# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 2 Materials and Methods
### Trim-Based Calculation of Aerodynamic Disturbance Set
Let \(n_{V}\) denote the number of representative points on the boundary of the airspeed envelope. Figure 3 shows exemplary representative query points in 2D for an ellipsoidal required aerodynamic velocity envelope. As the operation of the aircraft is considered around a steady trim state and during the transition from one trim state to another trim state, the aircraft is trimmed at all \(n_{V}\) query points using Equation (10). The aerodynamic forces and moments in these trim points already constitute one part of the disturbance moment set. However, as we are also interested in transitions/maneuvers from one trim state to another (e.g., acceleration or deceleration in the lateral direction), the disturbance for these maneuvers is calculated as well. In order to account for these maneuvers, we assume that, at each trim point, the attitude of the aircraft can vary from the trim point while the airspeed is kept constant. In order to model the transition phases and the obtained aerodynamic inflow angles  
Figure 3: Exemplary distribution of boundary points on the aerodynamic velocity envelope in the control frame (2D example for visualization purposes). realistically without being too conservative, a range of attitude deviation from the trim point obtainable via maneuvers is determined. For this, minimum and maximum allowed attitude angles for maneuvers can be derived from system requirements. Additionally, an overshoot over the trim pitch/roll angles is allowed in order to make the disturbances more realistic. Figure 4 shows the idea of the attitude variation for an exemplary 2D model. In the range of the minimum and maximum pitch and roll angles, different attitude deviations/combinations from the trim point are calculated. The allowed attitude region follows  
\[\Phi^{-} =\min(\Phi_{min},\Phi_{trim}-\Phi_{overshoot}), \tag{11}\] \[\Phi^{+} =\max(\Phi_{max},\Phi_{trim}+\Phi_{overshoot}),\] (12) \[\Theta^{-} =\min(\Theta_{min},\Phi_{trim}-\Theta_{overshoot}),\] (13) \[\Theta^{+} =\max(\Theta_{max},\Theta_{trim}+\Theta_{overshoot}). \tag{14}\]  
As for the query points on the boundary of the velocity envelope, query points in the bounded \(\Phi\)-\(\Theta\)-plane are used to obtain set points for various inflow angles during maneuvers from one trim state to another. Let \(n_{\Phi,\Theta}\) denote the total number of points in the attitude plane. Note that the maneuver disturbance here follows from attitude deviations and body rates are neglected for the sake of reducing computational complexity. The corresponding required moments to counter the disturbances follow by evaluating the aerodynamic model at the \(n_{\Phi,\Theta}\) query attitude points with the query inflow  
\[ \begin{bmatrix}L_{RMS,D}\\ M_{RMS,D}\\ N_{RMS,D}\end{bmatrix} _{B}=- \begin{bmatrix}L_{A}(\[PERSON],\[PERSON],\mathbf{V_{C}}) \\ M_{A}(\[PERSON],\[PERSON],\mathbf{V_{C}})\\ N_{A}(\[PERSON],\[PERSON],\mathbf{V_{C}})\end{bmatrix} _{B}, \tag{15}\]  
Figure 4: Illustration of attitude deviation for calculating disturbance moments and forces during maneuvers (2D example for visualization purposes). where the index \(D\) denotes the disturbance part. The requirement for the total propulsive thrust to compensate for the disturbances depends on the desired behavior during the maneuver. We considered transitions in the horizontal plane, as these transitions change the attitude/inflow angles, and therefore there will be arbitrary accelerations depending on the attitude in the horizontal direction. In the vertical direction, however, the propulsive system needs to counter the disturbance (e.g., negative lift of the wing when hovering forward with negative pitch angle). This requires a total zero force in the z direction of the control frame.