## 2 Mathematical Representation of Multirotor UAV Configurations
### Multirotor Configuration Control Allocation Scheme
To achieve the full rank of the allocation matrix, it is necessary to accordingly select the parameters of the configuration geometric arrangement, specifically the parameters of the rotor orientation. The allocation matrix for a hexarotor with passively tilted rotors (PTX6) is given:  
\[\mathbf{\Gamma}_{\text{PTX6}}=\left[\begin{array}{cccccc}k_{\text{f}}\text{ s}_{\text{\gamma}}&k_{\text{f}}\text{s}_{\text{\gamma}}&-\frac{1}{2}k_{\text{f}} \text{s}_{\text{\gamma}}&-\frac{1}{2}k_{\text{f}}\text{s}_{\text{\gamma}}&- \frac{1}{2}  
for the selected propulsion units. Five setups of EPUs shown in Table 1 were selected and further considered for the purposes of configuration parameter analysis. In this paper, the characterization of EPU was performed by the methodology presented in the last phase of the research [27]. The characterization can be based on the manufacturer's specifications, and for a more precise procedure, it is better to perform experimental measurements. The first step of characterization is the identification of parameters that can be divided into mechanical and electrical quantities. Mechanical quantities are further considered, which are related to aerodynamic effects and rotor revolutions per minute (RPM), e.g., angular velocity. Experimental measurements were performed using a RCbenchmark 1580 measurement system [30]. After processing, the measurement results are saved in data series and displayed as static maps. Figure 4 shows static maps of aerodynamic effects as functions of the input PWM signal. Respectively, Figure 4a shows the thrust forces, and Figure 4b the drag torques, for considered EPU setups. It can be seen that the thrust forces are the dominant aerodynamic effect which is an important aspect for the further implementation of parameter analysis. As expected, EPU setups with lower K\({}_{\
u}\), paired with larger propeller diameters, achieve higher aerodynamic forces and torques. The next step is the EPU characterization, which is carried out on the basis of static maps for five considered EPU setups. In this step, the aim is to present and estimate the factors of aerodynamic effects: the thrust force factor \(k_{\text{f}}\) and the drag torque factor \(k_{\tau}\), respectively. Figure 5 shows thrust force and drag torque characteristics with respect to rotor angular velocity. As can be seen from the obtained curves, the thrust forces and drag torques are approximately proportional to the square of the rotors' angular velocities; thus validating Equations (6) and (7). This fulfills the prerequisites for conducting a configuration performance analysis and enables computer simulations of the multirotor UAV behavior. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Setup**} & \multicolumn{2}{c}{**Fixed-Pitch Propeller**} & \multicolumn{2}{c}{**BLDC Motor**} & \multirow{2}{*}{**ESC**} & \multirow{2}{*}{**Energy Source**} \\ \cline{2-2} \cline{5-6}  & **Diameter** & **Pitch** & **Designation** & **Kv** & \\ \hline
1 & 7′′ & 2.4′′ & MN1806 & 1400 & Air 10 A & 12 V \\
2 & 9′′ & 4′′ & MN2214 & 920 & Afro 20 A & 12 V \\
3 & 10′′ & 4′′ & MN2214 & 920 & Afro 20 A & 12 V \\
4 & 17′′ & 6′′ & MN4014 & 400 & Air 40 A & 45 LiPo \\
5 & 17′′ & 6′′ & MN4014 & 400 & Air 40 A & 65 LiPo \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Considered EPU setups. Figure 4: EPU static maps: (**a**) thrust force; (**b**) drag torque.