# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 2 Materials and Methods
### Trim-Based Calculation of Aerodynamic Disturbance Set  
Now that an aircraft model is established, a very important next objective of this paper is to establish means to calculate different parts of required control sets. This section explains the proposed method of calculating the worst case aerodynamic disturbance throughout the aircraft mission. The presented analysis is carried out for a given altitude \(h\) and density of the atmosphere \(\rho\). At this point the aircraft is required to be able to fly up to certain kinematic speeds in various directions. A convenient frame to describe the three-dimensional velocities is the control frame C as it is indifferent to the heading \(\Psi\) but includes the down direction [18]. Figure 2 shows the axes of the C-frame. Besides the kinematic speed requirement, there can also be requirements for environmental conditions such as wind speeds. Combining both requirements yields a total airspeed envelope that the vehicle must be able to cover. For the sake of simplicity let us assume a cuboidal envelope of the shape  
\[u_{C,min} \leq u_{C}\leq u_{C,max}, \tag{7}\] \[v_{C,min} \leq v_{C}\leq v_{C,max},\] (8) \[w_{C,min} \leq w_{C}\leq w_{C,max}, \tag{9}\]  
with \(u_{C},v_{C},w_{C}\) denoting the components of the aerodynamic velocity given in the C frame \(\mathbf{V}_{C}^{3\times 1}\). For all points within this airspeed envelope the aircraft must be able to obtain steady state operation. In order for the aircraft to be in steady state attitude, the body rates \((p,q,r)\) and the corresponding accelerations \((\dot{p},\dot{q},r)\) need to be zero. The translational motion of the aircraft is then defined by the velocity point in the envelope \(\mathbf{V}_{C}\). The total hover thrust of the aircraft \(T\) tilted by the attitude \((\Phi,\Theta)\) needs to offset the aerodynamic and gravitational forces in order to reach no translational acceleration and steady state velocities in the C frame. For the sake of simplicity it is assumed that the hover thrust direction coincides with the negative body z direction which yields the force trim balance equation  
\[ \begin{bmatrix}X_{A}(\[PERSON],\Phi,\mathbf{V}_{C})\\ Y_{A}(\[PERSON],\Phi,\mathbf{V}_{C})\\ Z_{A}(\[PERSON],\Phi,\mathbf{V}_{C})\end{bmatrix} _{B}+ \begin{bmatrix}0\\ 0\\ -T\end{bmatrix} + \begin{bmatrix}-\sin(\Theta)\\ \cos(\Theta)\sin(\Phi)\\ \cos(\Theta)\cos(\Phi)\end{bmatrix} mg\ \overset{! }{=}\ \begin{bmatrix}0\\ 0\\ 0\end{bmatrix} . \tag{10}\]  
Note that for applying this method an aerodynamic model defining \(X_{A},Y_{A},Z_{A}(\Theta,\Phi,\mathbf{V}_{C})\) over the full envelope of angles of attack and angles of sideslip is required. As these models  
Figure 2: Visualization of the Control frame C, which is a North–East–Down frame O, rotated by the heading \(\Psi\) around the z-axis. are nonlinear, Equation (10) needs to be solved numerically, e.g., via Matlab's fsolve() [19] or more aircraft tailored trim solutions [20]. In order to determine the maximum possible aerodynamic disturbance hull, only the boundary of the aerodynamic velocity envelope is considered, as the dynamic pressure and therefore also possible disturbance forces/moments reach their highest values here, see Figure 3. As individual points are required for the moment set and assembly to a convex hull, we selected a number of query points that represents our envelope boundary. Unlike for simplified approaches when calculating an AMS where the number of set points is finite, for this nonlinear aerodynamic disturbance set the number of points can be chosen to be arbitrarily large. Higher resolution in the query points on the envelope boundary will increase the accuracy of the obtained RMS but also increase the amount of points that need to be analyzed. The later Results section will show the results for different chosen resolutions of the RMS. As the mission hover envelope in most cases follows simple geometrical shapes (e.g., cuboid or ellipsoid) in the control frame, the points can be chosen as linearly spaced on the boundary.