# Robust Near-Optimal Aerocapture Guidance Method Based on Saturation Function
## 4 Simulations and Result Analysis
### Algorithm Verification and Analysis
#### 4.1.1 Preliminary Monte Carlo Simulation  
The Monte Carlo simulation considers random changes in the entry state, vehicle mass, aerodynamic coefficients, and atmospheric density. Tables 2 and 3 show the dispersions in the entry state and spacecraft parameters. Only the spacecraft's mass follows a uniform distribution, others follow a Gaussian distribution. Note that the mean value of all random variables is zero. Compared with literature [35], dispersions of entry state have doubled in this paper, and the lift and drag coefficients' \(3\sigma\) dispersions also increase from 15% to 20%. Regarding atmospheric density dispersions, ideally, they can be modeled by the Earth Global Reference Atmospheric Model (GRAM) [39]. Without access to it, a simplified analytical model of atmospheric density is used to simulate the dispersion characteristics,  
\[\rho_{\text{true}}=\rho_{\text{nom}}\left\{1+B_{\text{atm}}\,+\left[M_{1}+M_{2 }\sin(\omega_{2}h)\right]\sin(\omega_{1}h+\theta_{\text{atm}})\right\} \tag{26}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**State** & **Distribution** & **Dispersion** \\ \hline Longitude & Gaussian & \(3\sigma=0.4\) deg \\ Latitude & Gaussian & \(3\sigma=0.4\) deg \\ Velocity & Gaussian & \(3\sigma=60\) m/s \\ Flight-path angle & Gaussian & \(3\sigma=0.02\) deg \\ Heading angle & Gaussian & \(3\sigma=0.1\) deg \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Entry-state dispersions for the Monte Carlo simulation. Figure 4: Nominal results for Earth aerocapture: (**a**) relative velocity versus altitude; (**b**) bank angle versus time. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Distribution** & **Dispersion** \\ \hline Lift coefficient & Gaussian & \(3\sigma=0.07398\) (20\%) \\ Drag coefficient & Gaussian & \(3\sigma=0.274\) (20\%) \\ Mass & Uniform & \(\pm 5\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Vehicle parameter dispersions for the Monte Carlo simulation. where \(\rho_{\rm nom}\) is the nominal atmospheric density, \(B_{\rm atm}\) is a constant bias subject to a uniform distribution, \(M_{1}\) and \(M_{2}\) are amplitudes subject to a uniform distribution, \(\omega_{1}\) and \(\omega_{2}\) are parameters representing low and high frequencies, respectively, which reflect the disturbances' periodic changes in altitude, and \(\theta_{\rm atm}\) is the phase. Based on the above dispersions, 100 Monte Carlo simulations are carried out in the Earth aerocapture scenario, and the simulation results are shown in Figures 5 and 6. Figure 5 shows the relationship between relative velocity and altitude. It can be seen that the decrease in velocity, that is, energy dissipation, mainly occurs in a narrow 15 km altitude zone from 60 to 75 km. In addition, the exit velocities are all between 7750 and 7850 m/s. In Figure 6, the bank angle control profile for Monte Carlo simulations is shown. It can be seen that the bank angle time history basically conforms to the bang-bang structure, ensuring the algorithm's near-optimal performance. In most of the simulations, the bank reversal happens in about 200 s. Due to various parametric disturbances, the bank angle in some cases finally saturates, while in others, it remains within a certain range. To further verify the algorithm's performance through comparison, the method proposed in this article is designated as Mode 1, and the single-impulse guidance method proposed in the literature [24] is designated as Mode 2. Monte Carlo simulation is carried out under the same conditions, including the same scenario, spacecraft model, entry conditions, dispersions, and operational limits. The statistical results are shown in Table 4.  
Figure 5: Relative velocity versus altitude for Earth aerocapture. Figure 6: Bank angle versus time for Earth aerocapture. From the table, the total \(\Delta V\)'s mean value and standard deviation in Mode 1 are smaller than those in Mode 2. The total \(\Delta V\)'s mean value in Mode 1 is reduced by 42.5% compared to that in Mode 2.