## 5 Implementation and Results  
### The Plant Model and the Moment Sets  
The proposed framework is applied to an eVTOL to optimize its eight hover propellers' installation angles. The outline of the propellers' arrangement is displayed below in Figure 7:  
The Cartesian coordinate in blue represent the body-fixed frame, denoted by \(B\). The eight discs denoted as \(P_{1}\) to \(P_{8}\) sketch the eight propellers with an arrow representing the directions of forces they produce. The figure demonstrates the propellers respective relative positions and alignment in the \(B\)-frame. Since the propellers are installed on two hanger frames (black bars in Figure 7b) which are parallel to the \(x\)-\(B\) axis, the propellers constrained to rotate around the \(y\)-\(B\) axis. Therefore, all the propellers are tilted symmetrically around  
Figure 7: Outline of propeller positions of the considered plant: (**a**) 3\(D\) view, (**b**) Top view. In (**b**), the black bars represent two hanger frames where the propeller are installed on. This leads to a natural constraint on the propeller rotation direction, which is only allowed to rotate around the \(x\)-\(B\) axis. the \(x\)-\(B\) axis which also accounts for the otherwise limited yaw authority. Note that the figure is only a conceptual illustration and does not represent actual sizing of the plant considered. Generally, a linear relationship between the forces and moments produced and the square of rotational rate of the propellers can be assumed [17; 18]. For hover control, the most relevant forces and moments are the vertical force along the \(z\)-\(B\) direction and three moments. Here we choose the generalized forces of the vertical load factor and angular accelerations as the axis of the moment sets, since they are algebraically equivalent to the vertical force and moments, and they present a more straightforward relationship between the most often controlled variables, e.g., vertical speed and attitude. The mathematical model we consider here is therefore:  
\[\left[ \begin{array}{c}n_{z}\\ \dot{p}\\ \dot{q}\\ \dot{r}\end{array} \right]_{B}=\mathbf{B}(\mathbf{p})\left[ \begin{array}{c}\omega_{1 }^{2}\\ \omega_{2}^{2}\\ \vdots\\ \omega_{3}^{2}\end{array} \right],\ \omega_{i}^{2}\in\left[0,\ \omega_{\text{max}}^{2}\right] \tag{12}\]  
On the LHS, \(n_{Z}\) is the vertical load factor; \(\dot{p},\ \dot{q},\ \dot{r}\) are the angular accelerations, subscript \(B\) denotes the quantities in the \(B\)-frame. On the RHS, \(\mathbf{B}\) is the effectiveness matrix, which is a function of the parameters, \(\mathbf{p}\) to be optimized. \(\omega_{1,\ldots,8}^{2}\) are the square of rotation rate of the propellers. \(\omega_{\text{max}}\) is the maximum rotational speed. The preliminary design tilts the propellers around the \(x\)-\(B\) axis symmetrically, as can be seen in Figure 7. Assume the tilt angle is \(\phi\), then the vector of preliminary parameters is \(\mathbf{p}=[\phi,-\phi,\phi,-\phi,-\phi,\phi,-\phi,\phi]\) for propellers \(P_{1}\) to \(P_{8}\). Figure 8 shows the projections of the \(\mathbb{R}^{4}\) AMS and RMS into \(6\)\(\mathbb{R}^{2}\) spaces, when \(\phi\) is \(5^{\circ}\), \(15^{\circ}\) and \(25^{\circ}\) respectively. Note that when the angle is small, the system shows very limited authority in yaw thus a very small marge with respect to the RMS. When the angle increases, the yaw authority is correspondingly enlarged but the achievable load factors and roll accelerations are noticeably degraded. For all the three cases, the pitch authority in the AMS is slightly changed and remains highly redundant compared to its counterpart in the RMS. Given  
Figure 8: Comparison of RMS and AMSs when the symmetrically tilted angles of propellers is \(5^{\circ}\), \(15^{\circ}\) and \(25^{\circ}\) respectively. these observations, we choose to optimize a \(\mathbb{R}^{3}\) subset of the \(\mathbb{R}^{4}\) AMS, using a sub-model of Equation (12):  
\[\left[ \begin{array}{c}n_{z}\\ \ddot{p}\\ \dot{r}\end{array} \right]_{B}=\mathbf{B}_{\text{sub}}(\mathbf{p})\left[