Noname manuscript No.
(will be inserted by the editor)
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
F. Topputo
Received: date / Accepted: date
Abstract In this paper two-impulse Earth–Moon transfers are treated in the restricted four-body problem
with the Sun, the Earth, and the Moon as primaries. The problem is formulated with mathematical means
and solved through direct transcription and multiple shooting strategy. Thousands of solutions are found,
which make it possible to frame known cases as special points of a more general picture. Families of solutions
are defined and characterized, and their features are discussed. The methodology described in this paper is
useful to perform trade-off analyses, where many solutions have to be produced and assessed.
Keywords Earth–Moon transfer · low-energy transfer · ballistic capture · trajectory optimization ·
restricted three-body problem · restricted four-body problem
1 Introduction
The search for trajectories to transfer a spacecraft from the Earth to the Moon has been the subject of
countless works. The Hohmann transfer represents the easiest way to perform an Earth–Moon transfer. This
requires placing the spacecraft on an ellipse having the perigee on the Earth parking orbit and the apogee
on the Moon orbit. By properly switching the gravitational attractions along the orbit, the spacecraft’s
motion is governed by only the Earth for most of the transfer, and by only the Moon in the final part.
More generally, the patched-conics approximation relies on a Keplerian decomposition of the solar system
dynamics (Battin 1987). Although from a practical point of view it is desirable to deal with analytical
solutions, the two-body problem being integrable, the patched-conics approximation inherently involves
hyperbolic approaches upon arrival. This in turn asks for considerable costs (i.e. ∆v maneuvers) to inject
the spacecraft into the final orbit about the Moon.
Low energy transfers have been found in the attempt to reduce the total cost for Earth–Moon transfers.
They exploit the concept of temporary ballistic capture, or weak capture, which is defined in the framework
of n-body problems (Belbruno 2004). It has been demonstrated that low-energy transfers may lower the
arrival ∆v , so entailing savings on the transfer cost (Belbruno and Miller 1993; Topputo et al. 2005b). A low
energy transfer was used in the rescue of the Hiten mission (Belbruno and Miller 1993; Belbruno 2004), and,
more recently, a similar transfer has been executed in the mission GRAIL (Hoffman 2009; Roncoli and Fujii
2010; Chung et al. 2010; Hatch et al. 2010). Low energy transfers are being preferred to standard patchedconics transfers for their appealing features, such as the lower cost, the extended launch windows, and the
property of allowing for a fixed arrival epoch for any launch date within the launch window (Biesbroek
and Janin 2000; Chung et al. 2010; Parker et al. 2011; Parker and Anderson 2011). The extended transfer
Francesco Topputo
Department of Aerospace Science and Technology
Politecnico di Milano
Via La Masa, 34 – 20156, Milano, Italy
Tel.: +39-02-2399-8351
Fax: +39-02-2399-8334
E-mail: [email protected]
2
F. Topputo
time also benefits operations, check-outs, and out-gassing1. It has been demonstrated that the cost for low
energy transfers can be further reduced by using low-thrust propulsion (Belbruno 1987; Mingotti et al. 2009a,
2012). The mission SMART-1 performed such a combination successfully (Schoenmaekers et al. 2001). The
low energy approach has also been extended to the case of interplanetary transfers (Topputo et al. 2005b;
Hyeraci and Topputo 2010, 2013).
1.1 Low Energy Exterior and Interior Transfers
Low energy transfers can be classified into exterior and interior, according to the trajectory geometry. In the
exterior transfers the spacecraft is injected into an orbit having the apogee at approximately four Earth–
Moon distances. In this region, a small maneuver, combined with the perturbation of the Sun, makes it
possible to approach the Moon from the exterior, and to perform a lunar ballistic capture (Belbruno and
Miller 1993). It has been shown that the performances of the transfer can be improved by avoiding the
mid-course maneuver (Belbruno and Carrico 2000) and by introducing a lunar gravity assist at departure
(Mingotti et al. 2012). The exterior transfers exploit the gravity gradient of the Sun in a favorable way
(Yamakawa et al. 1992, 1993). It has been found that these transfers can take place when the orbit is
properly oriented with respect to the Sun, the Earth, and the Moon. More specifically, in a reference frame
centered at the Earth, with the x-axis aligned with the Sun–Earth line, and with x growing in the anti-Sun
direction, the exterior transfers can take place when the apogee lies either in the second or in the fourth
quadrant (Belló-Mora et al. 2000; Circi and Teofilatto 2001; Ivashkin 2002; Circi and Teofilatto 2006).
Exterior transfers can be also viewed in the perspective of Lagrangian points dynamics (Belbruno 1994;
Koon et al. 2000). In the coupled restricted three-body problem approximation, the four-body problem that
governs the dynamics of the exterior transfers, is split into two restricted three-body problems having Sun,
Earth and Earth, Moon as primaries, respectively. It can be shown that the first portion of the exterior
transfer is close to the stable/unstable manifolds of the periodic orbits about either L1 or L2 of the Sun–
Earth problem. Analogously, the second part (i.e. the lunar ballistic capture) is an orbit defined inside the
stable manifold of the periodic orbits about L2 in the Earth–Moon problem (Gómez et al. 2001; Koon et al.
2001; Parker 2006; Parker and Lo 2006). With this approach, second and fourth quadrant transfers are
obtained by exploiting the dynamics of the Sun–Earth L1 and L2 , respectively.
In the interior transfers most of the trajectory is defined within the Moon orbit. These transfers exploit
the dynamics of the cislunar region in the Earth–Moon system (Conley 1968; Belbruno 1987; Miele and
Mancuso 2001). As the equilibrium point is approached asymptotically (to keep the energy of the transfer
low), interior transfers usually have long durations (Bolt and Meiss 1995; Schroer and Ott 1997; Macau
2000; Ross 2003). Also, multiple maneuvers are required to raise the orbit starting from low-altitude Earth
parking orbits (Pernicka et al. 1995; Topputo et al. 2005a). Although for such long transfers perturbations
may in principle produce significant effects, the solar perturbation is not crucial here, and interior transfers
may be defined in the Earth–Moon restricted three-body problem. By postulating that the spacecraft would
transit from the Earth region to the Moon region at the energy associated to that of L1 , it is possible to
quantify the minimum cost for a transfer between two given orbits (Sweetser 1991). Other kinds of interior
transfers exploit the resonances between the transfer orbit and the Moon. These require shorter flight times,
though their energy level is greater than that of L1 (Yagasaki 2004a,b; Mengali and Quarta 2005).
1.2 Statement of the Problem
In this paper, two-impulse Earth–Moon transfers are studied. These are defined as follows. The spacecraft
is initially in a low-altitude Earth parking orbit. A first impulse, assumed parallel to the velocity of the
parking orbit, places the spacecraft on a translunar orbit. At the end of the transfer, a second impulse inserts
the spacecraft into a low-altitude Moon orbit. This second impulse is again parallel to the velocity of the
Moon parking orbit. Thus, the transfer is accomplished in a totally ballistic fashion; i.e. neither mid-course
maneuvers nor other means of propulsion are considered during the cruise. A two-impulse transfer strategy
involves maneuvering in the proximity of the Earth and Moon. Maneuvering in the direction of the local
velocity maximizes the variations of spacecraft energy (Pernicka et al. 1995). Moreover, a mission profile
1
http://moon.mit.edu/design.html, retrieved on 13 February 2012
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
3
including a low-Earth orbit is consistent with major architecture requirements (e.g., use of small/medium
size launchers, in-orbit assembly of large structures, etc. (Perozzi and Di Salvo 2008)).
