## 4 Comparison of the Comprehensive Modeling  
This study was compared with 3 DOF, aiming to identify the advantages and differences arising from comprehensive modeling in optimization problems. In this context, 3 DOF model outputs were obtained for both the hybrid last-stage launch vehicle and the original Minotaur-I launch vehicle using 6 DOF inputs. Furthermore, the differences due to detailed subsystem modeling were also highlighted. The percentage differences in altitude, velocity and flight path angle in flight mechanics when using a 6 DOF model are generally smaller compared to those using 3 DOF, especially for complex maneuvers and simulations that involve precise positioning and interactions with the environment [44]. For advanced flight simulations, such as those used in aerospace engineering, a 6 DOF model is preferred because it can provide highly accurate results, often within a few percentage points of reality. For example, [PERSON] and [PERSON] indicate that the relative error between the 3 DOF and 6 DOF models for maximum range envelope calculations is 3% and 12% at altitudes of 1 and 15 km, respectively [60]. Therefore, as the flight simulation time increases, the difference between 3 DOF and 6 DOF models will gradually become more pronounced. To demonstrate the accuracy of the results, we compare the 3 DOF and 6 DOF simulations obtained for the same design point, Design ID 20245, and the original Minotaur-I launch vehicle, and discuss the differences between these simulations. A comparison is made with the results of Design ID 20245, as it provided results quite close to the reference values given for the trajectory of the original Minotaur-I launch vehicle, and this study aims to examine the difference between the 3 DOF outputs and those of 6 DOF. The altitude and velocity graphs obtained over time from the 3 DOF and 6 DOF simulations are presented for the hybrid last stage as the output of the optimization process in Figure 7, and for the original solid last stage that is taken as the reference data of the Minotaur-I launch vehicle in Figure 8, and the percentage differences \(\left(\left|\frac{\text{value}-\text{targetvalue}}{\text{targetvalue}} \right|\times 100\right)\) between the target altitude and velocity are given in Tables 7 and 8, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Minotaur-I** & **Minotaur-I-Like (Hybrid)** \\ \hline Propellant type & HTPB & HTPB/LOX \\ \(m_{p},\text{kg}\) & 770.2 & 416.2 \\ \(L^{*},\text{m}\) & 1.34 & 2.93 \\ \(T,\text{kN}\) & 36.9 & 34.48 \\ \(l_{sp},\text{s}\) & 287 & 308.95 \\ \(h_{bum},\text{s}\) & 66.8 & 36.65 \\ \(I_{t}/w_{0}\) & 253.3 & 257.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Propulsion comparison of optimization results and reference points. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Target** & **Optimization Results** & **Absolute Error** \\ \hline \(h,\,\text{km}\) & 741.2 & 747.6 & 0.863\% \\ \(V,\text{m}/\text{s}\) & 7482.2 & 7425 & 0.763\% \\ \(\gamma,\text{deg}\) & 0 & \(-\)0.00742 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Trajectory comparison of optimization results and reference points. As seen from Figures 7 and 8, it can be observed that the difference between the 3 DOF and 6 DOF simulations widens as flight time progresses. Therefore, since the values obtained from the 6 DOF simulation, especially for the original Minotaur-I launch vehicle, align with the reference, it can be said that, especially for long-duration missions, the \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **6 DOF** & **3 DOF** & **Target** &
\begin{tabular}{c} **\% Difference** \\ **(6 DOF/3 DOF)** \\ \end{tabular} \\ \hline \(V\), m/s & 7425 & 6567 & 7482.2 & 0.76/11.55 \\ \(\gamma\), deg & \(-0.0074\) & 7.2708 & 0 & - \\ \(h\), km & 747.6 & 807.6 & 741.2 & 0.86/8.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The comparison summary of 3 DOF and 6 DOF results for the hybrid last stage.