## 2 Mathematical Representation of Multirotor UAV Configurations
### Performance Analysis Procedure  
The distribution of propulsion aerodynamic forces and moments on the control vector of the aircraft depends primarily on the configuration of the geometric arrangement. Since the paper considers fully actuated configurations, a mathematical model shows that a full degree of actuation can be achieved by choosing proper configuration parameters. As can be seen from the propulsion units' characteristics, the dominant role in the aircraft dynamics is played by the thrust forces. The distribution of the thrust forces is determined by the orientation of the rotors which, alongside rotor position, determines the distribution of thrust forces and drag torques on the control moments. Configurations with passively tilted rotors are considered, whose signs of the tilt angles are chosen in order to achieve the full rank of the allocation matrix in a fully actuated system. When choosing a geometric arrangement, it is important to keep the balance of the aircraft, in other words, the static equilibrium. In this paper, a procedure for analyzing multirotor configuration parameters is proposed. Specifically, the influence of the geometric arrangement of passively tilted rotors on the distribution of thrust forces is numerically analyzed with the MATLAB software package. The analysis procedure is an iterative algorithm based on the implementation of a derived inverse control allocation scheme. In every iteration, aircraft forces, \(f_{\chi}\), \(f_{\chi}\) and \(f_{\mathrm{Z}}\), are independently and successively incremented until the calculated thrust force on any rotor exceeds the experimentally determined maximum value. A simplified flowchart of the analysis procedure is shown in Figure 6, where an iterative algorithm is accomplished with three nested loops, one for each force iteration. At the beginning of the iterative algorithm, the maximum value for \(f_{\mathrm{Z}}\) is obtained from the known maximal thrust force and rotors orientation. Since observed rotors cannot spin in the reverse direction, the \(f_{\mathrm{Z}}\) value will always be positive or 0 when the rotors do not rotate. While in the \(f_{\mathrm{Z}}\) loop, the algorithm will enter the \(f_{\chi}\) loop and increment \(f_{\chi}\) until the maximal thrust force is reached on any rotor. For every increment of \(f_{\chi}\), an \(f_{\mathrm{Y}}\) loop is entered, in which \(f_{\chi}\) is increased until the maximal thrust force is reached. Upon exiting the \(f_{\mathrm{Y}}\) loop, the values of forces in all three directions are saved as 3D point coordinates, respectively. When \(f_{\mathrm{Y}}\) and \(f_{\mathrm{X}}\) reach maximum values, \(f_{\mathrm{Z}}\) will increase and \(f_{\mathrm{X}}\) and \(f_{\mathrm{Y}}\) loops will be repeated again. When the maximum value of \(f_{\mathrm{Z}}\) is reached, the iteration is finished, and the 3D cloud of points is generated. The iterative algorithm shown in the simplified flowchart starts at forces set to zero and successively increases them. Hence, one run of the algorithm will provide the full results for \(f_{\mathrm{Z}}\), but only partial results for \(f_{\mathrm{X}}\) and \(f_{\mathrm{Y}}\) (as can be seen in Figure 7). Since multirotors can move forward and backward, and left and right, considering only positive values of \(f_{\mathrm{X}}\) and \(f_{\mathrm{Y}}\) will give limited results in only one quadrant. The iteration algorithm can easily be extended to results in all four quadrants by including another \(f_{\chi}\) loop below the existing one, which will now decrease \(f_{\chi}\) until the thrust limit is reached, and similarly, another for \(f_{\mathrm{Y}}\) loop which will now decrease for \(f_{\mathrm{Y}}\) and save the values of all three forces when the thrust force limit is reached on any rotor. Figure 5: Aerodynamic effects with respect to rotor angular velocity: (**a**) thrust force; (**b**) drag torque. However, for some cases, even limited results can provide the full information regarding the distribution of thrust forces.