For the sake of a better comprehension of the results presented in this work, the departure and arrival
orbits are chosen consistently with the existing literature. A 167 km circular Earth parking orbit, as well as
a 100 km circular orbit about the Moon have been extensively used to compute Earth–Moon transfers (see
Section 1.3). Thus, although the results presented in this paper can be easily extended to other orbits, for
which no qualitative difference is expected, quantitatively, the two-impulse Earth–Moon transfers treated
in this paper refer to
• an initial circular Earth orbit of altitude hi = 167 km;
• a final circular Moon orbit of altitude hf = 100 km.
1.3 Summary of Previous Contributions
A number of authors have dealt with Earth–Moon transfers between the two orbits defined above (Sweetser
1991; Yamakawa et al. 1992, 1993; Belbruno and Miller 1993; Pernicka et al. 1995; Yagasaki 2004a,b; Topputo
et al. 2005a; Mengali and Quarta 2005; Assadian and Pourtakdoust 2010; Peng et al. 2010; Mingotti et al.
2012; Da Silva Fernandes and Marinho 2011). Table 1 represents an attempt of harmonization of all the
solutions known to the author. For each solution, the cost (∆v ) and the time-of-flight (∆t) are reported2 .
The two values are given with accuracy of m/s and day, respectively. As different dynamical models can be
used to compute Earth–Moon transfers, the reported solutions have to be framed into the model they are
achieved with; this is also reported in Table 1. The solutions are labeled in terms of transfer type (interior
or exterior), as discussed in Section 1.1; the presence of lunar gravity assists (LGS) is also indicated. The
reader can refer to the works cited in Table 1 for details.
1.4 Motivations
Figure 1 reports the solutions of Table 1 in the (∆t, ∆v ) plane. Each solution taken from the literature
corresponds to a point in this plane. As expected, the cost of transfers tends to reduce for increasing
transfer time. Moreover, it can be noted that the solutions of different studies tend to cluster together. It
is conjectured that these points represent just traces of a more intricate structure, which characterizes the
(∆t, ∆v ) space of solutions of Earth–Moon transfers. Thus, questions arise. Is it possible to reduce known
examples to special points of a more general picture? What does the solution structure of two-impulse
Earth–Moon transfers look like in the (∆t, ∆v ) plane? No matter if the transfers are interior or exterior;
being different solutions of the same problem, they have to be thought of as special branches of a common
picture, which may even contain unknown results.
1.5 Assumptions and Scopes
The aim of this work is to characterize the two-impulse Earth–Moon transfers between two given orbits
within a restricted four-body model. In doing so, the same nonlinear boundary value problem as in Yagasaki
(2004b) is numerically solved to obtain the results. A number of transfer families are found, including many
optimal ones previously reported. The focus is to reveal the structure underlying the points in Figure 1.
Thus, patched-conics, interior, and exterior transfers have to be obtained in a common dynamical framework.
This means that the dynamical model should suffice to allow the reproduction of all these solutions. As
patched-conics, interior, and exterior transfers are defined in the two-, three-, and four-body problems,
respectively, a four-body model is to be used. On the other hand, the model has to be sufficiently simple
to allow the derivation of a general solutions set, without introducing specific mission parameters (e.g.,
launch dates, angular parameters on departure and arrival orbits, etc). With reference to Table 1, the
planar bicircular restricted four-body problem is chosen. This model is deemed appropriate to catch the basic
dynamics characterizing the Sun–Earth–Moon problem (Simó et al. 1995; Yamakawa et al. 1992; Yagasaki
2004b; Mingotti et al. 2012; Mingotti and Topputo 2011; Da Silva Fernandes and Marinho 2011).
2 The patched-conics solutions (Hohmann, bielliptic, and biparabolic) have been calculated by assuming the gravitational
parameters of the Earth and Moon equal to 3.986 × 105 km3 s−2 and 4.902 × 103 km3 s−2 , respectively.
4
F. Topputo
Table 1 Known examples of Earth–Moon transfers between a hi = 167 km circular orbit about the Earth and a hf = 100 km
circular orbit about the Moon. References are ordered by year of publication. The cost (∆v) is either the sum of the two
maneuvers (for two-impulse transfers) or the sum of all required maneuvers (for multi-impulse transfers); ∆t is the time-offlight. When more than four solutions are published (this is the case of Yamakawa et al. (1992); Yagasaki (2004a,b); Mengali
and Quarta (2005); Mingotti and Topputo (2011); Da Silva Fernandes and Marinho (2011)), the four most representative
results in terms of (∆v, ∆t) are reported. The fourth column indicates the model in which these solutions are obtained.
The fifth column specifies the type of transfer according to the geometry of the trajectory (I: Interior; E: Exterior; LGA:
Lunar Gravity Assist).
(†) Apogee radius of 1.5 × 106 km; (∗) h = 200 km; (‡) ∆v of 573 m/s and 676 m/s added to the results published in Yamakawa
i
et al. (1992) and Yamakawa et al. (1993), respectively, to achieve a 100 km circular orbit around the Moon; (◦) speed unit
of 1024.5 m/s and time unit of 4.341 days used to extract data from scaled values available.
Reference
∆v [m/s]
∆t [days]
Model
Biparabolic
Hohmann
Bielliptic(†)
3947
3954
4221
∞
5
90
Patched two-body, 3D
Sweetser (1991)
3726
–
Restricted three-body, 2D
Yamakawa et al. (1992)(∗),(‡)
3806
3783
3749
3741
93
160
111
180
Bicircular restricted four-body, 2D
Belbruno and Miller (1993)
3838
160
Full ephemeris n-body, 3D
Yamakawa et al. (1993)(∗),(‡)
3761
3776
3797
3896
83
95
134
337
Restricted four-body, 2D
Pernicka et al. (1995)
Type
E
I
E
I
E
E+LGA
E+LGA
E+LGA
E+MGA
E+LGA
E+LGA
E+LGA
E+LGA
3824
292
Restricted three-body, 2D
I
(2004a)(◦)
3916
3924
3947
3951
32
31
14
4
Restricted three-body, 2D
E
I
I
I
Yagasaki (2004b)(◦)
3855
3901
3911
3942
43
32
36
24
Bicircular restricted four-body, 2D
Topputo et al. (2005a)
3894
3899
255
193
Restricted three-body, 2D
I
I
Mengali and Quarta (2005)
3861
3920
3950
4005
85
68
14
3
Restricted three-body, 2D
I
I
I
I
Assadian and Pourtakdoust (2010)
3865
3880
4080
111
96
59
Restricted four-body, 3D
E
E
E
Peng et al. (2010)(∗)
3908
3908
3915
3920
112
121
113
108
Patched restricted three-body, 2D
E
E
E
E
Mingotti and Topputo (2011)
3793
3828
3896
3936
88
51
31
14
Bicircular restricted four-body, 2D
E+LGA
E
I
I
Da Silva Fernandes and Marinho (2011)
3857
3866
3901
3944
87
86
32
14
Bicircular restricted four-body, 2D
E
E
I
I
Yagasaki
E+LGA
E
E+LGA
I
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
5
4100
Hohmann
Theoretical Min, Sweetser (1991)
Yamakawa et al (1992)
Belbruno and Miller (1993)
Yamakawa et al (1993)
Pernicka et al (1995)
Yagasaki (2004a)
Yagasaki (2004b)
Topputo et al (2005)
Mengali and Quarta (2005)
Assadian and Pourtakdoust (2010)
Peng et al (2010)
Mingotti et al (2011)
Da Silva and Marinho (2011)
4050
4000
∆v (m/s)
3950
3900
3850
3800
3750
3700
0
50
100
150
∆t (days)
200
250
300
Fig. 1 Solutions of Table 1 reported on the (∆t, ∆v) plane. The horizontal line represents the theoretical minimum (Sweetser
1991); the vertical line separates the region investigated in this paper (white) from the rest of the solutions space.
