MGA T RAJECTORY M ODELS
Algorithm 2.1. Computing the entire trajectory in position formulation
1: For l 1 nlegs 2: For i 1 narcs
l 3: Compute Lambert arc 4: End For5: For i1narcs
l 16: Compute of the DSM between arc i and vl i, i1 7: End For
8: End For
9: For l1nlegs 1
10: Compute powered swing-by between leg l and l 1 11: Find cost of the lth swing-by vlSB
12: End For
2.3.4 Discussion
The position formulation model is flexible concerning the number of planetary swing-bys, as well as the number of DSMs in each leg. In fact, it is possible to vary the number of DSMs or swing-bys without any substantial change to the structure of the trajectory, and only at the cost of adding 4 more variables to the solution vector for each additional DSM, and 1 variable for each additional leg. As it will be shown in the following, especially with regard to the number of DSMs, this is something that cannot be achieved with the velocity formulation.
An advantage of this formulation is that the complexity of the MGA problem grows polynomially with the number of swing-bys and with the number of DSMs.
In the following subsections, we will briefly discuss these two features of the position formulation model. In addition, a few considerations about using powered and unpowered swing-bys are presented.
USING MORE DSMS PER LEG
There are essentially two advantages in using more than one manoeuvre in each leg.
The first is that each DSM splits the Lambert arcs, thus enabling multiple revolution legs without solving a multiple revolution Lambert problem. For example, let us assume to have one leg with no DSM. If we solve the arc using a single revolution Lambert arc, then spacecraft cannot perform more than one revolution during that leg. Now imagine introducing a DSM in the leg: in this case, the leg will be composed of two single-revolution Lambert arcs, and so the spacecraft will be able to perform up to 2 complete revolutions. Note that shorter solutions are still available even with an arbitrary number of DSMs.
This can be seen as a way to tackle multiple revolution legs without the need for an integer to represent the number of revolutions in the Lambert problem, and thus avoiding to have a mixed integer-real search space.
The second advantage is that it allows a better distribution of the along the leg. For some types of legs, two impulses are necessary to reach the desired target
v
orbit with the correct phase. In addition, if we fraction the v provided by a single DSM into more manoeuvres, the amount of v of each manoeuvre is smaller. This means that even a smaller level of thrust, hence a smaller engine, is needed to obtain the velocity change. The more DSM we consider for each leg, the more the leg approximates a low-thrust arc, in which the v is provided continuously and is distributed along the entire arc.
The following example illustrates how the total cost of the trajectory can take benefit from multiple DSMs. Two types of EVM transfers are considered: both of them have no DSM in the first leg; in the second leg, instead, the former has one DSM, while the latter has got two. If we identify each DSM with a d in the sequence, the two sequences can be identified with EVdM and EVddM, respectively.
For both instances, the objective is to minimise the total v , which is the sum of the launch excess velocity, the relative velocity at arrival at Mars (assuming that we would like to rendezvous with Mars), and the DSMs. An optimisation can be run for the two cases, using the bounds presented in Table 2.2: the best solutions found for either case are represented in Fig. 2.18. Although the two transfers appear identical, the one with two manoeuvres is 50 m/s cheaper.
MODEL COMPLEXITY
As we mentioned already, one important advantage of this formulation is that the model complexity is polynomial with respect to the number of DSMs and the number of swing-bys. This is a direct consequence of the fact that the position formulation allows us to compute each arc independently of the other arcs.
Table 2.2. Bounds for EVM transfer, both in case of one and two DSMs in the second leg.
EVdM EVddM Variable
LB UB LB UB t , d, MJD2000 0 4452.5 4492.5 = =
T1, d 152.29 192.29 = =
T2, d 677.61 717.61 = =
r 1 0 1 = =
1, rad 0 2 = =
1, rad –0.1 0.1 = =
1 0.01 0.6 0.01 0.3
r1 / / 0 1
1, rad / / 0 2
1, rad / / –0.1 0.1
1 / / 0.3001 0.6
-3 -2 -1 0 1 2
Fig. 2.18. Best solutions found for the two instances of the EVM transfer problem.
(a) EVdM, the total Δv is 8.14 km/s; (b) EVddM, the total Δv is 8.09 km/s.
