# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 3 Results  
This section investigates the results obtained from the described analysis carried out for the above specified model of a transition eVTOL configuration. Figure 5 shows the results for the RMS and AMS acceleration set.  
The chosen discretization numbers were (\(n_{u}=5\), \(n_{v}=5\), \(n_{w}=5\), \(n_{\Phi}=10\), \(n_{\Theta}=10\), \(n_{\Theta}=10\), \(n_{\beta_{1}}=12\), \(n_{\dot{\beta}_{2}}=12\), \(n_{\dot{\beta}_{3}}=24\)), the airspeed envelope was defined via (\(u_{C,min}=-8\) m/s, \(v_{C,min}=-8\) m/s, \(w_{C,min}=-3\) m/s, \(u_{C,max}=8\) m/s, \(v_{C,max}=8\) m/s, \(w_{C,max}=3\) m/s), the maneuver attitude deviations were (\(\Phi_{min}=-20\) deg, \(\Phi_{max}=20\) deg, \(\Theta_{min}=-20\) deg, \(\Theta_{max}=20\) deg, \(\Phi_{overshoot}=5\)), the scales were chosen to (\(n_{z,scale}=2\), \(\dot{p}_{scale}=1000\) deg/s\({}^{2}\), \(\dot{q}_{scale}=1000\) deg/s\({}^{2}\), \(t_{scale}=120\) deg/s\({}^{2}\)) and the maneuverability limitations were chosen to (\(\Delta n_{z,max}=3/g\) m/s\({}^{2}\), \(\dot{p}_{max}=90\) deg/s\({}^{2}\), \(\dot{q}_{max}=90\) deg/s\({}^{2}\), \(\dot{q}_{max}=30\) deg/s\({}^{2}\)).  
For visualization we chose the six possible pair combinations of the four chosen acceleration channels and plotted the cross section of the corresponding channel pair plane with the hull defined via the query directions lying in the plane. This visualization provides a very fast and intuitive understanding of possible problematic directions. The purple area represents the maximum possible aerodynamic disturbance accelerations while the orange areas on top are the additional accelerations corresponding to required maneuverability. The sum of both yields the total RMS (red line). As the right bottom yaw/body z acceleration plot in Figure 5 shows, we can immediately identify the critical directions of requirement violation. The cuboid shape of the maneuverability required moment set leads to higher required moments than the AMS can provide in cases of  
Figure 5: Results of the calculation of attainable moment set (AMS) and required moment set (RMS) hulls.  
simultaneous yaw and body z acceleration demands. The controllability evaluation taking into account the RMS also shows clear advantages over the solely AMS-based evaluation here, as it directly identifies the critical direction that would not be available for a purely AMS-based evaluation.  
Another important effect which needs to be investigated when assessing the proposed methods is the effect of resolutions for calculating the control sets. The first important resolution to mention here is the number of chosen query directions to span the control sets \(n_{\beta_{1}},n_{\beta_{2}},n_{\beta_{3}}\). Table 1 shows the controllability analysis results for different chosen values corresponding to \(3.75^{\circ}\), \(7.5^{\circ}\), \(15^{\circ}\) and \(30^{\circ}\) resolution.  
To assess the controllability, three measures are shown. The first two measures are the minimum and mean value of the control set margins from Equation (32). The third measure is the percentage of control set query directions for which the attainable control is not greater or equal than required control \(m<0\). Decreasing the control set resolution down to \(3.75^{\circ}\) also increases the accuracy of the controllability measures.  
Another important resolution effect to investigate is the chosen resolution of bank and pitch angles when evaluating the disturbance set. Table 2 shows the controllability analysis results for various resolutions in the pitch and bank angle plane.  
When the corresponding resolution is increased, all three controllability measures converge. This shows that the method of assembling the aerodynamic disturbance set points via linearly spacing is suitable to obtain worst case estimates of the aerodynamic disturbance for the given simulation model.