MGA T RAJECTORY M ODELS
2.3 Position Formulation
2.3.2 Powered Swing-by
A powered swing-by manoeuvre exploits a combination of gravity and propulsive action to turn the relative incoming velocity vector onto the relative outgoing velocity vector [59]. The legs preceding and following the swing-by provide the incoming velocity v and the outgoing velocity v .
Assuming that the spacecraft approaches the planet from infinity ( r ), and considering that the planet moves with velocity , the relative incoming and outgoing velocities and
vP
v v can be computed as v / v / v . Since P and are computed independently one from the other, they do not necessarily have the same modulus and the gravity of the planet cannot be sufficient to turn the incoming asymptote onto the outgoing one. Therefore, an impulse manoeuvre is introduced at the pericentre of the inbound hyperbola leg (denoted with (1) in
v
v
Fig.
2.16) to generate an outbound hyperbola leg (denoted with (2) in Fig. 2.16) with the required . Since the velocity is maximum at pericentre, a manoeuvre in this point represents the most effective way to achieve the outgoing conditions [
v
103].
Fig. 2.16. Geometry of the powered swing-by with tangential manoeuvre at pericentre.
In theory, a single manoeuvre at the pericentre of the hyperbola, at distance , normal to the local position vector and in the plane of the hyperbola, should be sufficient to achieve the required outgoing velocity vector. However, the distance is limited to be above the surface of the planet (or above its atmosphere). The set of possible outgoing velocities is therefore limited. If no tangential manoeuvre can achieve the desired outgoing conditions, a non-tangential manoeuvre is required. In the next two sections, the two models for tangential and non-tangential manoeuvre are presented.
rp
rp
TANGENTIAL MANOEUVRE
If the manoeuvre is tangential to the orbit at pericentre, it does not change the position of the pericentre (see Fig. 2.16). Using the results found for the unpowered swing-by, the semimajor axes of the two legs are:
and the corresponding eccentricities, expressed as a function of the radius of the pericentre are: rp
The problem then is to find the radius of the pericentre (or equivalently, since is given, the targeting distance
rp
v ) such that the required deflection angle 1
is achieved. In particular, considering Fig. 2.16, we can state that the semi-deflection of each leg of the hyperbola is related to its eccentricity in the following way:
Thus, the total deflection angle should be:
1 2
This can be treated as a zero-finding problem, with limited from below.
Thus we can state the problem as:
rp
We can use a Newton-Raphson method, using rp min, as a starting point:
If a value of is found, then there is no need to investigate further. The cost of the powered swing-by in terms of
rp
At each iteration, we also have to monitor whether r becomes smaller than p i . If so, then this strategy cannot find a feasible solution to the problem (i.e., a too small value of is necessary), and the loop can be aborted. Should this happen, the following strategy is started.
, p min
r
rp
NON-TANGENTIAL MANOEUVRE
If a solution cannot be found with a tangential manoeuvre at pericentre, then the following strategy is used to find a suitable non-tangential manoeuvre at the pericentre of leg (1).
The value for the radius of the pericentre of leg (1) is set to rp,1rp m, in. In fact, this allows exploiting the maximum possible deviation from the natural dynamics of the swing-by, thus minimising the propelled . v
Since the manoeuvre is not tangential, the pericentre of leg (2) changes. With reference to Fig. 2.17, we can consider that the line of apsides of leg (2) is rotated by an angle with respect to leg (1). The problem in this case is to find the value of such that leg (2) passes through the pericentre of leg (1). The polar equation for the hyperbola leg (2) can be written as:
2
22
2
1
1 cos
r a e
e
. (2.22)
To force the passage, we can impose that:
p,1r r and Eq. (2.22) becomes:
2
22 1
p,1 2 p,1cosg a e r e r 0. (2.23)
Fig. 2.17. Geometry of the powered swing-by with a non-tangential manoeuvre at pericentre.
,2
v
v
,1
rp
2
,2
vp
,1 ,1
vp
1
So the problem is:
This time, there are no physical constraints on the value of , nevertheless care must be taken as there are two singularities in Eq. (2.23) for each period of . Once has been found, for example using once more a Newton-Raphson iterative method, then the can be computed. The modulus of the two velocity vectors at , before and after the manoeuvre, can be easily computed, from the energy equation, as:
The former is purely transversal, as it is at the pericentre of leg (1), while the latter is not (see Fig. 2.17). Its transversal and radial components can be calculated from the angular momentum:
2
At this point, we can compute the total change in velocity needed:
p,2, p,1
2 2p,2,v v v v
r .
Before concluding this section, let us remark that unpowered swing-bys can be used in the position formulation. It was shown above that the powered swing-by is a manoeuvre to match the incoming and outgoing velocity vectors at the planet. If the unpowered swing-by is used, instead, Eqs. (2.4), (2.7) and (2.10) define non-linear constraints that cannot be solved explicitly, but they have to be taken into account as non-linear constraints in the trajectory model. This ensures that the modulus of the incoming and outgoing relative velocity vectors are the same, and the deflection angle is small enough that it can be achieved with a radius of pericentre above the lower limit, as required by the unpowered swing-by model.
2.3.3 Overall Trajectory Parameterisation
Given a sequence of planetary swing-bys, a launch planet and an arrival planet, the entire trajectory is composed of a set of legs, which connect the planets, from departure to arrival, through all the swing-bys. Each leg may contain one or more DSM.
Table 2.1. Solution vector for a trajectory in position formulation.
The parameters needed for modelling a trajectory with the position formulation are summarised in Table 2.1. The launch date determines when to depart from the departure planet. The timing of all the swing-bys and arrival to the last planet are defined through the time of flight of each leg . The ephemerides provide the position of all the planets at the given times. Position and timing of DSMs are given through
and the total number of variables in the solution vector to fully characterise the trajectory is:
1nlegs 4nDSM. The solution vector for this trajectory is therefore:
1 1 1 1 2 2 2 2
Algorithm 2.1 presents briefly the loops needed to compute the entire trajectory. The overall trajectory is built by computing first all the deep space legs (and the DSMs between each couple of arcs), and then the swing-bys (and their cost).
The total cost of the trajectory is obviously the sum of all the DSMs and the swing-by impulses. The launch excess velocity and relative velocity at the target planet of the sequence can be computed taking the difference between the velocities at the bounds of the trajectory and planetary velocities.