## 2 Mathematical Representation of Multirotor UAV Configurations
### Dynamics of a Multirotor UAV  
In order to obtain a representative aircraft model, in addition to mathematical description of multirotor configurations, a brief description of the multirotor UAV dynamic model is given. To begin with, it is necessary to define the aircraft coordinate system (called Body frame, \(\mathcal{F}^{\text{B}},\{\text{O}_{\text{B}},\text{ X}_{\text{B}},\text{ Y}_{\text{B}},\text{ Z}_{\text{B}}\}\)) as shown in Figure 2a. The Body frame is fixed to the multirotor UAV body where the origin coincides with the COG, and axes coincide with the multirotor main axes of inertia. In the Body frame, the velocities for each DOF of the aircraft are defined as \(\mathbf{v}=\left[ \begin{array}{cccccc}u&v&w&p&q&r\end{array} \right]^{\text{T}}\), respectively, translational \(\mathbf{v}^{\text{B}}=\left[ \begin{array}{cccccc}u&v&w\end{array} \right]^{ \text{T}}\) and rotational \(\mathbf{\omega}^{\text{B}}=\left[ \begin{array}{cccccc}p&q&r\end{array} \right]^ {\text{T}}\) velocities. It follows that the equations of motion are also defined in the Body frame. A system model of six second-order differential equations  
Figure 1: Generic conventional multirotor configurations. is obtained based on the Newton-Euler approach, described in more detail in previous research [27; 28]. The equations of motion given in the matrix form are defined by the following expression:  
\[\mathbf{M_{B}\dot{v}+C_{B}(v)v=\Lambda,} \tag{1}\]  
where \(\mathbf{M_{B}}\) represents a rigid body inertia matrix which comprises multirotor UAV mass \(m\), and the body's inertia matrix \(\mathbf{I}=\text{diag}\left\{ \begin{array}{cc}I_{xx},&I_{yy},&I_{zz}\\ \end{array} \right\}\). The next element on the left side of the equation represents the Coriolis and centripetal matrix \(C_{B}(v)\), which describe the inertial forces with respect to the rotating body frame [29]. To the right of the equation is the vector of forces and moments \(\Lambda\) acting on the aircraft rigid body. The dynamic of a multirotor UAV is influenced by the forces and moments of the environment and the propulsion of the aircraft. In this brief description of the dynamics, four elements are considered which affect the aircraft dynamics. Overall, the vector of forces and moments is given with the following expression:  
\[\Lambda=\mathbf{d}+\mathbf{g_{B}}+\mathbf{o_{B}}+\mathbf{u_{B}}, \tag{2}\]  
where \(\mathbf{d}\) is the disturbance vector which contains external disturbances, for instance, wind gusts and unmodelled dynamics. The gravitational force vector \(\mathbf{g_{B}}\) acts only on translational dynamics and can be accurately described since it acts in the vertical axis of the Earth frame. As a consequence of the rotors' velocities, a gyroscopic moment vector \(\mathbf{o_{B}}\) is included in the model which acts only on rotational dynamics. The last vector \(\mathbf{u_{B}}\) represents the forces and moments of the propulsion system with respect to the Body frame. Changing the rotor angular velocities directly affects the multirotor UAV dynamics. Therefore, the propulsion vector is called a control vector and consists of three forces \(\mathbf{f}=\left[ \begin{array}{ccc}f_{X}&f_{Y}&f_{Z}\\ \end{array} \right]^{\text{T}}\) which act on translational dynamics, and three moments \(\mathbf{\tau}=\left[ \begin{array}{ccc}\tau_{\Phi}&\tau_{\Phi}&\tau_{\Psi} \\ \end{array} \right]^{\text{T}}\) which act on rotational dynamics. The control vector \(\mathbf{u_{B}}=\left[ \begin{array}{ccc}f_{X}&f_{Y}&f_{Z}&\tau_{\Phi}&\tau_{ \Phi}&\tau_{\Psi}\\ \end{array} \right]^{\text{T}}\) comprises six elements, one for each DOF, and is given by the following expression:  
\[\mathbf{u_{B}}=\mathbf{\Gamma_{B}\Omega,} \tag{3}\]  
where \(\mathbf{\Gamma_{B}}\) is the matrix of the control allocation scheme. The matrix contains multirotor UAV configurations parameters which can be divided into propulsion geometric arrangement parameters and propulsion unit characteristics.