# Optimal Gliding Trajectories for Descent on Mars
## 3 Results
### Uncertainties on Aerodynamic Coefficients: Sensitivity Analysis  
The planetary entry and descent are complex aeromaneuvers, because they require high precision, which is difficult due to the estimated uncertainty of some factors, such as the behavior of the atmospheric conditions. The atmosphere is a stochastic system that depends on solar activity, the magnetic field of the planet, and other variables. On the other hand, factors like uncertainties in the state vector and/or in the aerodynamic performance could reduce the precision of the controller. In order to analyze that, the changes in \(l/d\) on the trajectory were analyzed from a mean value of 0.96, assuming a normal distribution with 0.01 on standard deviation. In this study, 100 random values for the spacecraft's \(l/d\) ratio were generated, and the simulations were repeated for each value. The main goal is to see how this parameter affects the spacecraft's trajectories, particularly the location of the final or terminal point. Figures 6-12 show the results. Looking at all the figures, Figure 6 confirms that all the trajectories ended with the spacecraft on the surface of Mars. Thus, no escapes from the planet were obtained in the simulations, which means that the variations in the \(l/d\) ratio were not strong enough to cause the spacecraft to either leave Mars or collide. Figure 7 shows in detail the final azimuth and flight path angles. We see that most of the trajectories ended with flight path angles near \(-18.6\) degrees, so the variations in the \(l/d\) ratio did not affect this variable much. The opposite occurs for the final azimuth angle. They have a large dispersion, going from zero to more than 100 degrees. So, this variable is very sensitive to the \(l/d\) ratio, and it is impossible to make accurate predictions of this value without knowing the \(l/d\) ratio of the spacecraft very well. Figure 6: Final velocity and altitude for 100 trajectories. Figure 8: Final longitude and latitude for 100 trajectories. Figure 7: Final FPA and azimuth for 100 trajectories. Figure 9: Velocity and altitude as a function of time for 100 trajectories. Figure 11: History of AOA from 100 simulations. Figure 12: History of bank angle from 100 simulations. Figure 10: Longitude vs. latitude for 100 trajectories. Figure 8 is the most important one for the goals of the present research. It shows the location of the final points. The final latitude is around 30 degrees for almost all trajectories. The final longitude goes closer to 26 degrees for the majority of the trajectories, which is a better interval compared to the results obtained for the latitude. So, in general, the variations in the \(l/d\) ratio change the final point but keep them in a square of 5 by 2 degrees in terms of latitude and longitude. In a more detailed maneuver, this final approach is selected to begin at a higher altitude so that there is enough time to loiter and make the final adjustment before the landing. After that, systems like retrorockets, parachutes, or skids (used on the X-15) could be deployed. Figure 9 indicates that almost all the trajectories (in different colors) have similar behaviors when compared to the nominal one in terms of velocity and altitude. With very few exceptions, they start the motion towards Mars, bounce back when reaching a point of higher density in the atmosphere, and then the spacecraft falls again. The velocity has a slow decreasing phase during this bouncing motion and a fast decrease when out of this zone. Figure 10 shows the different trajectories that each value of \(l/d\) generates for the spacecraft. We see a large concentration of solutions near the nominal trajectories, not only at the landing point but also throughout the motion. The angle of attack (Figure 11) is also variable. Almost all the trajectories follow the nominal trajectory, and variations in \(l/d\) have lower effects.