# Changing the Formations of Unmanned Aerial Vehicles
## 3 Small Unmanned Aerial Vehicle
### The Mathematical Model of the Quadrotor  
The quadrotor structure is presented in Figure 4, including the corresponding forces, torques, and orientation angles (pitch--\(\theta\), roll--\(\phi\), and yaw--\(\psi\)). Forces and torques are generated in the body frame (Ox\({}_{\text{b}}\)y\({}_{\text{b}}\)z\({}_{\text{b}}\)) whereas the orientation angles are described as relations between the body frame and the Earth frame (Oxyz). The quadrotor has six degrees of freedom, so the movement of the quadrotor is rewritten by six equations (three translation motions and three rotation motions) [26].  
The translation motions are rewritten as follows [26; 27]:  
\[\text{m}(\mathbf{V_{B}}+\mathbf{\omega}\times\mathbf{V_{B}})=\mathbf{F_{b}}, \tag{1}\]  
Here, m is the mass of the quadrotor, \(\mathbf{F_{b}}=\begin{bmatrix}\text{F_{x}}&\text{F_{y}}&\text{F_{z}}\end{bmatrix}^ {\text{T}}\) is the force vector in the body frame, \(\mathbf{\omega}=\begin{bmatrix}\omega_{\text{x}}&\omega_{\text{y}}&\omega_{\text{ z}}\end{bmatrix}^{\text{T}}\) denote the angular rates in the body frame, and \(\mathbf{V_{b}}=\begin{bmatrix}\text{v}_{\text{xb}}&\text{v}_{\text{yb}}& \text{v}_{\text{xb}}\end{bmatrix}^{\text{T}}\) is the velocity in the body frame.  
The rotation motions are rewritten as follows [26; 27]:  
\[\text{I}\dot{\mathbf{\omega}}+\mathbf{\omega}\times\text{I}\mathbf{\omega}=\mathbf{M_{b}}, \tag{2}\]  
Here, \(\mathbf{M_{b}}=\begin{bmatrix}\text{M_{x}}&\text{M_{y}}&\text{M_{z}}\end{bmatrix}^ {\text{T}}\) is the vector of moments in the body frame and I is the matrix moment of inertia:  
\[\text{I}=\begin{bmatrix}\text{I_{xx}}&-\text{I_{yy}}&-\text{I_{xz}}\\ -\text{I_{yx}}&\text{I_{yy}}&-\text{I_{yz}}\\ -\text{I_{xz}}&-\text{I_{zy}}&\text{I_{zz}}\end{bmatrix}.\]  
The relationship between the angular velocity \(\mathbf{\omega}\) and rate of change of Euler angles (pitch, roll, and yaw) \(\dot{\mathbf{\eta}}=\begin{bmatrix}\dot{\phi}&\dot{\theta}&\dot{\psi}\end{bmatrix}^ {\text{T}}\) is depicted below:  
\[\dot{\mathbf{\eta}}=\text{J}\mathbf{\omega}.\]  
Figure 4: Body frame (Ox\({}_{\text{b}}\)y\({}_{\text{b}}\)z\({}_{\text{b}}\)) and Earth frame (Oxyz).  
Here,  
\[\mathbf{J}=\begin{bmatrix}1&\text{sin}\phi\text{tan}\theta&\text{cos}\phi\text{tan} \theta\\ 0&\text{cos}\phi&-\text{sin}\phi\\ 0&\text{sin}\phi&\text{cos}\theta&\text{cos}\phi/\text{cos}\theta\end{bmatrix}.\]  
The resultant thrust is created as a sum of forces generated by the driver units:  
\[\text{F}_{\text{w}}=\sum_{\text{i}=1}^{4}\text{F}_{\text{i}}.\]  
So, the vector force from Equation (1) is equal to the following:  
\[\mathbf{F_{b}}=\begin{bmatrix}0&0&\text{F}_{\text{w}}\end{bmatrix}^{\text{T}}.\]  
The moment vector from Equation (2) is equal to the following:  
\[\text{M}=\begin{bmatrix}\text{r}(\text{F}_{1}+\text{F}_{2}-\text{F}_{3}-\text{ F}_{4})\\ \text{r}(\text{F}_{1}-\text{F}_{2}-\text{F}_{3}+\text{F}_{4})\\ \sum_{\text{i}=1}^{4}\text{M}_{\text{i}}\end{bmatrix}. \tag{3}\]  
Here, r is the radius of the action force and \(M_{i}\) is the torque around the driver unit axis. The last element in Equation (1) is the influence of gravity on the quadrotor.