Consider a transfer from Earth to Jupiter via Venus, Earth, Earth swing-bys. We consider no DSM in the first leg (and thus the first leg is a ballistic arc), and the subdivision into levels shown in Table 3.2.
The domain bounds and the number of intervals for each variable are shown in Table 3.3. These bounds have been chosen because from [41] it is know that they define a domain D that contains an optimal solution.
The hyperbola pericentre radius is normalised with respect to the radius of the planet at which the swing-by is performed, and the intervals for this variable are not equally spaced over the domain, but they are smaller for low values of the radius. This is done because the outgoing velocity from a swing-by is much more sensitive to the variation of the radius, when the radius is small, than when it is large. This choice allowed the use of fewer intervals for the variable .
rp
rp
Table 3.3. Domain bounds and number of intervals for each variable.
Level Variable Lower
bound Upper bound No. of intervals (p ) i
t , MJD2000 0 3159 3559 19
1
T , d 1 120.4 220.4 19
2, rad 2.742 3.789 19
2
,2
r p 1.672 2.272 9
2 0.01197 0.2619 19
3
T , d 2 220.2 420.2 19
3, rad 2.127 3.174 19
4
,3
r p 1.466 2.066 9
3 0.01728 0.2772 19
5
T , d 3 630.4 830.4 19
4, rad 2.638 3.685 19
6
,4
r p 1.312 1.912 9
4 0.01740 0.3074 19
7
T , d 4 747.0 947.0 19
Considering the first level of the problem only, it is possible to plot what is commonly called pork chop plot, that is a plot of the needed at the first planet v (in this case, Earth) in order to reach the second planed (Venus) as a function of the starting date t0 and the time of flight T1 (Fig. 3.3). The grid in the plot is representative of the intervals in which the domain has been divided into: each hyper-rectangle of the grid is a node, and it is represented by its middle point. In this example, the function has been evaluated once, for each node, in the centre of the corresponding sub-domain.
If the solution space of the first level is pruned as shown before, setting a pruning criterion:
1 v1 5 km/s
,
the algorithm removes all the parts of this domain which exceed this limit. The only remaining areas are shown in Fig. 3.4.
T1, d t 0, d, MJD2000
140 160 180 200 220
3200 3250 3300 3350 3400 3450 3500 3550
5 10 15 20 25 30 35
v, km/s
Fig. 3.3. Level 1, no pruning done. Transfer cost (Δv) as a function of departure time t0 and time of flight T1. The black dots represent the grid sample points.
T1, d t 0, d, MJD2000
140 160 180 200 220
3200 3250 3300 3350 3400 3450 3500 3550
2.8 3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8
v, km/s
Fig. 3.4. Level 1, after pruning. Transfer cost (Δv) as a function of departure time t0
and time of flight T1. Only feasible nodes have been plotted.
The coloured areas in Fig. 3.4 represent the domain that survived after pruning at first level D . The second level is a swing-by, thus the v1 is not a good pruning criterion. Instead, by considering the aim of this particular swing-by, that is to increase the semimajor axis of the transfer orbit, it is possible to prune the second level by requiring that the Venus swing-by increases the semimajor axis by at least 27,986,077 km. That is:
2 a 27986077 km/s
where is the difference between the semimajor axis before and after the swing-by of Venus. This quantity can easily be computed as the swing-swing-by fully determines the conditions of the outgoing heliocentric orbit. The numerical value for pruning has been set from the various data in literature about similar trajectories [
a
v
41, 74].
Then, since in the third level there is an impulsive manoeuvre, the criterion is a limit on 2:
3 v2 0.5 km/s
.
The value is suitable for a DSM. The results of the pruning up to level 3, with these criteria, are shown in Fig. 3.5 to Fig. 3.7. Fig. 3.5 shows the pork chop plot of the first two variables. The number of feasible nodes is clearly decreased with respect to Fig. 3.4. This is due to the fact that the pruning of levels 2 and 3 has made part of the nodes at level 1 infeasible. The infeasible nodes are then removed from level 1 according to the back pruning strategy explained in Section 3.2.2.
T1, d t 0, d, MJD2000
140 160 180 200 220
3200 3250 3300 3350 3400 3450 3500 3550
3.8 4 4.2 4.4 4.6 4.8
v, km/s
Fig. 3.5. Level 1, after pruning up to level 3. Transfer cost (Δv) as a function of departure time and time of flight. Only feasible nodes have been plotted.
rp,2, R
Fig. 3.6. Level 2, after pruning up to level 3. Increment of semimajor axis (Δa) due to the swing-by of Venus, as a function of the rotation angle and the hyperbola pericentre radius. Only feasible nodes have been plotted.
T2, d
Fig. 3.7. Level 3, after pruning up to level 3. Cost of the deep space manoeuvre (Δv) as a function of the DSM position α2 and time of flight T2. Only feasible nodes are plotted.
In Fig. 3.6, the increment of the semimajor axis due to the Venus fly-by is represented, as a function of the variables of the second level, and fixed the values of the first level, as t03359 d, MJD2000, T1170.5 d.
By fixing also the values of the variables of the second level (23.265 rad
2
, ), the of the third level can be plot as a function of
,2 2.090
rp v2 and (T2
2
Fig.
3.7). The plot highlights that only nodes with a of about 320 days have survived the pruning process. Furthermore, the timing of the DSM is not particularly important for pruning the solution space, as almost any value of
T2
within the considered global bounds generates at least one solution, which satisfies all pruning criteria.
The whole search space can be pruned down, until to the last, level by continuing this process of incremental and backward pruning. Fig. 3.8 represents one of the possible feasible trajectories. The solution in Fig. 3.8 corresponds to the following solution vector:
3359,170.5,
3.265,2.090,0.05802,320.2, 2.651,1.766,0.06517,730.4, 3.162,1.612,0.02503,941.8
x
-6 -4 -2 0 2 4 6
-6 -4 -2 0 2 4 6
x, AU
y, AU
Arrival at J
Departure from E Swing-by at E Swing-bys of V
Fig. 3.8. One of the feasible trajectories after the pruning of the whole domain.
The total of this trajectory is about 5.34 km/s, without considering any v capture manoeuvre at Jupiter. As a comparison, the best known solution, for the specified bounds, requires 4.84 km/s. It is worth to note that no search for locally optimal solution was performed: the solution in Fig. 3.8 was found only as the result of the systematic search over all the nodes survived to the pruning process.