Another parameter to set is the maximum transfer duration. This has to be limited to avoid the indefinite
growing of possible solutions. In literature, solutions with a time-of-flight of up to 300 days exist (see Table
1 and Figure 1), and Earth–Moon transfers requiring as much as two years have been found (Bolt and Meiss
1995). However, in this work, a lunar transfer requiring hundreds of days is not deemed a viable option
for real missions. For this reason, the characterization of two-impulse Earth–Moon transfers is restricted to
those solutions whose duration is less than 100 days (i.e. the search is limited to the white area in Figure
1). Although this could appear too restrictive, it will be shown that the selected search space possesses
a sufficiently rich structure, which may contain novel transfer families. In this search space, the front of
Pareto-efficient solutions is reconstructed. (These are the solutions showing the best balance between cost
and transfer time; see Definition 2.) The aim is to build a catalogue from which preliminary estimates on
the transfer cost could be extracted if the transfer time was specified (or vice-versa).
Summarizing, the scopes of the paper are:
i. to optimize two-impulse Earth–Moon transfers in the planar bicircular restricted four-body problem;
ii. to reconstruct the solution space in the (∆t, ∆v ) plane, ∆t ≤ 100 days, and to extract the set of Pareto-
efficient solutions;
iii. to characterize the solution structure by identifying families of transfer orbits, some of which may be
unknown in literature.
The remainder of the paper is structured as follows. In Section 2 the background notions are recalled
(restricted three-/four-body problem and temporary ballistic capture). In Section 3 the optimization problem
is treated in terms of first guess solutions, optimization algorithms, and implementation issues. In Section
4 the optimized solutions are presented and discussed on a global basis. Specific results are characterized
and families of solutions are defined in Section 5. Concluding, final considerations are given in Section 6.
2 Background
2.1 Planar Circular Restricted Three-Body Model
In the Planar Circular Restricted Three-Body Problem (PCRTBP) two primary bodies P1 , P2 of masses
m1 > m2 > 0, respectively, move under mutual gravity on circular orbits about their common center of
mass. The third body, P , assumed of infinitesimal mass, moves under the gravity of the primaries, and
6
F. Topputo
in the same plane. The motion of the primaries is not affected by P . In the present case, P represents a
spacecraft, and P1 , P2 represent the Earth and the Moon, respectively. Let µ = m2 /(m1 + m2 ) denote the
mass ratio of P2 to the total mass.
The motion of P relative to a co-rotating coordinate system (x, y ) with the origin at the center of mass
of the two bodies, and in normalized distance, mass, and time units, is described by (Szebehely 1967)
ẍ − 2ẏ =
∂Ω3
,
∂x
ÿ + 2ẋ =
∂Ω3
,
∂y
(1)
where the effective potential, Ω3 , is given by
Ω3 (x, y ) =
1 2
1−µ
µ
1
(x + y 2 ) +
+
+ µ(1 − µ),
2
r1
r2
2
(2)
with r1 = [(x + µ)2 + y 2 ]1/2 , r2 = [(x + µ − 1)2 + y 2 ]1/2 as P1 , P2 are located at (−µ, 0), (1 − µ, 0), respectively.
The motion described by Eqs. (1) has five equilibrium points Lk , k = 1, . . . , 5, known as the Euler–
Lagrange libration points. Three of these, L1 , L2 , L3 , lie along the x-axis;, the other two points, L4 , L5 ,
lie at the vertices of two equilateral triangles with common base extending from P1 to P2 . The system of
differential equations (1) admits an integral of motion, the Jacobi integral,
J (x, y, ẋ, ẏ) = 2Ω3 (x, y ) − (ẋ2 + ẏ 2 ).
(3)
The values of the Jacobi integral at the points Li , Ci = J (Li ), i = 1, . . . , 5, satisfy C1 > C2 > C3 > C4 =
C5 = 3. Table 2 reports the collinear libration points location and their Jacobi energy for the Earth–Moon
system.
The projection of the energy manifold
J (C ) = {(x, y, ẋ, ẏ ) ∈ R4 |J (x, y, ẋ, ẏ ) = C},
(4)
onto the configuration space (x, y ) is called Hill’s region. In the PCRTBP, the motion of P is always confined
to the Hill regions of the corresponding Jacobi energy C . The boundary of Hill’s region the a zero-velocity
curve. Hill’s regions vary with the Jacobi energy C (see Figure 2). In this model, for P to transfer from P1 to
P2 (and vice versa), the condition J < C1 has to be respected. In particular, this can occur at four different
energy ranges: a) C2 ≤ J < C1 ; b) C3 ≤ J < C2 ; c) C4,5 ≤ J < C3 ; d) J < C4,5 . In case a), P can transit
through the small allowed region about L1 , and therefore interior transfers only are admitted (Figure 2(a)).
In case b), P2 may be reached from either L1 or L2 , and thus both interior and exterior transfers are allowed
(Figure 2(b)). In cases c), P can approach P2 from any direction (Figure 2(c)). In case d) P is allowed to
move in the whole physical space. As the mechanical energy of P increases in moving from case a) to d),
the cost of the transfers increase accordingly.
Table 2 Location of the collinear Lagrange points, xLi , and their Jacobi energy, Ci , i = 1, 2, 3, for µ = 0.0121506683.
xLi
L1
0.8369147188
L2
1.1556824834
L3 −1.0050626802
Ci
3.2003449098
3.1841641431
3.0241502628
2.2 Planar Bicircular Restricted Four-Body Model
The Planar Bicircular Restricted Four-Body Problem (PBRFBP) incorporates the perturbation of a third
primary, P3 , the Sun in our case, into the PCRTBP described above. P3 is assumed to revolve in a circular
orbit around the center of mass of P1 –P2 . This model is not coherent as all the primaries are assumed to
move in circular orbits. Nevertheless, it catches basic insights of the real four-body dynamics (Castelli 2011);
this holds as the eccentricities of the Earth and Moon orbits are 0.0167 and 0.0549, respectively, and the
Moon orbit inclination on the ecliptic is 5 deg.
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
7
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P1
L2
L3
(b) C3 ≤ J < C2 .
L1
L2
(c) C4,5 ≤ J < C3 .
Fig. 2 Hill’s regions (white) and forbidden regions (grey) for varying energies.
The equations of motion of the PBRFBP are (Simó et al. 1995; Yagasaki 2004b)
ẍ − 2ẏ =
∂Ω4
,
∂x
ÿ + 2ẋ =
∂Ω4
,
∂y
(5)
where the four-body effective potential, Ω4 , reads
Ω4 (x, y, t) = Ω3 (x, y ) +
ms
ms
− 2 (x cos(ωs t) + y sin(ωs t)).
ρ
r 3 ( t)
(6)
In Eq. (6) t is the time, ms is the mass of the Sun, ρ is the distance between the Sun and the Earth–Moon
barycenter, and ωs is the angular velocity of the Sun in the Earth–Moon rotating frame; the phase angle
of the Sun is θ(t) = ωs t. Thus, the present location of the Sun is (ρ cos(ωs t), ρ sin(ωs t)), and therefore the
Sun-spacecraft distance is
r3 (t) = [(x − ρ cos(ωs t))2 + (y − ρ sin(ωs t))2 ]1/2 .
(7)
It is worth pointing out that the Sun is not at rest in the reference frame adopted in Eqs. (5). This
causes the loss of autonomy, and makes the PBRFBP a time-dependent model (see Eqs. (6)–(7)). This is a
great departure from the PCRTBP as both the equilibrium points and the Jacobi integral vanish. Table 3
reports the value of the physical constants used in this study and their associated units of distance, time,
and speed according to the normalization of the PCRTBP.
Table 3 Physical constants used in this work (taken from Simó et al. (1995)). The constants used to describe the Sun
perturbation (ms , ρ, ωs ) meet the length, time, and mass normalization of the PCRTBP. The units of distance, time, and
speed are used to map scaled quantities into physical units.
