# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 2 Materials and Methods
### Underlying Aircraft Simulation Model  
The use of a simulation model-based assessment can be applied already in early aircraft design and can help lower development costs. One of the main applications of aircraft simulation models is model-based control design as in [9; 10]. In order to guarantee a close match to the real aircraft system, identification can be used to improve the accuracy of the models [11; 12].  
To demonstrate the RMS approach with the methods of this paper, a simulation model of a lift and cruise multirotor UAV is used. As the methods used in this paper do not require time simulation, this section only shows the force and moment balance parts of the equations of motion. The translational and rotational equations of motion read [13].  
Figure 1: Overview of required and attainable moment sets and the proposed methods for obtaining the required moment set.  
\[\begin{bmatrix}u_{K}^{G}\\ \dot{v}_{K}^{G}\\ \dot{w}_{K}^{G}\end{bmatrix}_{B}^{II}=\frac{1}{m}\begin{bmatrix}X_{A}^{G}\\ Y_{A}^{G}\\ Z_{A}^{G}\end{bmatrix}_{B}+\frac{1}{m}\begin{bmatrix}X_{P}^{G}\\ Y_{P}^{G}\\ Z_{P}^{G}\end{bmatrix}_{B}+\begin{bmatrix}-\sin(\Theta)\\ \cos(\Theta)\sin(\Phi)\\ \cos(\Theta)\cos(\Phi)\end{bmatrix}_{B}g,\\ \begin{bmatrix}\dot{q}\\ \dot{r}\end{bmatrix}=(I_{BB}^{G})^{-1}\begin{bmatrix}L_{A}^{G}\\ M_{A}^{G}\\ N_{A}^{G}\end{bmatrix}_{B}+(I_{BB}^{G})^{-1}\begin{bmatrix}L_{P}^{G}\\ M_{P}^{G}\\ N_{P}^{G}\end{bmatrix}_{B}-(I_{BB}^{G})^{-1}\begin{bmatrix}p\\ q\\ r\end{bmatrix}_{B}\times I_{BB}^{G}\begin{bmatrix}p\\ q\\ r\end{bmatrix}_{B}, \tag{2}\]  
where \(u,v,w\) denote the three velocity components, \(K\) the kinematic nature, \(G\) the center of gravity, the index \(II\) the inertial acceleration, \(m\) the vehicle mass, \(X,Y,Z\) and \(L,M,N\) forces and moments due to aerodynamics \(A\) or propulsion \(P\). The inertia around the center of gravity in the body frame is denoted with \(I_{BB}^{G}\).  
In order to provide an aerodynamic model for flow from all possible directions, a multi-compartment aerodynamic model comparable to [14] is used. Assembling all forces and moments in the center of gravity then yields  
\[\begin{bmatrix}X_{A}^{G}\\ Y_{A}^{G}\\ Z_{A}^{G}\end{bmatrix}_{B}=\frac{1}{2}\rho V_{A}^{2}S_{ref}\begin{bmatrix}C_{X}( \alpha_{A},\beta_{A})\\ C_{Y}(\alpha_{A},\beta_{A})\\ C_{Z}(\alpha_{A},\beta_{A})\end{bmatrix}_{B}, \tag{3}\] \[\begin{bmatrix}L_{A}^{G}\\ M_{A}^{G}\\ N_{A}^{G}\end{bmatrix}_{B}=\frac{1}{2}\rho V_{A}^{2}S_{ref}\begin{bmatrix}sC_{ I}(\alpha_{A},\beta_{A})\\ \varepsilon C_{m}(\alpha_{A},\beta_{A})\\ sC_{n}(\alpha_{A},\beta_{A})\end{bmatrix}_{B}, \tag{4}\]  
where \(S_{ref}\) denotes the wing reference area, \(s\) the wing span, \(\bar{c}\) the chord, \(V_{A}\) the magnitude of the true airspeed and \(C_{i}\) the force/moment coefficients. The dependency of the coefficients with respect to the aerodynamic angle of attack \(\alpha_{A}\) and angle of sideslip \(\beta_{A}\) are realized with lookup tables [15].  
The multirotor propulsion system is modeled independent of the aerodynamics; propulsion airframe interactions are neglected. For each propulsive unit, the propeller forces and moments are modelled as linear in the squares of the angular rates [16; 17]  
\[T_{i} =C_{T}D^{4}\rho\frac{1}{4\pi^{2}}\omega_{i}^{2}, \tag{5}\] \[N_{i} =C_{N}D^{5}\rho\frac{1}{4\pi^{2}}\omega_{i}^{2}, \tag{6}\]  
where \(T_{i}\) denotes the thrust of the \(i\)th propeller, \(N_{i}\) the corresponding torque around the corresponding rotational axis, \(\omega_{i}\) the rotational rate in rad/s and \(D\) the propeller diameter. As the aircraft only has propellers of the same type the thrust coefficient \(C_{T}\) and the torque coefficient \(C_{N}\) are the same for all propellers.