# Required Moment Sets: Enhanced Controllability Analysis for Nonlinear Aircraft Models
## 2 Materials and Methods  
### Separation Into Individual Required Control Subsets  
In order to calculate the required moment set in a structured manner and in order to perform the proposed RMS-based controllability assessment in a traceable manner, the required forces and moments are split into individual subsets. We call the first part the disturbance part, which constitutes the required propulsive forces and moments that are required to counter disturbances. This paper focuses on the aerodynamic disturbance, as in hover mode the aerodynamic forces and moments have to be compensated for via the propulsive system and are not part of the flight control as they are for wingborne mode.  
In order to determine the maximum possible aerodynamic disturbance that can act during the operation, the aircraft is first trimmed at the boundary points of the aerodynamic velocity envelope, as there the aerodynamic disturbance obtains the highest magnitudes. In all possible trimmed directions of flight, multiple attitude deviations from the trim points are used to evaluate aerodynamic forces and moments. With this approach we obtain a realistic set of possible aerodynamic forces and moments that need to be countered by the propulsive system.  
For the sake of simplicity the calculation process and moment set approach include a configuration with a fixed direction of hover thrust (no tilt configuration); for this reason, the required moment set is defined by the three roll, pitch and yaw moments \((L,M,N)\) or corresponding angular accelerations \((\dot{p},\dot{q},\dot{r})\) and the total aircraft thrust in the body z-direction \(T\) or the corresponding body z load factor \((n_{z})_{B}\).  
Figure 1 shows an overview of the different parts of the required moment set that is used throughout this paper. Besides countering aerodynamic disturbance, the propulsive system also needs to provide the desired maneuverability. The corresponding desired quantities are attainable accelerations (angular and translational), which from a kinematic point of view and provide sufficient dynamics for fulfilling higher level mission requirements. We use the angular body accelerations \((\dot{p},\dot{q},\dot{r})\) and the inertial translational acceleration in  
the body z direction \((a_{z})_{B}\) as direct requirements. These acceleration limits again define a convex set (e.g., a four dimensional cube).  
In order to sum up the disturbance and maneuverability parts of the required forces/ moments, we use query directions in the four-dimensional set space and find the cross sections with the convex hull of the corresponding sets. This procedure simplifies the summation process of disturbance and maneuverability sets, which finally yields the full set of required forces/moments.  
In order to finally assess the controllability, the attainable and required control sets need to be compared. To evaluate this, first the AMS is calculated by applying all possible combinations of minimum or maximum propulsive inputs and assembling them in a convex set [7; 8]. In a second step, the convex set intersections in the direction of the query directions are determined. Afterwards, attainable and required forces/moments are available in the same query directions and can be directly compared to assess the controllability margins and to identify critical directions/channels.