Symbol
Value
µ 1.21506683 × 10−2
ms
3.28900541 × 105
ρ
3.88811143 × 102
ωs −9.25195985 × 10−1
lem
3.84405000 × 108
ωem
2.66186135 × 10−6
Re
6378
Rm
1738
hi
167
hf
100
DU
TU
VU
3.84405000 × 108
4.34811305
1.02323281 × 103
Units
Meaning
—
—
—
—
m
s−1
km
km
km
km
Earth–Moon mass parameter
Scaled mass of the Sun
Scaled Sun–(Earth+Moon) distance
Scaled angular velocity of the Sun
Earth–Moon distance
Earth–Moon angular velocity
Earth’s radius
Moon’s radius
Altitude of departure orbit
Altitude of arrival orbit
m
days
m s−1
Distance unit
Time unit
Speed unit
8
F. Topputo
2.3 Ballistic Capture
For the analysis below it is convenient to define the ballistic capture of P by P2 ; i.e. the ballistic capture of
the spacecraft by the Moon. Let x(t) be a solution of Eqs. (5), x(t) = (x(t), y (t), ẋ(t), ẏ(t)), and let H2 (x)
be the Kepler energy of P with respect to P2 ,
H2 =
1 2
µ
v − ,
2 2 r2
(8)
where v2 is the speed of P relative to a P2 -centered inertial reference frame3 ,
v2 = [(ẋ − y )2 + (ẏ + x + µ − 1)2 ]1/2 .
(9)
Definition 1 (Ballistic captures) P is ballistically captured by P2 at time t1 if H2 (x(t1 )) ≤ 0. It is temporary
ballistically captured by P2 for finite times t1 , t2 , t1 < t2 , if it is ballistically captured at any time t ∈ [t1 , t2 ] and
H2 (x(t)) > 0 for t < t1 and t > t2 . It is permanently captured by P2 if it is ballistically captured for all times
t ≥ t3 , for a finite time t3 .
These are the standard definitions of capture (see Belbruno (2004)). In the frame of Earth–Moon transfers, P can experience temporary ballistic captures only. For the capture to be permanent, a dissipative
force must act upon P (De Melo and Winter 2006). To assess if Earth–Moon transfers exhibit temporary
ballistic capture upon Moon arrival it is sufficient to check the sign of the right-hand side of Eq. (8) at
the final time. (See Belbruno (2004); Belbruno et al. (2008); Topputo et al. (2008); Topputo and Belbruno
(2009); Belbruno et al. (2010); Circi (2012) for more details on ballistic capture).
The angular momentum is another two-body indicator that can be used to understand the motion upon
Moon arrival. As the motion is planar, the angular momentum h2 of P with respect to P2 is perpendicular
to the plane (x, y ), and its magnitude is given by
h2 = (x + µ − 1)(ẏ + x + µ − 1) − y (ẋ − y ).
(10)
At arrival, direct or retrograde orbits with respect to the Moon can be detected by simply checking the sign
of the right-hand side of Eq. (10) at the final time.
3 Optimization of Two-Impulse Earth–Moon Transfers
Two impulse transfers are defined as follows. The spacecraft is assumed to be in a circular orbit about the
Earth of scaled radius ri = (Re + hi )/DU . At the initial time, ti , an impulse of magnitude ∆vi injects the
spacecraft
into a translunar trajectory; ∆vi is aligned with the local circular velocity whose magnitude is
p
(1 − µ)/ri . At the final time, tf , an impulse of magnitude ∆vf inserts the spacecraft into the final orbit
around the Moon. This orbitphas scaled radius rf = (Rm + hf )/DU ; ∆vf is aligned with the local circular
velocity whose magnitude is µ/rf . The total cost of the transfer is ∆v = ∆vi + ∆vf and the transfer time
is ∆t = tf − ti .
3.1 Definition of the Optimization Problem
Let xi = (xi , yi , ẋi , ẏi ) be the initial transfer state. This is at a physical distance ri from the Earth, and has
the velocity aligned with the local circular velocity if
(xi + µ)2 + yi2 − ri2 = 0,
(xi + µ)(ẋi − yi ) + yi (ẏi + xi + µ) = 0,
(11)
(12)
are both verified (Yagasaki 2004b). (The reader can refer to Eq. (34) in Appendix 1 for the transformation
from the synodic to the Earth-centered, inertial frame.) For brevity, these two initial conditions are indicated
3
See Appendix 1 for the coordinate transformation behind (9) and (10).
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
9
as ψ i (xi ) = 0, where ψ i is the two-dimensional vector valued function of xi given by the left-hand sides of
(11)–(12). Under these initial conditions, the cost for the initial maneuver is
r
p
1−µ
∆vi = (ẋi − yi )2 + (ẏi + xi + µ)2 −
,
(13)
ri
which can be thought as a scalar, nonlinear function of xi ; i.e. ∆vi (xi ).
Analogously, let xf = (xf , yf , ẋf , ẏf ) be the final transfer state. In order to arrive at a circular orbit of
radius rf around the Moon, with the velocity vector aligned with the local circular velocity, xf must satisfy
(xf + µ − 1)2 + yf2 − rf2 = 0,
(xf + µ − 1)(ẋf − yf ) + yf (ẏf + xf + µ − 1) = 0.
(14)
(15)
Equations (14)–(15) are denoted as ψ f (xf ) = 0; the two-dimensional vector valued function ψ f is defined
by the left-hand sides of (14)–(15). Under these final conditions, the arrival maneuver can be quantified by
r
q
µ
∆vf = (ẋf − yf )2 + (ẏf + xf + µ − 1)2 −
,
(16)
rf
which is a scalar, nonlinear function of the final state; i.e. ∆vf (xf ). We can now proceed to state the
optimization problem as follows.
Statement of the problem. Let x(t) = ϕ(xi , ti ; t) be a solution of (5), integrated from (xi , ti ) to t ≥ ti .
The optimization problem for two-impulse Earth–Moon transfers consists of finding {xi , ti , tf }, tf > ti , such that
ψ i (xi ) = 0, ψ f (xf ) = 0, where xf = ϕ(xi , ti ; tf ), and the function
∆v (xi , ti , tf ) = ∆vi (xi ) + ∆vf (ϕ(xi , ti ; tf ))
(17)
is minimized.
3.2 Forming an Initial Guess
From (17) it can be inferred that a two-impulse Earth–Moon transfer is uniquely specified once the six
scalars {xi , ti , tf } are given. In principle, one can guess the initial state, the initial time, and the final time,
and check if the boundary conditions (11)–(12) and (14)–(15) are satisfied, and the objective function (17)
is minimized. In the most likely event in which either the boundary conditions are violated or the objective
function is not minimized, the guess can be used to define an initial solution, or first guess solution. This
can be treated in an optimization framework, where both the boundary conditions and the minimization of
the objective function are enforced.
Since this work aims at reconstructing the totality of solutions of two-impulse Earth–Moon transfers,
thousands of initial guesses have to be processed to find locally optimal solutions. This can be done by
performing a systematic search (i.e. a grid sampling) within the space in which the six scalars are defined.
However, systematic searches suffer from the curse of dimensionality, and the high nonlinearities characterizing most Earth–Moon transfers would require a fine sampling of the variables. From this perspective, it
would be desirable to guess as few numbers as possible. A transformation has been developed to define first
guess solutions by specifying four scalars only. With reference to Figure 3, these quantities are:
α, an angle defined on the Earth circular parking orbit;
β , the initial-to-circular velocity ratio;
ti , the initial time;
δ , the transfer duration.
Given
{α, β, ti , δ}, the construction of initial guess solutions is straightforward. Let r0 = ri and v0 =
p
(1 − µ)/r0 , the initial transfer state, x0 (α, β ) = (x0 , y0 , ẋ0 , ẏ0 ), reads
β
x0 = r0 cos α − µ,
y0 = r0 sin α,
ẋ0 = −(v0 − r0 ) sin α,
ẏ0 = (v0 − r0 ) cos α.
