# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 4 Numerical Simulations
### Orbit Injection with Dispersed Initial Conditions and Errors in Perturbation Estimation  
The aim of this section is to show the effectiveness of feedback orbit control in the presence of dispersed initial conditions, which are representative of orbit injection errors at the planetary capture, and errors in estimating the orbit perturbations.  
To model this, random displacements--with uniform distribution--on periares and apoares radii are assumed in place of semimajor axis and eccentricity. For each simulation, the latter orbit elements can be obtained once the periares and apoares radius is specified. The remaining orbit elements obey a normal distribution with mean value and standard deviation reported in Table 4, together with upper and lower bounds, for each variable.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Unit** & **Mean Value** & **Std Deviation** & **Lower Bound** & **Upper Bound** \\ \hline Periares radius & [km] & - & - & 3596 & 10,000 \\ \hline Apoares radius & [km] & - & - & 80,000 & 150,000 \\ \hline Inclination & [deg] & 92.29 & 5.00 & 87.29 & 97.29 \\ \hline RAAN & [deg] & 64.70 & 5.00 & 59.70 & 69.70 \\ \hline Argument of periares & [deg] & 342.39 & 5.00 & 337.39 & 437.39 \\ \hline True anomaly & [deg] & 180.00 & 5.00 & 175.00 & 185.00 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Monte Carlo 2: mean value, standard deviations, and bounds for the stochastic initial orbit elements.  
Figure 15: Monte Carlo analysis 1: correlation analysis.  
Moreover, errors in estimating the orbital perturbations are modeled as well. Each component of the perturbing acceleration is assumed to be  
\[a_{p,k}=a_{p,k}^{(est)}\left\{1+\vartheta_{k}\sin\!\left[\frac{2\pi}{T}t+\varphi_{ k}\right]\right\}\ \ \left(k=r,\theta,h\right) \tag{46}\]  
where \(\left\{\vartheta_{k}\right\}\) and \(\left\{\vartheta_{k}\right\}\) are six random variables, with uniform distribution in \([0,0.05]\) and \([0,2\pi]\), respectively, \(T\) is the time-varying orbital period associated with the instantaneous osculating orbit; symbol \(a_{p,k}^{(est)}\) denotes the estimated value of the perturbing acceleration component, which is compensated through feedback control. It is apparent that the oscillating term that appears in Equation (46) represents the unmodeled perturbing acceleration component, with a magnitude up to \(5\%\) of the nominal value.  
Even if the spacecraft starts its orbital transfer with nonnominal orbit elements, the closed-loop guidance strategy can be used to complete the orbit transfer. To prove this, a set of 1000 Monte Carlo simulations (campaign MC 2) is run with also the intent of obtaining information on propellant consumption and time of convergence.  
The results of this study are summarized in Figures 16-21: in all the 1000 simulations, the spacecraft successfully reaches the operational orbit. The orbit elements converge to the desired values, and the orbit acquisition is successfully completed. In Table 5, the mean value and the standard deviation of the convergence time and of the final mass ratio are reported. Figures 16-18 witness the effectiveness and robustness of the guidance strategy at hand. Moreover, an inspection of Figure 19 reveals that no impact with the Martial surface occurs (the minimum altitude equals 804 km, i.e., the final, desired periapse altitude). Finally, Figure 20 depicts the time of convergence to the target set for all the simulations, whereas Figure 21 portrays the propellant fraction.  
Figure 19: Monte Carlo analysis 2: radius (**a**) and related zoom (**b**).  
Figure 17: Monte Carlo analysis 2: inclination (**left**) and RAAN (**right**).  
Figure 18: Monte Carlo analysis 2: periareon argument (**left**) and mass ratio (**right**).