# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 4 Conclusions  
The results indicate that the calculation of a required moment set allows an intuitive controllability analysis based on system requirements. The assembly in a control set enables a simple comparison of AMS and RMS and at the same time keeps traceability of a source of a required moment (e.g., aerodynamic disturbance or maneuverability). The resolution analysis of control sets showed the convergence of simple measures of controllability when the underlying resolutions are increased. Consequently, the methods for assembling the aerodynamic disturbance set points and the query direction-based intersection set assembly are able to cover the full control set area. However, the price of a high accuracy of the presented results are higher computation times as the number of query directions rises with third power of the underlying resolution. As the introduced RMS and calculation approach were only carried out in an exemplary manner for a considered eVTOL, refinements can be investigated in the future. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Resolution** & **Mean (m)** & **Min (m)** & **Failure Percentage** \\ \hline \(n_{\beta_{1}}=n_{\beta_{2}}=2\) & \(0.391\) & \(-0.298\) & \(18.52\%\) \\ \(n_{\beta_{1}}=n_{\beta_{2}}=12,n_{\beta_{3}}=24\) & \(0.391\) & \(-0.367\) & \(15.50\%\) \\ \(n_{\beta_{1}}=n_{\beta_{2}}=24,n_{\beta_{3}}=48\) & \(0.398\) & \(-0.369\) & \(14.94\%\) \\ \(n_{\beta_{1}}=n_{\beta_{2}}=48,n_{\beta_{3}}=96\) & \(0.403\) & \(-0.371\) & \(14.69\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Controllability Analysis Results For Different Query Point Resolutions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Resolution** & **Mean (m)** & **Min (m)** & **Failure Percentage** \\ \hline \(n_{\phi}=2,n_{\theta}=2\) & \(0.414\) & \(-0.318\) & \(14.04\%\) \\ \(n_{\phi}=5,n_{\theta}=5\) & \(0.399\) & \(-0.365\) & \(14.83\%\) \\ \(n_{\phi}=10,n_{\theta}=10\) & \(0.398\) & \(-0.369\) & \(14.94\%\) \\ \  
For example, disturbance sources such as center of gravity and mass uncertainties can be included in the disturbance moment set as well. At the same time, means to include the effect of body rates on the disturbance forces/moments can be analyzed in future work. Another important aspect to mention here is the inclusion of model uncertainties in order to obtain probabilistic controllability statements. To obtain probabilistic judgements, the uncertainties first need to be quantified [23] and then included in the simulation model [15]. It is important to mention here that the inclusion of sample-based stochastic analysis will further increase the computation times and therefore requires an efficient algorithmic implementation. As this work mainly focuses on the introduction and calculation of the RMS, it opens various options to use the RMS, e.g., in design optimization and controller design. For design optimization, suitable measures based on the presented controllability margins can be investigated. For control design for a given configuration, the allowable airspeed envelope can be determined which guarantees sufficient maneuverability. Additionally, given a desired airspeed envelope, the maximum available maneuverability can be determined, which can be used for protective measures in flight control algorithms. Conceptualization, M.S. and J.Z. ; methodology, M.S. ; software, M.S. ; validation, M.S. and J.Z. ; investigation, M.S. ; writing--original draft preparation, M.S. ; writing--review and editing, N.B. ; visualization, M.S. ; supervision, F.H. ; project administration, F.H. ; funding acquisition, F.H. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Data Availability Statement: Data sharing not applicable. The authors declare no conflict of interest. The following abbreviations and nomenclature are used in this manuscript: