## 2 Mathematical Representation of Multirotor UAV Configurations
### Multirotor Configuration Control Allocation Scheme
The selection of components is an important step in the design of the aircraft since the dynamics of the aircraft depend on the characteristics of the components, as well as its energy consumption. Very often, component manufacturers provide detailed specifications for the recommended propeller and motor setups. BLDC motor is a permanent-magnet electric motor where electromagnets (armature) are located on the stator of the motor, while permanent magnets are located on the driven rotor shell. This type of motor is driven by ESC which represents an electronic commutation system with rectangular-shaped voltage signals. The ESC is an integrated power inverter that processes the input PWM signal from the FC in order to set the adequate speed of the transistors' switching sequence. It converts input DC voltage into the appropriate phase voltages by which it energizes particular phases of a motor and achieves the desired rotor angular velocity. With a higher duty cycle of the PWM signal, the voltage is increased, therefore rotor angular velocity also increases. Once the geometric arrangement of the rotor and the characteristics of the rotor are described, the force and moment vectors of the \(i\)-th rotor, with respect to the Body frame, can be defined. The rotor force vector is obtained by mapping the rotor thrust force through the orientation vector, and is given by the expression:  
\[\mathbf{f}_{i}=\big{(}k_{\mathbf{f}_{i}}\eta_{\mathbf{R}_{i}}\big{)}\omega_{i}{ }^{2} \tag{8}\]  
The rotor moment vector consists of two components, where the first one arises from the action of the thrust force and the second one arises from the action of the drag torque. The moment vector of the \(i\)-th rotor is derived using the matrix representation of the vector product \(\mathbf{a}\times\mathbf{b}=\mathbf{S}(\mathbf{a})\mathbf{b}\):  
\[\mathbf{\tau}_{i}=\big{(}k_{\mathbf{f}_{i}}\mathbf{S}\big{(}\mathbf{\xi}_{ \mathbf{R}_{i}}\big{)}\eta_{\mathbf{R}_{i}}+k_{\mathbf{\tau}_{i}}\eta_{\mathbf{ R}_{i}}\big{)}\omega_{i}{}^{2} \tag{9}\]  
The sign of drag torque factor \(k_{\mathbf{\tau}_{i}}\) depends on the rotor angular velocity direction. Rotors with a CW direction have a positive sign, while CCW rotors have a negative sign. The control allocation matrix (\(\mathbf{\Gamma}_{\mathbf{B}}\in\mathbb{R}^{6\times N}\)), which contains the propulsion geometric arrangement parameters and the EPU characteristics, is presented with the following expression:  
\[\mathbf{\Gamma}_{\mathbf{B}}=\left[\begin{array}{cccc}k_{\mathbf{f}_{1}} \eta_{\mathbf{R}_{1}}&\cdots&k_{\mathbf{f}_{N}}\eta_{\mathbf{R}_{N}}\\ k_{\mathbf{f}_{1}}\mathbf{S}\big{(}\mathbf{\xi}_  
vector and, as a result, the matrix rank is not of full order. Regardless of the number of rotors, such configurations are underactuated systems. The following matrix describes a conventional X-shaped hexarotor (X6):  
\[\mathbf{\Gamma}_{\text{X6}}=\left[ \begin{array}{cccccc}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ k_{\text{f}}&k_{\text{f}}&k_{\text{f}}&k_{\text{f}}&k_{\text{f}}&k_{\text{f}}&k_{ \text{f}}\\ -k_{\text{f}}\text{l}&k_{\text{f}}\text{l}&\frac{1}{2}k_{\text{f}}\text{l}&- \frac{1}{2}k_{\text{f}}\text{l}&-\frac{1}{2}k_{\text{f}}\text{l}&\frac{1}{2}k_{ \text{f}}\text{l}\\ 0&0&-\frac{\sqrt{3}}{2}k_{\text{f}}\text{l}&\frac{\sqrt{3}}{2}k_{\text{f}}\text{l }&-\frac{\sqrt{3}}{2}k_{\text{f}}\text{l}&\frac{\sqrt{3}}{2}k_{\text{f}}\text{l }\\ k_{\text{\tau}}&-k_{\text{\tau}}&k_{\text{\tau}}&-k_{\text{\tau}}&-k_{\text{ \tau}}&k_{\text{\tau}}\end{array} \right] \tag{11}\]  
The considered configurations have the same length (l) as the rotor arms on which the same EPUs with the factors of thrust force (\(k_{\text{f}}\)) and drag torque (\(k_{\text{\tau}}\)) are mounted. Regarding the control of conventional configurations, there are two basic cases. The first case is related to remote control where the operator controls thrust, roll, pitch, and yaw. The second case is related to the autonomous flight and position control [9] where aircraft position is controlled by changing propulsion moments.