# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
###### Abstract  
Attainable moment sets (AMS) are a powerful method to assess aircraft controllability. However, as attainable moment sets only take into account the achievable moment set of the control effectors, they do not assess the required moment set to fulfill the aircraft mission requirements. This paper proposes to calculate a corresponding required moment set (RMS) which defines a set of moments sufficient for fulfilling aircraft controllability requirements in the mission flight envelope. The paper applies the required moment set approach to a nonlinear simulation model of an electric vertical take of vehicle (eVTOL) transition drone in hover. By comparing the required moment set to the AMS of the aircraft model, moment set margins are derived and used to assess the controllability of the considered aircraft. The results indicate that the combined evaluation directly identifies critical moment channels and margins, which is advantageous when compared to a pure AMS-based evaluation. The proposed approach enables the execution of simulation-based assessments in aircraft design and flight control development. In the early stages of aircraft design, required moment sets can support sizing, positioning and tilting of control effectors (e.g., propulsive elements) to fit the AMS to the actual required force and moment set for the specific system.  
controllability; eVTOL; attainable moment set +
Footnote †: journal: Applied Sciences  
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For this reason, this paper proposes to evaluate a required moment set (RMS), which is a set containing the worst case required moments that need to be covered by the attainable moments in order for the aircraft to fulfill controllability requirements. Consequently, the RMS defines a minimum required AMS which can be used to assess and even to optimize the actual AMS of the vehicle, e.g., by tilting propulsive elements or sizing propulsive units.  
The main objective of this paper is to present methods to obtain a measure of controllability based on attainable and required control sets. As the AMS has already been studied thoroughly in current literature, this work's objective is to mainly focus on the RMS and to allow a direct comparison of AMS and RMS. As the proposed methods shall be applicable already in early stages of aircraft design, the used methods shall rely only on aircraft simulation models. Another very important objective is the traceability of RMS subsets, i.e., to be able to distinguish between different parts of the RMS, e.g., parts to obtain angular accelerations and parts to trim against aerodynamic disturbances.  
After this introduction, the paper focuses on the methods used to achieve above mentioned objectives. In a first step, the RMS is split up conceptually in different required parts, e.g., for maneuvering and for countering aerodynamic disturbances. As the proposed approach is based on a simulation model, the underlying physical assumptions as well as equations are presented. In a next step we derive a method to numerically evaluate the required moment set for aerodynamic disturbances. In order to guarantee a straightforward comparison of AMS and RMS the following subsection deals with the assembly in a convex hull-based control set. In order to complete the different types of control sets, the required control set and the attainable set are investigated as well. Finally a method to assess the controllability is shown. The paper then finishes with results for a transition capable multirotor configuration and derives conclusions about the applicability of the methods to achieve the above specified objectives.