(18)
In Eq. (18), α determines the location of the initial point on the circular orbit, and β specifies the initial
velocity, which is already aligned with the circular velocity. It is easy to verify that x0 (α, β ) automatically
satisfies Eqs. (11)–(12); i.e. ψ i (x0 (α, β )) = 0, ∀{α, β}.
10
F. Topputo
v0
x0 (α, β)
x̃f = ϕ(x0 , ti ; ti + δ)
α
Earth
r0
Moon
rf
Fig. 3 Direct shooting of initial guess solutions.
The final state is defined by integrating x0 in the time span [ti , ti +δ ]; i.e. x̃f (α, β, ti , δ ) = ϕ(x0 (α, β ), ti ; ti +
δ ). If ψ f (x̃f ) = 0 and ∆v (x0 (α, β ), ti , δ ) is minimum, then the combination {α, β, ti , δ} produces an optimal
solution; otherwise, it defines an initial solution for a subsequent optimization step (see Figure 3).
The search of first guess solutions can be further specialized by defining the set of admissible parameters.
The selection of lower and upper bounds for each parameter sets the four-dimensional hypercube to sample
(this is the reason why δ is the transfer duration and not the final time; (Armellin et al. 2010)). As for
√ the first
parameter, it is straightforward that α ∈ [0
√, 2π ]. The second parameter is allowed to be β ∈ [1, 2], where
β = 1 sets the circular velocity, while β = 2 defines the Earth-escape velocity. As for ti , setting the initial
time specifies the initial angular position of the Sun through θi = ωs ti (see Eqs. (6)–(7)). Thus, ti ∈ [0, 2π/ωs ]
yields θi ∈ [0, 2π ]; i.e. any possible relative configuration of Sun, Earth, and Moon is considered. The transfer
duration δ ∈ [0, T ] is bounded by T , which is the maximum prescribed transfer time. First guess transfers
∼ 23 T U (see Table 3).
with durations of up to 100 days are obtained by setting T =
Let Iα , Jβ , Kti , Lδ be the number of points in which the domains of α, β , ti , δ defined above are
discretized, respectively. A direct shooting of first guess solutions is implemented through Algorithm 1. The
generic first guess orbit, γ (ijkl) , can be used to initiate the subsequent trajectory optimization.
Algorithm 1 Pseudo code for first guess solution generation.
for i = 1 → Iα do
α(i) = 2π(i − 1)/(Iα − 1)
for j = 1 → Jβ do
√
β (j) = 1 + ( 2 − 1)(j − 1)/(Jβ − 1)
for k = 1 → Kti do
(k)
ti = 2π/ωs (k − 1)/(Kti − 1)
for l = 1 → Lδ do
δ(l) = T (l − 1)/(Lδ − 1)
(k) (k)
γ (ijkl) = {ϕ(x0 (α(i) , β (j) ), ti ; ti + δ), δ ≤ δ(l) }
end for
end for
end for
end for
3.3 Solution by Direct Transcription and Multiple Shooting
Optimal two-impulse Earth–Moon transfers are found with a direct transcription and multiple shooting
approach. Direct transcription maps a differential optimal control problem into an algebraic nonlinear
programming (NLP) problem; see Enright and Conway (1992); Betts (1998) for details. This method has
shown robustness and versatility when used to derive trajectories defined in highly nonlinear vector fields
(Mingotti et al. 2009a, 2011a,b, 2012). Moreover, it does not require the explicit derivation of the necessary
conditions of optimality, and it is less sensitive to variations of the first guess solution.
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
11
Let the PBRFBP Eqs. (5) be written in the first-order form ẋ = f (x, t), and let x(t) be a solution. This
is discretized over a uniform time grid, which is made up by N points such that t1 = ti , tN = tf , and
tj = t1 +
j−1
( t − t1 ) ,
N −1 N
j = 1, . . . , N.
(19)
Let xj = (xj , yj , ẋj , ẏj ) be the solutions at the j -th mesh point; i.e. xj = x(tj ), j = 1, . . . , N . The multiple
shooting method requires the dynamics (5) to be integrated within the N − 1 intervals [tj , tj +1 ] by taking
xj as initial condition (see Figure 4). The continuity of position and velocity requires the defects
ζ j = ϕ(xj , tj ; tj +1 ) − xj +1 ,
j = 1, . . . , N − 1
(20)
to vanish. Let the NLP variable vector be defined as
y = {x1 , x2 , . . . , xN , t1 , tN },
(21)
where the initial, final time, t1 , tN , are let to vary, to allow the optimizer to converge to an optimal initial
configuration and transfer duration. Let ψ 1 = ψ i (x1 ) and ψ N = ψ f (xN ). Then the vector of nonlinear
equality constraints is given by
c(y ) = {ψ 1 , ζ 1 , ζ 2 , . . . , ζ N −1 , ψ N }.
(22)
The trajectory optimization has to avoid solutions which impact either the Earth or the Moon. This is
imposed at each mesh point through
(Re /DU )2 − (xj + µ)2 + yj2 < 0,
(Rm /DU )2 − (xj + µ − 1)2 + yj2 < 0,
j = 1, . . . , N.
(23)
Moreover, the transfer duration has to be strictly positive; i.e. t1 − tN < 0. Thus, the vector of nonlinear
inequality constraints is
g (y ) = {η 1 , η 2 , . . . , η N , τ },
(24)
where η j = η (xj ) is the two-dimensional vector valued function defined by the left-hand sides of Eqs. (23)
and τ = t1 − tN . The scalar objective function is
f (y ) = ∆v1 + ∆vN ,
(25)
where ∆v1 = ∆vi (x1 ) and ∆vN = ∆vf (xN ), defined in Eq. (13) and (16), respectively. It is easy to verify that
y is (4N + 2)-dimensional, c is 4N -dimensional, and g is (2N + 1)-dimensional; with inequality constraints
satisfied, the problem has 4N + 2 variables and 4N constraints.
NLP problem. Optimal, two-impulse Earth–Moon transfers are found by solving the NLP problem:
min f (y ) subject to c(y ) = 0,
y
gj ( y ) < 0 , j = 1 , . . . , 2 N + 1 .
(26)
3.4 Continuation of Optimal Solutions
Once an optimal solution is found, it would be desirable to continue it in order to reconstruct, locally, the
whole solution space. This is crucial to derive the totality of solutions and characterize them. To do that,
it is convenient to use the transfer duration as continuation parameter.
Suppose an optimal solution to problem (26) has been found: y = {x1 , . . . , xN , t1 , tN }. This solution has
transfer duration ∆t = tN − t1 . The aim is to use y to derive a new optimal solution having a fixed duration
∆t′ = ∆t + δt, where δt has to be small enough to allow the convergence to a new optimal solution. Thus, the
new problem has to enforce the constraint on the transfer duration. This is done through the new condition
σ = tN − t1 − ∆t′ = 0,
(27)
that yields the augmented, (4N + 1)-dimensional vector of equality constraints
c′ (y ) = {ψ 1 , ζ 1 , ζ 2 , . . . , ζ N −1 , ψ N , σ}.
(28)
Continuation problem. Optimal solutions are continued in transfer duration by solving the NLP problem:
min f (y ) subject to c′ (y ) = 0,
y
gj ( y ) < 0 , j = 1 , . . . , 2 N + 1 .
(29)
12
F. Topputo
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xN −2 xN −1
x2
x1
xN
...
x3
rf
...
ri
Earth
x4
x
ζ1
x2
x1
t1
xN −2
x4
ζ3
ζ2
t2
Moon
ζ N −1
xN −1
...
x3
t3
t4
tN −2
ζ N −2 xN
tN −1 tN t
Fig. 4 Direct transcription and multiple shooting optimization scheme.
