## 2 Preliminary Description of a Multirotor UAV
### Multirotor UAV Propulsion System  
A common feature of all multirotor design variants is that they consist of \(N\) propulsion units (rotors). The multirotor performance is affected by the geometric arrangement and power of the propulsion units. Conventional multirotor configurations (so-called planar configurations) consist of an even number of propulsion units symmetrically arranged in one or more parallel planes. Additionally, half of the rotors turn in a clockwise (CW) direction, while the other half turns counterclockwise (CCW) in order to cancel the reactive moment around \(Z_{B}\). There are also configurations with the nonplanar geometric arrangement, where it is possible to achieve a full actuation of multirotor [21].  
The propulsion system configuration is defined with a control allocation scheme that describes the mapping of the rotor angular velocities to the multirotor control vector. The rows of the control allocation matrix,  
\[\boldsymbol{\Gamma}_{B}=\left[\begin{array}{ccc}k_{f_{1}}\eta_{R_{1}}&\dots &k_{f_{N}}\eta_{R_{N}}\\ k_{f_{1}}\mathbf{S}\Big{(}\underline{\epsilon}_{R_{1}}\Big{)}\eta_{R_{1}}+k_{ \tau_{1}}\eta_{R_{1}}&\dots&k_{f_{N}}\mathbf{S}\Big{(}\underline{\epsilon}_{ R_{N}}\Big{)}\eta_{R_{N}}+k_{\tau_{N}}\eta_{R_{N}}\end{array}\right], \tag{9}\]  
represent the degrees of freedom of motion, while the columns represent the rotors (propulsion units). Propulsion units are defined by two characteristic aerodynamics factors: thrust force (\(k_{f_{1}}\)) and drag torque (\(k_{\tau_{i}}\)). The geometric arrangement of each rotor (see Figure 2) is defined by the position \(\underline{\epsilon}_{R_{1}}\) and orientation \(\eta_{R_{i}}\).  
The position vector of the \(i\)-th rotor is given by the following expression  
\[\xi_{R_{i}}=\mathbf{R}_{R_{i}}{}^{\mathrm{T}}(\chi_{i},\ Z_{B})\left[\begin{array} []{c}l_{i}\\ 0\\ 0\end{array}\right], \tag{10}\]  
where \(\chi_{i}\) is the position angle of the \(i\)-th rotor defined in the horizontal plane of the aircraft (\(X_{B}Y_{B}\)), and \(l_{i}\) is the length of the \(i\)-th rotor arm. The orientation vector of the \(i\)-th rotor is given by the following expression  
\[\eta_{R_{i}}=\ \mathbf{R}_{R_{i}}{}^{\mathrm{T}}(\chi_{i},Z_{B})\mathbf{R}_{R_{i}}{ }^{\mathrm{T}}\!\left(\beta_{i},\ Y_{R_{i}}\right)\mathbf{R}_{R_{i}}{}^{ \mathrm{T}}\!\left(\gamma_{i},\ X_{R_{i}}\right)\mathbf{e}_{3}, \tag{11}\]  
where \(\beta_{i}\) is the cant angle of the \(i\)-th rotor, \(\gamma_{i}\) is the tilt angle of the \(i\)-th rotor, and \(\mathbf{e}_{3}=\left[\begin{array}{ccc}0&0&1\end{array}\right]^{\mathrm{T}}\) is a unit vector since the considered aerodynamic effects are represented in the rotor vertical axis.  
In order to separate the parameters of the geometric arrangement and the parameters of the propulsion units, a decomposition of the control allocation matrix is given as  
\[\mathbf{\Gamma}_{B}=\left[\begin{array}{c}\mathbf{H}\\ \Xi\end{array}\right]\mathbf{k}_{f}+\left[\begin{array}{c}\mathbf{0}_{3\times N }\\ \mathbf{H}\end{array}\right]\mathbf{k}_{\tau}. \tag{12}\]  
Multirotor configuration geometric arrangement is represented with \(\mathbf{H}=\left[\begin{array}{ccc}\eta_{R_{1}}&\cdots&\eta_{R_{N}}\end{array}\right]\) and \(\Xi=\left[\begin{array}{ccc}\mathbf{S}\!\left(\!\xi_{R_{1}}\!\right)\!\eta_ {R_{1}}&\cdots&\mathbf{S}\!\left(\!\xi_{R_{N}}\!\right)\!\eta_{R_{N}}\end{array}\right]\). The propulsion unit parameters are represented by a diagonal matrix of rotor thrust force factors  
\[\mathbf{k}_{f}=\mathrm{diag}\!\left\{\begin{array}{ccc}k_{f_{1}}&k_{f_{2}}& \ldots&k_{f_{N}}\end{array}\right\}\!, \tag{13}\]  
and a diagonal matrix of the rotor drag torque factors  
\[\mathbf{k}_{\tau}=\mathrm{diag}\!\left\{\begin{array}{ccc}k_{\tau_{1}}&k_{ \tau_{2}}&\ldots&k_{\tau_{N}}\end{array}\right\}\!. \tag{14}\]  
The sign of the drag torque factor depends on the direction of rotation of the rotor, CW rotors have a positive sign, while CCW rotors have a negative sign.