## 2 Preliminary Description of a Multirotor UAV
### Multirotor UAV Dynamics  
To describe the multirotor kinematics and dynamics, two right-handed Cartesian coordinate systems are defined as follows. Inertial coordinate system (called _Earth frame_, \(\mathcal{F}^{E}\)) is fixed to the considered stationary point on the Earth surface, where \(Z\) axis coincides with the normal and positive direction is defined upwards from the ground level. Aircraft coordinate system (called _Body frame_, \(\mathcal{F}^{B}\)) is fixed to the moving multirotor where the \(\mathcal{F}^{B}\) origin coincides with the center of gravity (COG), and axes coincide with the multirotor main axes of inertia. Multirotor position is determined by the vector \(\boldsymbol{\xi}=\left[ \begin{array}{cc}x&y&z\end{array} \right]^{\mathrm{T}}\) which connects the origin of \(\mathcal{F}^{E}\) with the origin of \(\mathcal{F}^{B}\) (see Figure 1). Multirotor orientation (attitude) is determined by a vector consisting of three Euler angles, \(\boldsymbol{\eta}=\left[ \begin{array}{ccc}\phi&\theta&\psi\end{array}  \right]^{\mathrm{T}}\). Rotation around the longitudinal axis (\(X\)) is defined as roll angle \(\phi\), while rotation around the lateral axis (\(Y\)) as pitch angle \(\theta\), and rotation around the vertical axis (\(Z\)) as yaw angle \(\psi\). The multirotor translational \(\boldsymbol{v}^{B}=\left[ \begin{array}{ccc}u&v&w\end{array} \right]^{ \mathrm{T}}\) and rotational \(\boldsymbol{\omega}^{B}=\left[ \begin{array}{ccc}p&q&r\end{array} \right]^{ \mathrm{T}}\) velocity vectors are defined in \(\mathcal{F}^{B}\) in which the equations of motion are described. Assuming that the multirotor is a rigid body, translational and rotational motion can be described as the motion of a particle located at the center of gravity of the body. The kinematics of a rigid body describes the mapping of the translational and rotational velocities from \(\mathcal{F}^{B}\) to \(\mathcal{F}^{E}\). It is defined by the following expression  
\[\left[ \begin{array}{c}\dot{\boldsymbol{\xi}}\\ \dot{\boldsymbol{\eta}}\end{array} \right]=\left[ \begin{array}{cc}\mathbf{R}& \boldsymbol{0}_{3\times 3}\\ \boldsymbol{0}_{3\times 3}&\boldsymbol{\Omega}_{B}\end{array} \right]\left[ \begin{array}{c}\boldsymbol{v}^{B}\\ \boldsymbol{\omega}^{B}\end{array} \right], \tag{1}\]  
where \(\mathbf{R}\) is the rotation matrix that maps the translational velocities, \(\boldsymbol{0}_{3\times 3}\) is the zero matrix, and \(\boldsymbol{\Omega}_{B}\) is the transformation matrix that maps rotational velocities. According to [PERSON]'s orientation theorem, the transformation from \(\mathcal{F}^{E}\) to \(\mathcal{F}^{B}\) is described by three consecutive rotations and the final orthogonal rotation matrix can be defined as  
\[\mathbf{R}=\left[ \begin{array}{ccc}c_{\phi}c_{\theta}&c_{\psi}s_{\theta}s_{ \phi}-s_{\psi}c_{\phi}&c_{\psi}s_{\theta}c_{\phi}+s_{\psi}s_{\phi}\\ s_{\psi}c_{\theta}&s_{\psi}s_{\theta}s_{\phi}+c_{\psi}c_{\phi}&s_{\psi}s_{ \theta}c_{\phi}-c_{\psi}s_{\phi}\\ -s_{\theta}&c_{\theta}s_{\phi}&c_{\theta}c_{\phi}\end{array} \right], \tag{2}\]  
where \(c_{i}=\cos(i)\), \(s_{j}=\sin(j)\). By resolving Euler angle rates, the transformation matrix is defined as  
\[\boldsymbol{\Omega}_{B}=\frac{1}{c_{\theta}}\left[ \begin{array}{ccc}c_{ \theta}&s_{\phi}s_{\theta}&c_{\phi}s_{\theta}\\ 0&c_{\phi}c_{\theta}&-s_{\phi}c_{\theta}\\ 0&s_{\phi}&c_{\phi}\end{array} \right] \tag{3}\]  
The multirotor equations of motion are based on the Newton-Euler approach where a system of six second-order differential equations is obtained. The equations of translational motion in \(\mathcal{F}^{B}\) are defined by the following expression  
\[m\dot{\boldsymbol{v}}^{B}+\boldsymbol{\omega}^{B}\times(mv^{B})=\mathbf{F}, \tag{4}\]  
where \(m\) is the mass of the multirotor system, and \(\mathbf{F}\) is the vector of forces acting on the body with respect to \(\mathcal{F}^{B}\), \(\mathbf{F}=\left[ \begin{array}{ccc}F_{X}&F_{Y}&F_{Z}\end{array} \right]^{ \mathrm{T}}\). Figure 1: Multirotor unmanned aerial vehicle (UAV) reference coordinate systems.