## 6 Result and Discussion
### Optimization Result
#### 6.1.1 Null Controllability  
This section presents the optimization result of the assumed UAV towards achieving actuator fault-tolerant capabilities in a single actuator total failure while hovering. The framework used Equation (15) to maximize and fit the sphere described in Equation (11)  
Figure 8: Comparison of proposed UAV preliminary design and UAV with tilting mechanism (**a**) preliminary design where all propellers tilting angles are zero (coplanar) (**b**) preliminary design with axis offsetting angle zero (**c**) \(\alpha_{i}\) tilted propeller configuration (**d**) configuration with axis offset angle \(\beta\).  
into the AMS, and Figure 9 shows the result as a plot of parameters and cost function against the number of iterations for actuator-1 total failure. The result showed that the parameters were converged to values that maximize the cost function within their constraint limits. The initial values, optimal values, and the resulting cost function computed by the optimization framework for each actuator's possible failure are listed in Table 2.  
A comparison of the preliminary designs of the AMS (yellow) and the configuration augmented with optimum parameters (aqua) for each actuator's failure is presented in Figure 10. In preliminary design, actuator-1 total failure results in an inability to produce a negative yaw moment and a negative roll moment simultaneously, and actuators-2 total failure results in an inability to produce a positive yaw moment and a positive roll moment. Likewise, the complete failure of actuators-3-4-5-6 degrades the system's controllability, so the system loses its attitude control. In contrast, owing to the vectorization of the vertical thrust force into the lateral force via optimum angle tilting and arm installation angle of the produced lateral force from symmetrically located actuators, the yaw moment was produced independently with a slight loss of roll moment in the optimal configuration. As a result, sufficient control was produced around the origin of the AMS, as shown on the optimized configuration AMS by origin-centered sphere geometry.  
The marginal evaluation result for actuator 1 failure optimization is depicted as shown in Figure 11. The actuator-1 failure in the preliminary design results in \(S=0\) and \(\gamma=0.462\), which indicates 46.2% of the required moment points outside the AMS envelope, as shown in Figure 11a. Given the marginal value of \(\zeta=1\), the coverage of all points within the prescribed margin was ensured through the magnification of \(g_{s}\) by \(S=3.26\). Furthermore,  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Fault Condition**} & \multicolumn{6}{c}{**Parameters (Angles in Degree)**} & \multirow{2}{*}{**Cost Function**} \\ \cline{2-2} \cline{4-9}  & \multicolumn{2}{c}{**Initial Value**} & \multicolumn{6}{c}{**Optimization**} \\ \hline Actuator 1 failed & \(\begin{bmatrix}0\\ 15\\ 15\\ 15\\ 15\\ 15\\ 15\\ \end{bmatrix}\) & \(\begin{bmatrix}30&0&-20&20&20&20&20&20\\ 30&20&0&20&20&20\\ 30&20&13.5&0&19&20&20\\ 30&20&13.5&0&19&20&20\\ 30&20&13&-16&0&20&20\\ 30&20&5&20&19&0&20\\ 30&20&13&20&20&-20&0\\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter optimizations result in null controllability, single actuator failure at a time.  
Figure 9: Optimization output: plot of actuator’s tilting angle and cost function trend against the number of iterations for actuator 1 failure and the null controllability performance requirement.  
Figure 11b shows the maximum achievable scaling factor \(S=3.886\) and the corresponding marginal value of \(\zeta=1.316\).  
Figure 10: AMS comparison of preliminary configuration with optimized configuration for each actuator failure.