###### Abstract  
In this paper, we discuss an attainable moment set (AMS) optimization methodology considering a system's required moment set (RMS). The AMS describes the achievable moments from a system, given its input limits. An RMS, like an AMS, is a convex set in the moment space, describing the required moments for a system to meet the designed mission profile and disturbance rejection requirements. Given the configuration of a system, its mission requirements, and the derived RMS, the proposed optimization maximizes coverage of the AMS onto the RMS, thus ensuring the system possesses the guaranteed controllability to fulfill its required missions from a design level. To achieve this goal, the variables to optimize are chosen as effector settings, such as the installation position and angle of propellers and control surfaces, which effectively change the AMS without a vast impact on major design parameters, such as mass and moment of inertia. Since the optimization includes massive geometry operations of rays intersecting polyhedron, an efficient intersection solver is proposed to speed up the optimization process. The described method is applied to an electric-vertical-take-of-landing vehicle (eVTOL) with eight hover propellers, which delivers a highly improved coverage of the RMS compared to its initial design. attainable moment set; required moment set; configuration optimization; eVTOL  
32021 3C 2021 15 April 2021 19 April 2  
mission profiles, as well as the capability of disturbance rejections. Often a limited AMS poses risks on the safe operation of the system in off-nominal situations, e.g., effector or power system failures. As a consequence, a system with relatively big and redundant AMS means better safety margin against extreme cases. What influences the AMS of a system is however fixed at the design phase by sizing and positioning of the effectors. Therefore, appropriately configuring the effectors at early design phase is critical to the system's performance. For conventional airplanes, the freedom of such effector designs is very limited given major design requirements such as stability and endurance. However, this freedom becomes largely expanded for novel aircraft architectures, such as electrical-vertical-take-off-landing (eVTOL) vehicles, which is often a hybrid system capable of both hover and high-speed flight, exploiting large number of redundant inputs in the form of distributed electric propulsion (DEP) for a bigger flight envelope and additional safety features. As a consequence, optimization can be utilized to find the best combination of effector settings without violating major design parameters such as mass and moment of inertia of the system. Several studies have incorporated optimization to refactor settings of effectors or to directly search for the optimal AMS of the system. In [4], a trim analysis is performed for a conventional aircraft enhanced with DEP over wingspan, showing that the size of the vertical tail can be reduced by 45% by exploiting differential thrust without adversely affecting directional stability. The idea is further explored in [5] together with a \(\mathcal{H}_{\infty}\) control methodology in a sequential optimization process, which reduces the area of vertical tail by 60% and lowers the actuator bandwidth, while maintaining the overall control loop bandwidth. In [6], a propeller optimization is applied to a DEP enhanced airplane configuration to decide for the optimal propeller diameter, which shows up to 80% of take-off distance reduction compared to the two-propeller conventional counterpart. As for AMS optimization, a generic AMS framework is proposed in [7] to aid visualize AMSs of complex systems during design phase, which is then conceptually applied to the Max Launch Abort System escape vehicle to help decide the optimal orientation of the thrust nozzles. In [3], a trim routine is developed for an unconventional box-wing configuration to maximize local AMS at operation point.