## 2 Overview of Multirotor UAVs  
### Multirotor UAV Configuration  
Multirotor UAVs are aerial vehicles that employ more than two rotors with fixed pitch spinning blades, so-called propellers. The spinning of each propeller through the air produces aerodynamic forces that are proportional to the square of their rotation rate \(\omega\). The thrust force \(f\) acts along the propeller's axis, where the drag moment \(\tau_{d}\) acts about the propeller's axis [12]. The thrust force of the \(i^{\text{th}}\) propeller is modeled as:  
\[f_{i}=k_{t}\omega_{i}^{2}, \tag{1}\]  
where \(k_{t}\) is thrust coefficient defined by propeller geometric characteristics. The drag moment that is generated in reaction to the air resistance around the propeller is given as:  
\[\tau_{i,d}=k_{d}\omega_{i}^{2}, \tag{2}\]  
where \(k_{d}\) is a constant of drag coefficient defined by propeller geometric characteristics. The rotors' number, geometrical distribution, and orientation characterize multirotor UAV configurations, as shown in Figure 1. The convectional design has single propellers arranged with an even number and alternating spinning directions to balance out the drag moment generated about the vertical axis of the airframe plane. However, according to design requirements, such as power consumption, size, weight, control ease, payload, and growing application in tasks requiring long flight time and complex maneuvers, various configurations of multirotor UAVs have been constructed. The limitation of the conventional design was resolved by introducing unconventional designs characterized by overlapping propellers and the nonparallel arrangement of propellers. In all multirotor configurations, the generalized effect of aerodynamic forces generated from each propeller on the overall airframe is governed by the propeller's position and orientation. Therefore, it is necessary to define the propeller's position and orientation relative to the origin of the body frame. The position of \(i^{\text{th}}\) propeller \(x_{i}\) can be given as:  
\[x_{i}= \begin{bmatrix}\cos\theta_{i}\\ \sin\theta_{i}\\ 0\end{bmatrix} \times l_{i} \tag{3}\]  
Figure 1: Various layouts of multirotor UAVs. where \(\theta_{i}\) is the \(i^{th}\) propeller position angle about the \(Z_{b}\) axis in the horizontal \((X_{b}Y_{b})\) aircraft plane, which is formed by the arm with a length \(l_{i}\) and \(X_{b}\) of the right-hand body coordinate. The orientation of the \(i^{\text{th}}\) propeller can be given as:  
\[q_{i}=\mathbf{R}_{Z_{b}}(\theta_{i})\mathbf{R}_{Y_{p}}\big{(}\alpha_{i,y}\big{)}\mathbf{R}_ {x_{p}}(\alpha_{i,x})e_{3} \tag{4}\]  
where \(\mathbf{R}_{Z_{b}}(\cdot)\) is the rotation matrix for the arm rotation \(\theta_{i}\) about \(Z_{b}\) the axis of the body frame; \(\mathbf{R}_{Y_{p}}(\cdot)\) is the rotation matrix for the propeller rotation \(\alpha_{i,y}\) about \(Y_{p}\) of the propeller coordinate; \(\mathbf{R}_{x_{p}}(\cdot)\) is the rotation matrix for the propeller rotation \(\alpha_{i,x}\) about \(Y_{p}\) of propeller coordinate; and \(e_{3}\) is a unit vector. The detailed computations and descriptions of the rotation matrices \(\mathbf{R}\), the generalized propeller's position matrix \(x\), and the orientation matrix \(q\) are presented in Appendix A. A vertically orientated propeller (\(\alpha_{i,y}=\alpha_{i,x}=0\)) applies all its generated force to lift the vehicle, and tilting the propellers results in the vectorization of vertical thrust into lateral force along the plane of the airframe. Moment force about the body frame is generated by virtue of the propellers being positioned some distance from the center of mass.