Earth–Jupiter Mission: Numerical Results and Comparisons
All mechanisms evolving the tags over subsequent generations were tested on the Earth-Jupiter problem. Mechanism A (with a mutation probability of 5%) generated the best solution and so its solution is presented here in detail. The solution of the first phase shows that the trajectory consists of 2 fly-bys around planets Venus and then Earth; the mission sequence is then Earth-Venus-Earth-Jupiter (EVEJ). The second phase results in adding one DSM, and the total mission cost is 10.1266 km/sec (the fuel consumption can be measured in velocity units). The resulting mission is detailed in Table 9.10.
Table 9.10: HGGA solution of Earth-Jupiter problem using Mechanism A.
Mission parameter |
Zero-DSM model (first step) |
MGADSM model (second step) |
Departure Date |
11 — Sep — 2016. 04 : 50 : 27 |
31 —Aug — 2016. 02 : 15: 15 |
Departure Impulse (km/s) |
3.6283 |
3.4414 |
DSM| date |
- |
26— Oct — 2016. 21 : 29 : 29 |
DSMi impulse (km/s) |
- |
0.036516 |
Venus flyby date |
07 — Sep — 2017. 01 : 58: 51 |
05 — Sep — 2017. 09 : 52 : 46 |
Post-flyby impulse (km/s) |
0.031108 |
0.001704 |
Pericenter altitude (km) |
1.333.7876 |
1255.2411 |
Earth flyby date |
03-Apr- 2019,09: 55: 30 |
29-Mar-2019,09 :56: 34 |
Post-flyby impulse (km/s) |
0.4685 |
0.4522 |
Pericenter altitude (km) |
637.7999 |
637.80(H) |
Arrival date |
25-Dec-2021_{:}23 : 05 : 00 |
25 — Dec — 2021. 18:23:09 |
Arrival impulse (km/s) |
6.2813 |
6.1948 |
TOF (days) |
360.88083.934.21184.636.66743 |
370.3177.570.0026.902.3518 |
Mission cost (km/s) |
10-4092 |
10-1266 |
Also both Logic C and Logic A were tested on this problem; logic A demonstrated superiority compared to logic C in this problem and hence only the results of logic A are here presented. The total cost for the mission is 10.1181 km/sec. The detailed results of both steps are presented in Table 9.11.
Table 9.11: HGGA solution of Earth-Jupiter problem using Logic A.
Mission parameter |
Zero-DSM model (first step) |
MGADSM model (second step) |
Departure Date |
01 —Sep —2016,22:58: 19 |
29-Aug — 2016, 16 : 04 : 38 |
Departure Impulse (km/s) |
3.4811 |
3.1398 |
DSM date |
- |
07-Oct-2016,05 : 57: 10 |
DSM impulse (km/s) |
- |
0.34746 |
Venus flyby date |
05-Sep-2017. 10:42:20 |
06 — Sep — 2017. 19: 14: 50 |
Post-flyby impulse (km/s) |
0.006200 |
l.9788e —05 |
Pericenter altitude (km) |
1330.9042 |
972.1739 |
Earth flyby date |
29-Mar-2019, 23 : 48 : 29 |
29 — Mar - 2019, 14: 31 : 32 |
Post-flyby impulse (km/s) |
0.4398 |
0.4396 |
Pericenter altitude (km) |
637.8000 |
637.8000 |
Arrival date |
19 — Sep — 2021,03 : 07 : 38 |
23-Sep-2021, 15:01 : 36 |
Arrival impulse (km/s) |
6.1999 |
6.1891 |
TOF (days) |
368.4889,570.5459,904.1383 |
373.1321,568.8033,909.0209 |
Mission cost (km/s) |
10.1299 |
10.1181 |
Comparing the results of Logic A with Mechanism A, we can see that the cost of the mission using Logic A is slightly better than that obtained using Mechanism A. The mission architecture is the same from both methods, while the values of the other variables are slightly different. The success rate of an algorithm is a measure for how many times the algorithm finds the best found solution in a repeated experiment (see Chapter 6). This experiment was repeated 200 times using Logic A and the success rate is 75.5%, as shown in Fig. 9.11.
Previous solutions in the literature for this problem can be divided into two categories. The first category of methods do not search for the optimal architecture; rather the trajectory is optimized for a given architecture. Reference [91], for instance, presents a minimum cost solution trajectory for this Earth-Jupiter mission, assuming a fixed planet sequence of EVEJ. The departure, arrival, and fly-bys dates were also assumed fixed, with a launch in 2016 and a mission duration of 1862 days. The primer vector theorem solution has four DSMs. Two DSMs are applied in the first two legs. The total cost for this solution is 10.267 km/s, which is about slightly higher than the obtained cost in this section. The method presented in this section, however, has the advantage of the autonomous search for the optimal architecture of the solution. The obtained solution here has the same planet sequence of EVEJ but a different DSM architecture compared to [91]. Reference [49] and Section 9.2 present the solution to this problem using the simple HGGA (without the tags concept). The solution in Section 9.2 also finds the planet sequence of EVEJ, and has a total mission cost of 10.182 km/sec, which is slightly higher than the cost obtained in this section. This problem was also solved using Mechanisms E and F (presented in Section 6.5) and the results were presented in [4]. The total cost obtained using Mechanism E is 10.1438
Figure 9.11: Success rate versus number of runs for the Earth-Jupiter trajectory optimization problem using Mechanism A and Logic A.
Figure 9.12: Mechanism A: EVEJ Trajectory for MGADSM Model.
km/s and using Mechanism F is 10.9822 km/s, which are higher than the cost obtained in this section. The mission trajectory obtained using Mechanism A is shown in Fig. 9.12.
Figure 9.13: Evolution of tags using Logic C in the Earth-Jupiter problem.
As a demonstration for how the tags evolve over subsequent generations, consider this Earth-Jupiter problem solved using Logic C. The population size is 300 and the number of generations is 100. Six tags are examined. Figure 9.13 shows the number of times each tag has a value of ‘ Г in each generation. For example, tag 6 takes a value of ‘ Г in all the population members in generations 55 and above. In the З0'^{л} generation, for instance, tag 6 takes a value of ‘ Г in only 40 chromosomes and takes a value of ‘0’ in the other 260 chromosomes. The other 5 tags converge to a value of ‘0’ in the last population in all the chromosomes.