To explain better this concept, let us assume, without loss of generality, that the problem is planar and that the distance of each DSM from the centre of the coordinate system is constant; then, the position Mi of each DSM can be identified by a single variable: the angle
Mi The time at which the DSM happens requires another variable,
Mi distinct possible positions on an ideal time-space grid for a DSM is equal to hk . The position of the planets is determined through the ephemerides, thus only one parameter, the epoch, has to be specified. Once again, it is assumed that there is a finite set of h possible epochs. This means that there are
h hk h k 2
). These arcs can be computed independently of the rest of the trajectory, once
1 1
1, M , M
t t are given. The same holds for connecting the last DSM to the arrival planet of the leg.
An arc connecting two consecutive DSMs is determined when time and position of the two DSMs is fixed (see Fig. 2.19 for an ideal representation). Thus, the total number of independent arcs is
2 2. hk hk h k
Once again, these arcs can be computed independently of the other parts of the trajectory. For the trajectory given as an example, with 2 DSMs, the total number of independent legs is
2 2 2 2 2 2 2
h k h k h k h k h k 2
Fig. 2.19. If we imagine to discretise position and time of two consecutive DSMs, then the possible arcs connecting the two are all and only those found by connecting a point on the grid DSM 1 to a point on the grid DSM 2. The arcs do not depend on any other parameter in the solution vector.
and in general, if nDSM DSMs are considered:
2 2
2h k nDSM 1 h k2.
Therefore, the position formulation does not suffer from any dependency on the previous legs and the growth of the number of solutions is polynomial.
COMPARISON BETWEEN POWERED AND UNPOWERED SWING-BYS
It was mentioned in Chapter 1 that the powered swing-by generates super-optimal solutions, with respect to the model with manoeuvres in deep space only. This is due to the fact that performing a corrective manoeuvre at the pericentre of the swing-by hyperbola is more efficient than performing the manoeuvre in deep space, even if the manoeuvre is very close to the planet ( ). As a result, the same 0 transfer trajectory may result to be less expensive (in terms of v ) when computed using powered swing-bys, than the using velocity formulation. This can be a problem because, as discussed, powered swing-bys pose serious constraints in the operations phase.
As an example, let us consider the Cassini mission. If the trajectory is modelled with the position formulation and no DSMs, the resulting minimum- trajectory is shown in
v
Fig. 2.20 (a): the total v , including launch excess velocity and all the powered swing-bys, resulted to be 4.45 km/s (the statement of this transfer optimisation problem, including bounds and objective function, can be found in [65]).
The same solution can be constructed using the velocity formulation with DSMs. A first guess is generated by extracting all the parameters necessary for the velocity formulation from the optimal trajectory in position formulation, except for the positions of the DSMs. These are initially set very close to the departure planet (i 0.01). An optimisation is then run to find a locally optimal solution in a
r2 DSM 2
r1
DSM 1
t2
k
t1
h
neighbourhood of the first guess. The resulting solution, in Fig. 2.20 (b), has a cost of 4.58 km/s, which is about 130 m/s more expensive than the analogous solution in position formulation.
(a) Position formulation (b) Velocity formulation
Fig. 2.20. ACT “Cassini1” optimal solution with position formulation (a) and velocity formulation (b).
2.4 Discussion
In this chapter, two different formulations for modelling an interplanetary MGA trajectory have been presented: the position formulation and the velocity formulation. Appendix B describes a paradigm to model virtually any type of trajectory, by decomposing it into building blocks. It is shown that the paradigm can be used to model the position and the velocity formulations of the trajectory, as well as a combination of the two. Furthermore, it is shown that it is possible to sort the blocks in a sequence such that it can be evaluated incrementally.
The incremental pruning method that will be presented in this thesis is based on the velocity formulation of MGA trajectories; some aspects of the position formulation were addressed by Becerra et al. in [104] and [105]. The main motivations for addressing the velocity formulation can be summarized as follows:
the velocity formulation gives an unconstrained problem with lower dimensionality with respect to the position formulation, which in turns gives a constrained problem with higher dimensionality; the model based on the velocity formulation is closer to the actual way trajectories are operated in space (e.g. no powered swing-bys are performed in real missions); finally, as mentioned above, a model based on the position formulation can generate solutions that are more energy efficient than a model based on the velocity formulation, therefore the latter is more conservative.
Nevertheless, an incremental approach based on the position formulation is possible, thanks to the findings in Appendix B.