## 3 Model and Research  
### Coordinates System and Kinematics  
The UAV kinematics model is designed to perform the simulation. The number of independent auxiliary relationships is equal to the difference between the number of coordinates in the set and six; all coordinate systems are Cartesian and right-handed NED coordinate systems. As a result, the primary vector of states looks like this:  
\[\mathrm{P}= \begin{bmatrix}\mathrm{X_{e}}&\mathrm{O_{b}}\end{bmatrix} = \begin{bmatrix} \mathrm{X}&\mathrm{Y}&\mathrm{Z}&\varphi&\theta&\psi\end{bmatrix}  \tag{1}\]  
where \(\mathrm{X_{e}}= \begin{bmatrix}\mathrm{X}&\mathrm{Y}&\mathrm{Z}\end{bmatrix} \) is the system coordinate in the earth frame, which is a vector of dimension \(3\times 1\), with dimension in meters. \(\mathrm{O_{b}}= \begin{bmatrix}\varphi&\theta&\psi\end{bmatrix} \), representing the orientation of the body frame relative to the earth frame, is described using the Euler angles \(3\times 1\) in radians. The kinematic model from [8] is used with some assumptions as a basis for the model. It is important to note that, in his work, the author describes the kinematics and orientation control of the quadcopter. To describe the navigation, the position of the aircraft is added to the model. As a result, the complete kinematic model is summarized in (2) and expanded in (3). \[ \begin{cases}\overline{\mathrm{F_{b}}}=m\left(\dot{\overline{\mathrm{V_{b}}}} +\overline{\omega}\times\overline{\mathrm{V_{b}}}\right)\\ \overline{\mathrm{T}}=\mathrm{I}\overline{\omega}+\overline{\omega}\times \left(\mathrm{I}\ast\overline{\omega}\right)\end{cases}  \tag{2}\]  
where \(\overline{\mathrm{V_{b}}}\) is a linear velocity vector, \(\overline{\omega}\) is angular rate vector, \(\mathrm{m}\) is the mass of the aircraft and \(\mathrm{F}\) denotes the force vector. There, \(\mathrm{I}\) is an aircraft inertia matrix. When considering the vector \(\overline{\mathrm{V_{b}}}\) defined for all components in the direction x, y and z and \(\overline{\omega}\) for roll \(\varphi\), pitch \(\theta\) and yaw \(\psi\). The following system (3) describes the flight dynamics of aircraft. \[ \begin{bmatrix}\dot{\mathbf{V_{bx}}}\\ \dot{\mathbf{V_{by}}}\\ \dot{\mathbf{V_{bz}}}\\ \dot{\mathbf{v_{x}}}\\ \dot{\mathbf{\omega_{y}}}\\ \dot{\mathbf{\omega_{z}}}\end{bmatrix} = \begin{bmatrix}\frac{\mathbbm{1}}{ \mathbbm{m}}\mathbf{F_{bx}}+\left(\mathbf{\omega_{z}}\mathbf{V_{by}}-\mathbf{ \omega_{y}}\mathbf{V_{bz}}\right)\\ \frac{\mathbbm{1}}{\mathbbm{m}}\mathbf{F_{by}}+\left(\mathbf{\omega_{x}} \mathbf{V_{bz}}-\mathbf{\omega_{z}}\mathbf{V_{bx}}\right)\\ \frac{\mathbbm{1}}{\mathbbm{r}}\mathbf{F_{bz}}+\left(\mathbf{\omega_{y}} \mathbf{V_{bx}}-\mathbf{\omega_{x}}\mathbf{V_{by}}\right)\\ \frac{\mathbbm{1}}{\mathbbm{r}}\mathbf{T_{x}}+\left(\frac{\mathbbm{1}- \mathbbm{1}}{\mathbbm{r}}\right)\mathbf{\omega_{z}}\mathbf{\omega_{y}}\\ \frac{\mathbbm{1}}{\mathbbm{r}}\mathbf{T_{y}}+\left(\frac{\mathbbm{1}- \mathbbm{1}}{\mathbbm{r}}\right)\mathbf{\omega_{x}}\mathbf{\omega_{z}}\\ \frac{\mathbbm{1}}{\mathbbm{r}}\mathbf{T_{z}}+\left(\frac{\mathbbm{1}- \mathbbm{1}}{\mathbbm{r}}\right)\mathbf{\omega_{x}}\mathbf{\omega_{y}}\end{bmatrix}  \tag{3}\]  
The system (3) presents UAV kinematics in general form. The body forces and the torques are generated via three different sources, which are standard for all flying objects: gravity force, motor forces and aerodynamics. The last two sources are typical for the control of an aircraft, but the system under study uses only motors for this aim. The following Equation (4) presents the described above forces and torques in the system. \[\left\{ \begin{array}{c}\overline{\mathrm{F_{b}}}=\overline{\mathrm{F_{mb}}} +\overline{\mathrm{F_{a}}}+\overline{\mathrm{C_{b}}}\\ \overline{\mathrm{T}}=\overline{\mathrm{T_{mb}}}+\overline{\mathrm{T_{a}}} \end{array}