# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 2 Materials and Methods
### Convex Hull-Based Control Set Assembly  
The last subsection contributed to the objective of calculating clearly traceable parts of the RMS. This subsection proposes a method to add these individual RMS subsets and to compare them to the AMS. After the query points for the disturbance set are assembled, the convex hull [7] of the set can be computed in order to obtain a worst case estimation of the aerodynamic disturbance. Note that this convex hull-based approach is chosen here as it can be performed generically for any set of points regardless of the physical origin. An efficient algorithm for determining the convex hull of an arbitrary set of points is the quickhull algorithm [8], which is implemented in the Matlab function convhulln(). It is important to mention here that, due to the nonlinearities in the aerodynamic coefficients, the aerodynamic disturbance set is in general no convex set. The goal of this paper, however, is a comparison of the RMS with the AMS. For this reason, assembling the RMS as a convex set does not produce overly conservative results. Any convex AMS that contains a non-convex RMS will also contain a more conservative convex version of the RMS.  
In order to be able to sum up the different independent elements of the RMS (disturbance and maneuverability parts) and compare them to the AMS, we propose to calculate the maximum attainable acceleration in the query directions of the four-dimensional acceleration space. This reduces computational effort and produces magnitudes in predefined directions that do not depend on the configuration but are chosen by the user. This approach also allows comparison of different configurations with respect to each other very easily. A disadvantage of this method is that for small numbers of query directions, the actual set might not be resolved accurately enough. In order to resolve the moment set with higher accuracy, the number of query directions increases very fast as the moment set is four-dimensional.  
The maximum of the accelerations in the query directions is determined by the points in the query direction that lie on the convex hull. An efficient algorithm to determine the cross sections with the convex is presented in [21].  
To parameterize the four-dimensional directions, we use an n-spherical coordinate system [22] which, for the four-dimensional case, requires three angles for the parameterization. Including a scaling along the four-dimensions which leads to  
\[d_{nz} =n_{z,scale}\cos(\beta_{1}) \tag{20}\] \[d_{\hat{p}} =\dot{p}_{scale}\sin(\beta_{1})\cos(\beta_{2})\] (21) \[d_{\hat{q}} =q_{scale}\sin(\beta_{1})\sin(\beta_{2})\cos(\beta_{3})\] (22) \[d_{r} =\dot{\pi}_{scale}\sin(\beta_{1})\sin(\beta_{2})\sin(\beta_{3})\] (23) \[\mathbf{d} =\frac{1}{\sqrt{d_{nz}^{2}+d_{\hat{p}}^{2}+d_{\hat{q}}^{2}+d_{ \hat{r}}^{2}}}\begin{bmatrix}d_{n_{z}}\\ d_{\hat{p}}\\ d_{\hat{q}}\\ \end{bmatrix}, \tag{24}\]  
with the normalized query direction \(\mathbf{d}\in\mathbb{R}^{4\times 1}\) and the three parameterization angles \(\beta_{1},\beta_{2}\in[0...\pi]\) and \(\beta_{3}\in[0...2\pi)\). As for the velocity envelope and the attitude deviation, we again use linearly spaced parameterization angles. Let \(n_{\beta_{1}},n_{\beta_{2}},n_{\beta_{3}}\) denote the number of the corresponding angles for the linear spacing, then \(n_{\beta}=n_{\beta_{1}}n_{\beta_{2}}n_{\beta_{3}}\) is the total number of query directions. Finally, all directions are assembled in a direction matrix \(\mathbf{D}\in\mathbb{R}^{4\times n_{\beta}}\).  
Applying this query direction approach to the convex hull obtained from the aerodynamic disturbance moments in all directions contained in \(\mathbf{D}\) and assembling the results in a row vector yields the required acceleration magnitudes for disturbance rejection \(\mathbf{a}_{RMS,D}\in\mathbb{R}^{1\times n_{\beta}}\).