3.5 Implementation and Verification
The optimization of two-impulse Earth–Moon transfers described above has been implemented within a
numerical framework. The initial value problems (e.g., ϕ(xj , tj ; tj +1 )) are solved with a variable-order,
multi-step Adams–Bashforth–Moulton scheme (Shampine and Gordon 1975) with absolute and relative
tolerances set to 10−14 . This stringent tolerance is imposed to have accurate solutions within the N − 1
intervals in which the solution is split (when the N − 1 pieces are put together, the solution has to be still
accurate — provided that the equality constraints are satisfied with equally accurate tolerance).
The first guess solutions have been generated with the algorithm sketched in Section 3.2. The fourdimensional search space has been sampled with Iα = 361, Jβ = 101, Kti = 13, and Lδ = 101. Thus, about
4.7 × 107 first guess orbits γ (ijkl) have been generated.
The NLP problems (26) and (29) have been solved with a sequential quadratic programming algorithm
implementing an active-set strategy (Gill et al. 1981) with tolerance on equality and inequality constraints
of 10−10. The gradient of the objective function (25) as well as the Jacobians of the equality and inequality
constraints (Eqs. (22), (24), (28)) are calculated using a second-order central difference approximation. Once
a solution to problem (26) is obtained, the entire family is computed by continuation (problem (29)) with
values of δt of about 5 × 10−2 T U . In Figure 5 the convergence history of a sample solution is reported. It can
be seen that, although the final position of the initial guess misses the Moon by more than one Earth–Moon
distance, the algorithm is still able to converge to an optimal solution.
Optimal solutions are verified a posteriori. If the solution is deemed sufficiently accurate, it is stored,
otherwise it is discarded. The check is made by numerically integrating the optimal initial condition within
the optimal time interval; i.e.
xf = ϕ(x1 , t1 ; tN ).
(30)
The final state xf has to be as close as possible to the optimal final state xf . For a solution to be accepted
it is required that max{|ψ 1 (x1 )|, |ψ N (xf )|} ≤ 10−8 ; i.e. the real trajectory violates the boundary conditions
by less than 10−8 scaled units (∼ 3 m in length and ∼ 0.01 mm/s in speed). Moreover, it is also checked that
the real optimal solution impacts neither the Earth nor the Moon. As the impact-free constraint is imposed
only on the mesh points through Eqs. (23), the verification is done from end to end, by re-integrating the
solution starting from the optimal initial condition; i.e.
ηj (x(t)) < 0,
t ∈ [ t1 , tN ] ,
j = 1, 2,
with x(t) = ϕ(x1 , t1 ; t), and η (x) as from Section 3.3, Eqs. (23).
(31)
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
2
13
1.5
1.5
1.5
1
1
1
0.5
0.5
y
0
0
y
y
0.5
−0.5
−0.5
−1
−1
−1.5
−1.5
0
−0.5
−1
−1.5
−2
−2
−3
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
x
x
x
(a) Iter 0; max cj = 1.866.
(b) Iter 3; max cj = 0.475.
(c) Iter 6; max cj = 0.203.
1.5
2
1.5
1
1
1
0.5
0.5
0.5
0
y
y
y
0
−0.5
−1
−1
−1
−1.5
0
−0.5
−0.5
−1.5
−1.5
−2
−2
−2
−1
0
1
2
−2
−1
0
1
2
−2
−1
0
1
2
x
x
x
(d) Iter 10; max cj = 0.078.
(e) Iter 20; max cj = 0.397.
(f) Iter 29; max cj = 3.9e-11.
Fig. 5 Iterative solutions of the NLP problem (26). Each figure reports the iteration and the maximum equality constraint
violation (max cj ). The first guess (Iter 0) is in Figure 5(a). This is generated with the following set of parameters: α = 1.5π,
β = 1.41, ti = 0, δ = 7 (see Section 3.2). This solution has N = 23 mesh points (red bullets in the figures); this yields 94
variables and 92 equality constraints. Convergence has been achieved in 29 iterations. The computational time is 32.4 s on
an Intel Core 2 Duo 2 GHz with 4 GB RAM, Mac OS X 10.6.
4 Optimal Solutions
4.1 The Global Picture
The optimal solutions found are reported in Figure 6. Each point of the figure is associated to an optimal
solution with duration ∆t and cost ∆v (these two quantities are re-scaled using the time and speed units in
Table 3). In total, approximately 5 × 105 optimal solutions have been found, many of which are not displayed
in Figure 6 (having ∆v > 4150 m/s or ∆t > 100 days). Figure 6 suggests that the solutions are organized
into families, and these families constitute a well-defined structure in the (∆t, ∆v ) plane, thus confirming
the conjecture in Section 1.4. The families have been achieved by continuation with the procedure shown in
Section 3.4. This procedure generates also spurious points not belonging to any known structure. These are
the points in which continuation of solutions fails, and are not considered in the analysis below. It is noted
that the existence of optimal solutions is not guaranteed for every combination of (∆t, ∆v ). This explains
the existence of empty areas between families.
A question arises: how do these optimal solutions relate to those found in previous studies? To answer this
question, the reference solutions in Figure 1 are superimposed on the points of Figure 6. A good matching is
found between the solutions found and those taken from the literature. In particular, the reference solutions
appear to be sample points of the reconstructed solutions set, and, most importantly, almost all of them
are located in the regions about the local minima of the function ∆v (∆t).
For fixed transfer time, the solutions having a lower cost are of more interest in the investigation below.
To ease the analysis, the total set of solutions in Figure 6 is subdivided into two regions that enclose “Short”
and “Long”transfers (‘S’ and ‘L’ squares in Figure 6, respectively). Considering one region at a time helps
characterizing the families of solutions, and allows us to better focus on the solutions deemed of interest.
This is done in Section 5.
14
F. Topputo
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Fig. 6 Optimal two-impulse Earth–Moon transfers, shown in the (∆t, ∆v) plane. The two regions S and L enclose “Short”
and “Long” transfer solutions, respectively. The reference solutions in Figure 1 are superimposed (legend in Figure 1).
4.2 Pareto-Efficient Solutions
Although Figure 6 is generated within a simplified, planar gravitational model, the total solutions set may
serve to mission designers as a catalogue: first guess values for duration and cost can be extracted for
preliminary estimations, and the corresponding solutions may be refined in more complete models. In this
perspective, it would be desirable to extract those solutions for which the balance between ∆t and ∆v
is optimal. In the framework of multi-objective optimization, this means computing the Pareto front of
solutions, made up by the nondominated points.
Definition 2 (Pareto optimality) Let y i and y j be two solutions of Problem (26) (or Problem (29)) defined
as in (21); let ∆v (y) = f (y ) in (25) and let ∆t(y) = tN − t1 . The solution y i strictly dominates y j , or y i ≻ y j ,
if the following conditions are both satisfied:
i) ∆t(y i ) ≤ ∆t(y j ) and ∆v (yi ) ≤ ∆v (y j ),
ii) ∆t(y i ) < ∆t(y j ) or ∆v (yi ) < ∆v (y j ).
The Pareto front is the set of points that are not strictly dominated by another point.
In Figure 7, the Pareto front for the set of solutions in Figure 6 is reported. In practice, the points
belonging to the Pareto front correspond to the solutions having the best balance between cost and time.
This is worthwhile for real Earth–Moon transfers, provided that the mission constraints do not prevent
the choice of a solution belonging to the Pareto front. As a matter of example, the coordinates of the ten
sample Pareto-efficient solutions shown in Figure 7 are reported in Table 4. Moving from left to right, the
solutions range from the Hohmann-like transfer (i) to a particular exterior low energy transfer (x), which is
the cheapest solution found in the entire work (∆v = 3769.3 m/s). The transfers corresponding to the points
in Figure 7, except for solution (vii) whose shape is similar to that of (viii), are reported in Figure 8 in
the Earth-centered, inertial frame. The conversion to the Earth-centered inertial frame as well as the initial
condition, xi = (xi , yi , ẋi , ẏi ), and the initial, final time, ti , tf , for each of these ten solutions, are reported
in Appendix 1. These numerical values are given with high accuracy (15 decimals) to allow an independent
reproduction of the results.
