## 2 Methodologies  
### Dynamic and Kinematic Equations  
This work considers a quadcopter as an UAV. To describe the motion of a quadcopter, the reference frames and the vectors with the quadcopter model are displayed in Figure 1. The inertial frame is defined by \(\hat{\mathbf{n}}_{k}\) for \(k=1,2\), and \(3\), and the body frame fixed on the center of mass of the quadcopter is defined by \(\hat{\mathbf{b}}_{k}\). The quadcopter's position and velocity vectors expressed in the body frame are defined as \(\mathbf{r}\in\mathcal{R}^{3}\) and \(\mathbf{v}\in\mathcal{R}^{3}\), respectively. The quadcopter's attitude is defined using the 3-2-1 set of the Euler angles, and the yaw, pitch, and roll angles of the quadcopter with respect to the inertial frame are defined as \(\psi\), \(\theta\), and \(\phi\), respectively. In addition, the quadcopter's angular velocity vector with respect to the inertial frame is defined by \(\mathbf{\omega}\in\mathcal{R}^{3}\). The direction cosine matrix from the inertial frame to the body frame of the quadcopter is defined as [17]  
\[C_{BN}= \begin{bmatrix}\cos\theta\cos\psi&\cos\theta\sin\psi&-\sin\theta\\ \sin\phi\sin\theta\cos\psi-\cos\phi\sin\psi&\sin\phi\sin\theta\sin\psi+\cos \phi\sin\psi&\sin\phi\cos\theta\\ \cos\phi\sin\theta\cos\psi+\sin\phi\sin\psi&\cos\phi\sin\theta\sin\psi-\sin \phi\cos\psi&\cos\phi\cos\theta\end{bmatrix} . \tag{1}\]Using Equation (1), the position and velocity vector expressed in the inertial frame are defined as  
\[{}^{N}\mathbf{r} =C_{BN}^{\mathrm{T}}\mathbf{r}, \tag{2}\] \[{}^{N}\mathbf{\hat{r}} =C_{BN}^{\mathrm{T}}\mathbf{v}, \tag{3}\]  
where a vector with the superscript \(N\) denotes the vector expressed in the inertial frame. Note that the superscript \(B\) is omitted in the vectors with the body frame expression. The motion of the quadcopter is composed of rotational and translational motions. The rotational motion of the quadcopter with respect to the inertial frame is defined as [17]  
\[J\dot{\mathbf{\omega}}+[\mathbf{\omega}^{\times}]\ \mathbf{J}\mathbf{\omega}=\mathbf{\tau}_{b}-\mathbf{ \tau}_{g}, \tag{4}\]  
where \(J\in\mathcal{R}^{3\times 3}\) is the inertia matrix, \([\mathbf{\omega}^{\times}]\) is the skew-symmetric matrix composed of the elements of the angular velocity vector (\(\omega_{1}\), \(\omega_{2}\) and \(\omega_{3}\)), \(\mathbf{\tau}_{b}\in\mathcal{R}^{3}\) is the external torque, and \(\mathbf{\tau}_{g}=[\mathbf{\omega}^{\times}][0,\ 0,\ I_{t}\Omega_{r}]^{\mathrm{T}}\) is the gyroscopic torque generated by the quadcopter's rotors. Here, \(I_{r}\) is the inertia of the rotor, and \(\Omega_{r}\) is the sum of the rotors' speed, which is defined as \(\Omega_{r}=\Omega_{1}-\Omega_{2}+\Omega_{3}-\Omega_{4}\). Here, \(\Omega_{i}\) for \(i=1,2,3\), and \(4\) represents the rotation speed of the \(i\)-th rotor. The quadcopter's translational motion is defined as [18]  
\[m\dot{\mathbf{v}}+[\mathbf{\omega}^{\times}]m\mathbf{v}=\mathbf{f}_{g}+\mathbf{f} _{b}, \tag{5}\]  
where \(m\) is the mass of the quadcopter and \(\mathbf{f}_{g}\in\mathcal{R}^{3}\) is the gravitational force defined as  
\[\mathbf{f}_{g}=C_{BN}\ ^{N}\mathbf{g}, \tag{6}\]  
where \({}^{N}\mathbf{g}=[0,\ 0,\ -mg]^{\mathrm{T}}\) is the gravitational force vector expressed in the inertial frame with the gravitational constant \(g\). Furthermore, the thrust force vector is defined by \(\mathbf{f}_{b}=[0,\ 0,\ f_{th}]^{\mathrm{T}}\), and the thrust force along with the \(\mathbf{\hat{b}}_{3}\) axis generated by the quadcopter's rotors is expressed as  
\[f_{th}=k_{f}(\Omega_{1}^{2}+\Omega_{2}^{2}+\Omega_{3}^{2}+\Omega_{4}^{2}), \tag{7}\]  
where \(k_{f}\) is the aerodynamic force constant. In addition, the control input vector for controlling the quadcopter is defined as  
\[\mathbf{u}= \begin{bmatrix}f_{th}\\ \tau_{1}\\ \tau_{2}\\ \tau_{3}\end{bmatrix} =