## 2 Mathematical Representation of Multirotor UAV Configurations
### Multirotor Configuration Control Allocation Scheme  
Propellers mounted on rotors by their rotation generate the aerodynamic forces and moments required for the multirotor UAV motion. The control allocation scheme defines the multirotor configuration in a way that describes the mapping of the rotors' angular velocities to the aircraft control vector. The multirotor propulsion configuration, which consists of N rotors, is determined by the rotors' geometric arrangement parameters (EPUs) and their characteristics. The geometric arrangement is defined by the vector of the position and orientation of each rotor. For this purpose, it is necessary to determine for each rotor its coordinate system (\(\mathcal{F}^{\mathrm{R}_{i}}\), \(\left\{\mathrm{O}_{\mathrm{R}_{i}},\ X_{\mathrm{R}_{i}},\ Y_{\mathrm{R}_{i}}, \ Z_{\mathrm{R}_{i}}\right\}\)) whose \(\mathrm{X}_{\mathrm{R}_{i}}\mathrm{Y}_{\mathrm{R}_{i}}\) plane is parallel to the aircraft plane \(\mathrm{X}_{\mathrm{B}}\mathrm{Y}_{\mathrm{B}}\), as it is shown in Figure 2a. The propulsion configuration position vector of \(i\)-th rotor \(\xi_{\mathrm{R}_{i}}\) is defined by the length of the rotor arm \(\mathrm{I}_{i}\) and the angle of the rotor arm \(\mathrm{X}_{i}\), as shown in Figure 2. The equation is defined based on a rotation matrix \(\mathbf{R}_{\mathrm{R}_{i}}{}^{\mathrm{T}}(\mathrm{X}_{i},\ Z_{\mathrm{B}})\) which describes the rotation of the rotor arm around the aircraft axis \(Z_{\mathrm{B}}\) for the angle \(\mathrm{X}_{i}\). The position vector of the \(i\)-th rotor is given by the following expression:  
\[\xi_{\mathrm{R}_{i}}=\mathbf{R}_{\mathrm{R}_{i}}{}^{\mathrm{T}}(\mathrm{X}_{ i},\ Z_{\mathrm{B}})\left[ \begin{array}{c}\mathrm{I}_{i}\\ 0\\ 0\end{array} \right]. \tag{4}\]  
This expression is characteristic of configurations whose rotors are bblocated in the \(\mathrm{X}_{\mathrm{B}}\mathrm{Y}_{\mathrm{B}}\) plane, which is typical for conventional configurations with a planar arrangement in one plane. If the rotors are arranged on parallel planes, then the third coordinates of the input vector depend on the distance of the rotor's plane \(\mathrm{X}_{\mathrm{R}_{i}}\mathrm{Y}_{\mathrm{R}_{i}}\) from the aircraft plane \(\mathrm{X}_{\mathrm{B}}\mathrm{Y}_{\mathrm{B}}\) as \(\left[ \begin{array}{cc}\mathrm{I}_{i}&0&\mathrm{z}_{\mathrm{R}_{i}}\end{array}  \right]^{\mathrm{T}}\). As previously described, conventional configurations are characterized by a planar arrangement of rotors which means that the rotational axes of all the rotors are parallel to the aircraft axis \(Z_{\mathrm{B}}\); therefore, the orientation vector of every rotor is a unit vector \(\eta_{\mathrm{R}_{i}}=\mathbf{e}_{3}=\left[ \begin{array}{cc}0&0&1\end{array}  \right]^{\mathrm{T}}\). Since there are also non-planar configurations, where special emphasis is placed on fully actuated configurations, it is necessary to define the orientation vector. The orientation vector definition is given by the following expression:  
\[\eta_{\mathrm{R}_{i}}=\mathbf{R}_{\mathrm{R}_{i}}{}^{\mathrm{T}}(\mathrm{X}  
same angular velocity (RPM), the rotor assembly will create a greater thrust force which is modeled as proportional to the square of the rotor angular velocity \(\omega_{i}\). The thrust force of the \(i\)-th rotor is defined with the following expression:  
\[f_{\mathbf{R}_{i}}=k_{\mathbf{f}_{i}}\omega_{i}{}^{2}, \tag{6}\]  
where \(k_{\mathbf{f}_{i}}\) is the thrust force factor which depends on the propeller geometry and the air density \(\rho\). As a consequence of the greater thrust force, the resistance to rotation is greater, therefore producing a greater motor drag torque. The load on the \(i\)-th motor is imposed by drag torque and can be defined with the following expression:  
\[\tau_{\mathbf{R}_{i}}=k_{\mathbf{\tau}_{i}}\omega_{i}{}^{2}, \tag{7}\]  
where \(k_{\mathbf{\tau}_{i}}\) is the drag torque factor which depends on the geometry of the propeller, the air density, and the motor power.