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
15
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Fig. 7 Pareto front (black) associated to the global set of optimal solutions (grey). The coordinates of the ten sample
Pareto-efficient points (red dots) are reported in Table 4.
Table 4 Example of Pareto-efficient solutions in Figure 7; ∆t in days, ∆v in m/s. The initial condition and initial, final
time for each of these solutions are reported in Appendix 1.
Point
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
∆t
∆v
4.6
3944.8
14.4
3936.6
30.6
3892.7
31.9
3887.4
43.6
3840.0
51.3
3828.6
61.9
3828.9
64.6
3805.9
66.7
3801.4
82.6
3769.3
4.3 Angular Momentum at Final Time
The problem statement and, more specifically, the final boundary conditions (14)–(15), enforce the final
transfer state, xf = (xf , yf , ẋf , ẏf ), to be at a physical distance rf from the Moon and to have the velocity
parallel to the local circular velocity. Yet, these conditions do not specify the direction of the final velocity,
so allowing the possibility of having both direct and retrograde final orbits around the Moon. These are
discriminated by computing the angular momentum with respect to the Moon associated to xf , h2,f =
h2 (xf ), through Eq. (10). Direct orbits are those orbits for which h2,f > 0; retrograde orbits have instead
h2,f < 0. The sign of h2,f for the total solution set in Figure 6 is highlighted in Figure 9.
It can be noticed that, for fixed transfer time, solutions belonging to the same family have different
costs according to the sign of h2,f . Moreover, there is no apparent way to establish a priori whether the
direct or retrograde final orbits are more convenient without having fixed the transfer time. For instance,
by considering the local minima in Figure 9(a), we can infer that acquiring a direct final orbit around the
Moon is more convenient for ∆t . 25 days; retrograde final orbits become more convenient in the range
25 . ∆t . 72 days (see also Figure 9(b)). For 72 . ∆t . 90 days the ∆v of direct final orbits is lower than
that of retrograde orbits, and this conditions is again reversed for 90 . ∆t . 100 days. In summary, the
analysis of the final angular momentum at the Moon shows that the total set of solutions is organized into
a nontrivial structure, which can be understood by a thorough characterization of the families.
4.4 Kepler energy at Final Time
The spacecraft may (or may not) experience ballistic capture upon Moon arrival. To detect those solutions
that exhibit ballistic capture it is sufficient to compute the Kepler energy with respect to the Moon at the
final time, H2,f = H2 (xf ), through Eqs. (8)–(9). If H2,f < 0, then P is ballistically captured by the Moon
16
F. Topputo
(a) Sol (i).
(b) Sol (ii)
(c) Sol (iii)
(d) Sol (iv)
(e) Sol (v)
(f) Sol (vi)
(g) Sol (viii)
(h) Sol (ix)
(i) Sol (x)
Fig. 8 The Pareto-efficient Earth–Moon transfers indicated in Figure 7 shown in the Earth-centered inertial frame (the
Moon orbit is dashed). Solution (vii) is very similar to solution (viii) and therefore it is not shown.
according to Definition 1 (it is straightforward that the Kepler energy with respect to the Moon is positive
at departure). If H2,f > 0, there is no ballistic capture at arrival.
In Figure 10(a) the sign of H2,f is highlighted for the total set of solutions in Figure 6 (ballistic captured
solutions are reported in red). It can be noticed that only a subset of the total solution set performs lunar
ballistic capture at arrival. These are those cases located in the bottom-right part of the (∆t, ∆v ) plane; i.e.
the solutions with long transfer time and low cost. With reference to Figure 8, only solutions (8), (9), and
(10) are ballistically captured at the Moon. While solution (10) belongs to the well-known class of exterior
low energy transfers (Belbruno and Miller 1993; Belbruno 2004; Parker et al. 2011; Parker and Anderson
2011; Yamakawa et al. 1992, 1993), the other two (solutions (8) and (9)) deserve further analysis.
4.5 Jacobi energy at Final Time
In the PBRFBP, in which the optimal two-impulse Earth–Moon transfers are achieved, the Jacobi energy
is no longer an integral of motion. However, computing the Jacobi energy at the final time, Jf = J (xf ) via
Eq. (3), helps understanding the geometry of Hill’s regions upon Moon arrival, where the perturbation of
the Sun can be neglected. Moreover, Jf is an indicator of the energy level of interior transfers, where the Sun
perturbation has little influence (Yagasaki 2004a,b; Mengali and Quarta 2005). In addition, the exterior low
energy transfers can be formulated with the patched restricted three-body problem approximation, which
fixes the Jacobi energy in the Earth–Moon system (Koon et al. 2001; Parker 2006; Parker and Lo 2006;
Mingotti and Topputo 2011; Mingotti et al. 2011b, 2012).
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
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(a) S region.
(b) L region.
Fig. 9 Angular momentum with respect to the Moon at the final time, h2,f , in the (∆t, ∆v) plane. Direct (blue) and
retrograde (red) final orbits around the Moon.
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(a) Kepler energy at final time (H2,f < 0 red, H2,f > 0 blue). (b) Jacobi energy at final time (case b) red, c) green, d) blue).
Fig. 10 Kepler and Jacobi energies at the final time.
The value of Jf for the total solution set in Figure 6 has been calculated through Eq. (3), and the four
energy ranges mentioned in Section 2.1 (Figure 2) have been highlighted in Figure 10(b). First of all, it can
be noticed that no solutions belonging to case a) are found. This is due to the fact that, in the PCRTBP,
orbits defined at energies C2 < J < C1 , do not approach low altitude Earth orbits (Bolt and Meiss 1995;
Schroer and Ott 1997; Macau 2000; Ross 2003). For the Earth–Moon transfer to occur at this energy level, a
multi-impulse strategy has to be considered (Pernicka et al. 1995; Topputo et al. 2005a). Transfers belonging
to case b) are shown red in Figure 10(b). These are associated to the bottom-right part of the (∆t, ∆v ) plane,
where low energy transfers exist. Thus, with the patched restricted three-body problems approximation,
where energies C3 < J < C2 are required, a subset of the ballistic captured solutions (H2,f < 0, Figure
10(a)) can be obtained. Case c) (green points in Figure 10(b)) is associated to solution not belonging to
the Pareto front (Figure 7); these transfers are characterized in Section 5. Finally, case d) (blue points in
Figure 10(b)) encloses the vast majority of solutions found. This means that the interior and a portion
of the exterior transfers have final energies that allow P to move in the entire physical space; thus, P can
approach the Moon from any direction. By comparing Figures 7 and 10(b), it can be observed that solutions
(7)–(9) belong to case d); i.e. solutions of this kind cannot be found with the patched restricted three-body
problems approximation.
18
F. Topputo
5 Characterization of Transfers
In this section the optimal two-impulse Earth–Moon solutions are characterized. Given the global set in
Figure 6, only those solutions showing the best balance in terms of cost and transfer time are considered for
brevity’s sake. In Figure 6 many of the computed optimal two-impulse transfers are observed to construct
some sets of similar curves. Using this structure, we classify the optimal transfers into some families below.
The solutions are parameterized by using the four scalars {α, β, ti , δ}. In Section 3.2 these numbers are
used to define a first guess solution. In this section they are extrapolated a posteriori from the optimal
solution at hand. This is done to present the optimal solutions in a concise form, using parameters having
geometrical or physical meaning. (Alternatively, x1 , t1 , tN could have been given, but this involves reporting
six numbers – instead of four – with no direct meaning). These four numbers are given with high accuracy
for some sample solutions in each family. This allows an independent reproduction of the optimal solutions
through the approach sketched in Section 3.2.
Let x1 = (x1 , y1 , ẋ1 , ẏ1 ), t1 , tN be the optimal initial condition and the initial, final time, respectively.
