents the mass of the quadrotor, \(I\) denotes the inertial matrix of the quadrotor in the inertial coordinate system, given by \(I=diag\left(I_{xx},I_{yy},I_{zz}\right)\), \(R_{B}^{E}\) signifies the rotation matrix between the body-fixed coordinate system and the inertial coordinate system for the quadrotor's translation, and \(R_{\Theta}\) represents the rotation matrix for the quadrotor's rotation. These matrices satisfy the following conditions:  
\[R_{B}^{E}=\left[ \begin{array}{ccc}cos\theta cos\psi&sin\phi sin\theta-cos \phi sin\psi&sin\phi sin\psi+cos\phi sin\theta cos\psi\\ cos\theta sin\psi&cos\phi cos\psi+sin\phi sin\psi sin\psi&cos\phi sin\phi sin \psi-cos\psi sin\phi\\ -sin\theta&sin\phi cos\theta&cos\phi cos\theta\end{array} \right] \tag{2}\]  
\[R_{\Theta}=\left[ \begin{array}{ccc}1&sin\phi tan\theta&cos\phi tan\theta \\ 0&cos\phi&-sin\phi\\ 0&sin\phi/cos\theta&cos\phi/cos\theta\end{array} \right] \tag{3}\]  
_dv_ represents step-like increasing values of external disturbances. The variables \(F_{d}\) and \(M_{d}\) represent bounded variations of aerodynamic force and moment, respectively. \(T_{B}\) satisfies \(T_{B}=T_{K}E_{3}-mg\left(R_{B}^{E}\right)^{T}E_{3}\), where \(g\) denotes the gravitational constant. \(T_{K}\) represents the total thrust generated by the quadrotor at time \(t_{0}\), and it can be expressed as \(T_{K}=T_{1}+T_{2}+T_{3}+T_{4}\). Each \(T_{i}(i=1,2,3,4)\) represents the thrust \(T_{i}=k_{\omega}\omega_{i}^{2}\) generated  
Figure 1: Quadrotor unmanned aerial vehicle (UAV) coordinates in body fixed and inertial frame.