## 5 Discussion and Conclusions  
In the present work, a multidisciplinary design optimization of a launch vehicle with a hybrid last stage was carried out and compared with the existing all-solid-stages launch vehicle, Minotaur-I, for performance improvements. The MDO problem is formulated to maximize the total impulse-to-weight ratio while satisfying target orbit parameters such as \(\gamma\), \(h\), and \(V\) criteria for a better \(I_{sp}\) value than the solid stage. Within the MDO process, the propulsion, aerodynamics, environment, and flight dynamics subsystems were modeled using MATLAB 2017b in a 6 DOF simulation. The nonlinear coupling between subsystems was considered to address the research gap, making it possible to solve the complex optimization problem and determine the hybrid motor characteristics and various trajectory parameters, including the TVC angle, necessary to achieve the target orbit for a specified PM and mission. Aerodynamic coefficients were modeled using DATCOM v2011, while structural properties were obtained from CATIA V5R19. In the optimization process, all subsystems, including variations in length (\(L^{*}\)) and m, are updated in each iteration. This comprehensive optimization approach leads to the determination of the optimal solution. Based on the results, the optimized design point 20245 achieves the target orbit more quickly compared to the reference trajectory. There are no significant changes in \(h\) of the first three stages. However, the decrease in \(V\) of the third stage during unpowered flight is smaller than in the optimized design due to the reduction in \(m_{F}\) resulting from the optimization process. It should be noted that the structural mass of the fourth stage remains the same as the reference to facilitate a comparison between the optimized propulsion system and the reference values. The value of \(\gamma\) initially starts near 90 degrees and gradually decreases using the optimized TVC angles, eventually approaching zero. The absolute errors in \(h\) and \(V\) are less than 1%, as expected. According to the results in Table 6, the optimized design operates with lower \(T\), shorter \(t_{\text{burn}}\), higher \(I_{sp}\), and a greater \(I_{t}/w_{0}\). Despite the improvement, the optimized system is slightly longer in \(L^{*}\). Although \(L^{*}\) of the stage and the total vehicle is increased, this \(L^{*}\) increase is not critical due to the updates in the subsystems considering the subsystem couplings during the optimization iterations. Consequently, any unfavorable conditions would have been observed in the optimization results, resulting in a limited number of feasible points. Overall, it can be concluded that the performance and efficiency of the launch vehicle are enhanced by incorporating a hybrid propulsion system. The results also demonstrate that the optimized design reaches the target orbit faster, with a smaller \(V\) decrease during the unpowered flight of the third stage. As a result, the mission \(t\) and \(m\) consumption are reduced through the modification of the last stage with hybrid propulsion. The TVC angles of the first and last stages are also obtained to ensure they remain within the control deflection limits. In addition, the comparison between the 3 DOF and 6 DOF models provided valuable insights. The 3 DOF model lacks complex angular velocity calculations present in the 6 DOF model, and it excludes essential factors like moment of inertia, center of mass, and moment changes. As the simulation progresses, the disparity between these two models widens. Specifically, the percentage differences between the 3 DOF and 6 DOF models were found to be 11.55% in velocity and 8.02% in altitude. While the 6 DOF model closely aligns with the targeted parameters, there is evidence that the 3 DOF model results deviate from the targeted parameters when run with the parameters entered into the 6 DOF model. Further analysis of the results for the hybrid last stage reveals significant differences when specific parameters are neglected.