As already mentioned, we cannot afford the necessary resources to make a full Monte Carlo simulation.
Therefore, changing one parameter at a time, we can study its impact in the output in order to choose the most critical to be used in our final simulation. We must have in mind that even if changing one parameter does not present a significant influence in the output, its effect may be enhanced by changing simultaneously another parameter from which the former is dependent on. Hence, the following tests are merely a general approximation to predict the critical inputs that present a great impact in the output when changing inside a restricted interval.
The uncertainties in the trajectory inputs may arise from sudden fluctuations in the atmospheric conditions or due to the errors when guessing the value of a given property. In Table 7.3 are presented all the parameters that contribute to these uncertainties in the trajectory (indirectly through the atmospheric and drag coefficient models) and their respective influence in the landing site coordinates. The uniform
Parameter Distribution Landing site
µx(m) µy(m) σx(m) σy(m) Local pressure (mbar) N(1027,12) −7089 7096 0.065 0.065
Pressure rate (m−1) U(0.1,3) −7090 7096 0.427 0.425
Local temperature (◦C) N(21,22) −7089 7096 0.537 0.537 Temperature rate (m−1) U(0.1,4) −7089 7096 0.835 0.840 Local windspeed (m s−1) N(4.3,12) −7500 7507 2078 2077 Surface roughness length –z0(cm) U(4,10) −7095 7102 286 286 Downwind (from north, clockwise) (◦) N(45,72) −7153 6903 956 989
Table 7.3: Mean (µ) and standard deviation (σ) of the landing site in the two directions (x– Southing;
y – Easting) changing each input within a distribution of 20 samples.
distribution is used to pick up values for the parameters that are unknown within an specific interval. On the other hand, the normal distribution is used to generate values for the parameters that are measured but may present some disturbances. In this case, the mean represents the expected value under the current conditions and the standard deviation specifies how much the measured value may deviate from the mean. The minimum and maximum thresholds in the uniform distributions from the pressure and temperature rates are defined, respectively, in a way that the atmospheric profiles match the CIRA-86
model at about 500 m of altitude and at the apogee. The standard deviations for the local pressure and temperature are defined as the greatest variation occurred in a 30 min interval during April 15, 2015 in Lisbon [66]. In the case of the wind speed’s normal distribution, its standard deviation is defined as the necessary amplitude that changes the local wind to another class according the Beaufort scale [62] (see fourth paragraph in Section 6.3). Finally, the standard deviation for the wind direction is defined as the variation occurring at the measured wind speed found from a dataset taken during daytime in a flat grass surface [67].
Comparing the mean landing coordinates with the standard deviations, in Table 7.3, we conclude the uncertainties from the pressure and temperature measured at the launch site, and their respective rates to match the atmospheric model along the altitude, present a negligible impact under the defined conditions.
On the other hand, the impact from the remaining parameters’ uncertainties is much more considerable, which leads us to neglect the first four parameters from Table 7.3. With only three parameters left, we still have to compute a billion iterations if we use samples of 1000 values for each parameter, as illustrated in Figure 7.1. Therefore, for the remaining normal distributions (wind speed and direction), we take samples of 200 values since, after testing, we concluded that a larger sample would not improve the landing site dispersion but only saturate the graphic. In respect to the surface roughness length, since we do not know any characteristic that helps to define it other than the terrain type (defined as high grass), it can take any value within the range of 4 cm to 10 cm (see Table 3.1). Hence, in order to reduce the simulation time, we decided this parameter should only take the worst case scenarios, i.e., 4 cm and 10 cm. Nevertheless, it is still required to replace the CD model by a constant value since it slows down each iteration and it is not expected to impact the results much. Therefore, theCD used in the simulation is the mean value of 0.70 calculated from
CD=
Z tapogee
0
CD(t)
tapogee , (7.2)
whereCD(t) is the drag coefficient function found from the developed model after simulating a trajectory under the same conditions used in this chapter.
Figure 7.2 shows the outcome of the Monte Carlo simulation resulting from the input conditions mentioned above. Since the rocket follows the wind speed direction, it is expected that the uncertainties from the local wind velocity and the surface roughness length (both only influencing the wind intensity) change the landing site along this direction, describing a straight line. Analogously, the changes in the wind direction, due to its uncertainties, make the landing points to describe an arc of a circumference that gets larger as the distance to the launch site increases.
The dispersion of the landing points, in Figure 7.2, clearly shows the influence of the normal distribu- tions used to generate the local wind direction and speed applied in each iteration. Since a mean of 45◦ was defined for the wind direction’s normal distribution, the orange dots are heavily concentrated along the northeast direction. In the same way, the rocket lands less often apart from the northeast direction as there are fewer inputs that deviate more from the mean. Also due to this principle, the wind speed uncertainties contribute to the lower density of points near the closest and furthest distances from the
Figure 7.2: Landing positions and respective confidence ellipses from the Monte Carlo simulation.
launch site. This pattern can also be seen in Figure 7.3 where the probability density function of the landing sites is represented. Furthermore, this figure enables us to distinguish better the most proba- ble landing area since Figure 7.2 gets saturated in this region due to the great amount of trajectories simulated.
The confidence ellipses in Figure 7.2 are rather eccentric. Their major axes (aligned northeastward) measure 10.4 km, for a confidence of 95%, and 9.1 km, for 90% of confidence. As the furthest landing site, collinear with the major axes, is 14.6 km away from the launch location, the major axis from the lowest confidence level represents almost two thirds of the furthest distance. Thus, we conclude the measured wind speed, allied with the surface roughness length range, brings a strong uncertainty to the landing site estimation under the specified conditions. Regarding the minor axes compared to their distances from the launch site, we see that the wind direction also induces a considerable uncertainty.
0 0.02 0.04 0.06
Figure 7.3: Probability density function of the landing coordinates resulting from the Monte Carlo simulation.