## 6 Result and Discussion
### Optimization Result
#### 6.1.2 Maneuver Requirement  
In this case, the proposed framework was used to find the optimum design parameters that would allow the system to execute its assigned mission in the event of an actuator failure. The required moment data to track mission trajectory were obtained from the assumed UAV model simulation at nominal conditions. The distribution of moment data points in its three directions of moment space \(\mathbb{R}^{3}\) was portrayed geometrically by constructing the SDE using Equations (12)-(14). The framework used Equation (17) to maximize and fit the SDE described by Equation (14) into the AMS, and Figure 12 shows the result as a plot of parameters and cost function against the number of iterations. Similarly, the parameters were converged to values that maximize the cost function within their constraint limits. Similarly, a comparison of the preliminary design of the AMS (yellow) and the configuration augmented with optimum parameters (aqua) for each actuator's failure is presented in Figure 13. Unlike the hovering operation, the maneuver requires different control authorities in all moment directions in this operation. In this case, the optimization fits Equation (14), which describes the required moments to meet the assigned maneuver into the AMS using the formulation in Equation (17). Similarly, the results demonstrated that sufficient control authority was obtained in all directions, based on their relative weight. The initial values, optimal values, and the resulting cost function computed for each actuator's possible failure are listed in Table 3.  
Using Algorithm 1 given in Section 4, the orientation of points can be defined using the exclusion ratio \(\gamma\) and confidence level \(p_{r}(S)\) given marginal value \(\zeta\) for an arbitrary value of scale factor \(S\), as shown in Figure 14a. In the preliminary design, failure in actuator one results in a loss of controllability in one of the directions; hence, the geometry will have zero radii that result in \(S=0\) and the corresponding \(p_{r}(S)=0\). In this circumstance, about \(\
icefrac{{1}}{{3}}\) of the moments required to perform the needed operation were present outside of the AMS envelope. As \(S\) increases, the number of points flowing into the AMS polytope increases, whereas the number of points outside the envelopes decreases, as indicated by decreasing of \(\gamma\). At \(S=5\) the confidence level reaches a maximum \(p_{r}(S)=1\), which shows the existence of all points within the ellipsoid and hence in the AMS envelope. However, 5.2% of points reside outside the AMS's prescribed margin. Further magnification of the ellipsoid results in the enlargement of the AMS and crossing of the remaining points across the specified margin inside the AMS. At \(S=6.35\) all points were orientated inside of the requested margin. Figure 12: Optimization output: plot of actuator’s parameters and cost function trend against the number of iterations for actuator 1 failure and prescribed maneuver performance requirement. Moreover, the maximum marginal value that can be imposed is depicted in Figure 14b. At \(\zeta=0\) all points are orientated inside the AMS polytope without marginal specification. For \(\zeta>0\), the polytope must be enlarged to keep \(\gamma=0\). As a result of imposed constraint on the parameter, the maximum marginal value that can be achieved was \(\zeta=2.05\), which corresponds to the maximum scale factor (\(S=7.747\)). Therefore, the computed parameters can be used to reconfigure the UAV to tolerate the considered fault and perform the desired maneuver. Figure 13: AMS comparison of preliminary configuration with optimized configuration for each actuator failure and prescribed maneuver performance requirement.