The angle that locates the initial point on the Earth parking orbit is
y1
α = arctan
,
(32)
x1 + µ
whereas the initial-to-circular velocity ratio is
r
q
r0
r0 + ẋ21 + ẏ12 .
β=
1−µ
(33)
For the analysis below it is convenient to keep track of the initial phase of the Sun (in place of the initial
time ti ). This is given by θi = ωs t1 , modulo 2π . The last parameter is simply δ = tN − t1 .
A thorough investigation has been conducted to analyze the optimal solutions. In particular, for each
solution the following quantities have been tracked (beside {α, β, θi , δ}): angular momentum and Kepler
energy with respect to the Moon at final time (h2,f and H2,f ), Jacobi energy at final time (Jf ), magnitude
of the second impulse (∆vf ), and total cost (∆v ). These quantities are reported in the tables below in
dimensionless units except for ∆v and ∆vf which are given in m/s. Dimensional values can be achieved
through the units in Table 3. Solutions giving rise to similar transfers that differ only by the sign of h2,f
are deemed to belong to the same family. Direct and retrograde final orbits are discriminated by using the
superscript ‘+’ and ‘−’, and called ‘positive’ and ‘negative’, respectively (see Section 4.3). For each family
both positive and negative solutions are shown, provided that both of them exist. Some figures are reported
in Appendix 2 for brevity. It is worth pointing out that only those families deemed of interest have been
characterized. Many other solutions exist, which are not shown below.
5.1 The S Region
The S region contains solutions whose transfer time is less than 50 days (see Figure 6). The dynamics of
the solutions lying in this region are mostly governed by the Earth and Moon gravitation attractions. For
this reason, each of the sample solutions reported is shown in both the Earth–Moon rotating frame and
the Earth-centered, inertial frame. In the latter reference system, it is useful to define nr as the number of
revolutions of P (the spacecraft) around P1 (the Earth).
5.1.1 Family a
Family a is made up of transfers with nr = 0; i.e. Earth–Moon transfers that do not perform a complete
revolution around the Earth. In the (∆t, ∆v ) plane, family a has two local minima, located at ∆t ≃ 5 days,
∆t ≃ 44 days, respectively (Figure 11(a)). Ten sample solutions are reported in Table 5 (five positive, five
negative). The solution a+
1 corresponds to a Hohmann-like transfer with prograde final orbit around the
Moon (∆t = 4.59 days, ∆v = 3944.8 m/s). When moving along the family for increasing ∆t, the transfers
approach the Moon in no favorable conditions, and this explains the local maximum at about ∆t ≃ 27
days. Then, the cost of the solutions decreases due to the presence of lunar close encounters (see Figures 12
−
and 24(a)). This gives rise to the minimum represented by solutions a+
5 and a5 . These solutions cost up to
105 m/s less than the Hohmann transfer with a transfer time of approximately 44 days. Some examples in
Yagasaki (2004a,b); Mengali and Quarta (2005) fall into family a (see Figure 6).
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
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(a) Families a, b.
(b) Families c, d.
Fig. 11 Families a, b, c, d in the ∆t, ∆v plane.
Table 5 Data of transfer samples belonging to family a (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
a+
1
a+
2
a+
3
a+
4
a+
5
4.25229089
5.63387979
0.84297937
3.34893923
4.20833847
1.40215710
1.40494588
1.40676446
1.40851449
1.40204421
1.66738357
6.14451517
1.35668877
3.84224935
2.57748965
1.05792443
2.61596283
4.21874246
6.90369808
10.00079555
0.0114
0.0118
0.0120
0.0121
0.0110
0.3056
0.5158
0.6268
0.6608
0.0928
2.3752
1.9557
1.7340
1.6662
2.8000
810.4
898.9
944.4
958.2
717.3
3944.8
4055.0
4114.7
4142.2
3850.9
a−
1
a−
2
a−
3
a−
4
a−
5
4.06517864
5.36112426
1.19258419
3.64846498
4.20411724
1.40344247
1.40453947
1.40711159
1.40818454
1.40204448
1.35319824
2.70928637
4.83929965
4.18482537
2.57735436
0.80536682
2.32554805
4.60012948
7.36164303
10.04653597
-0.0116
-0.0117
-0.0120
-0.0120
-0.0109
0.3894
0.4637
0.6182
0.5875
0.0682
2.1616
2.0126
1.7030
1.7645
2.8053
846.0
877.3
940.9
928.4
706.4
3990.5
4030.2
4114.0
4109.8
3839.9
5.1.2 Family b
Family b is formed by transfers with nr = 1; i.e. orbits that perform one complete revolution around the
Earth. For each of the two branches (positive and negative) family b shows three local minima and two
local maxima in the (∆t, ∆v ) plane (see Figure 11(a)). The three minima are represented by the solutions
±
±
±
b±
1 , b3 , b5 , whose parameters are reported in Table 6. The first minimum (b1 ) corresponds to solutions
having the same cost of the Hohmann transfer but ∆t three times higher. Again, when moving from this
minimum toward increasing ∆t, the cost increases accordingly until lunar close encounters appear and the
two local minima arise. The geometry of these solutions can be seen in Figures 13 and 24(b). Reference
solutions belonging to family b can be found in Yagasaki (2004a,b); Mengali and Quarta (2005); Mingotti
and Topputo (2011); see Figure 6.
5.1.3 Families c, d
In families c, d the particle P performs two revolutions around the Earth, and thus nr = 2. These two
families are defined in the range ∆t ≃ 24–34 days (see Figure 11(b)). No positive family d is found. Family c
has two local minima (at ∆t ≃ 24 and 32 days) and two local maxima (both at ∆t ≃ 28 days). Family d has
a local minimum at ∆t ≃ 31. The parameters of families c, d are reported in Table 7 and the corresponding
trajectories are shown in Figures 14 and 25. The trajectories in family d are very similar to those of family c,
and thus they are not reported. The examples in Yagasaki (2004a,b); Mingotti and Topputo (2011) can be
deemed belonging to family c, d; see Figure 6. All of the solutions shown so far may be classified within the
class of interior transfers. Solutions similar to those of families a+ , b+ , and c+ have been found in Yagasaki
(2004a,b). The families a− , b− , c− , and d− presented in this paper further enrich the set of two-impulse
Earth–Moon interior transfers.
20
F. Topputo
5
x 10
2
4
1.5
a+
2
0.5
a+
1
a+
3
0
a+
1
−2
−0.5
a+
5
−1
−1.5
−2
−1
0
x (adim.)
a+
3
−4
a+
5
−6
a+
4
−2
−3
a+
2
a+
4
0
y (km)
y (adim.)
1
2
1
−8
2
(a) Rotating frame.
−6
−4
−2
0
x (km)
2
4
6
5
x 10
(b) Earth-centered inertial frame.
Fig. 12 Positive solution samples belonging to family a. Different scales of grey are used from now on to ease the understanding of the figures.
Table 6 Data of transfer samples belonging to family b (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
b+
1
b+
2
b+
3
b+
4
b+
5
0.21510709
3.27491402
4.16580666
3.25940457
4.15756374
1.40255827
1.40545258
1.40184677
1.40740655
1.40196534
3.84979016
3.75654396
1.88698381
3.75017462
5.35862484
3.25979107
6.60786839
8.27446872
6.13644217
7.15636510
0.0115
0.0118
0.0112
0.0121
0.0113
0.3273
0.5180
0.2230
0.6506
0.2271
2.3318
1.9512
2.5400
1.6865
2.5319
819.7
899.8
774.7
954.1
776.5
3957.2
4059.9
3906.7
4129.4
3909.4
b−
1
b−
2
b−
3
b−
4
b−
5
0.36971517
2.44816652
4.17458021
3.04668668
4.00950307
1.40244726
1.40488455
1.40184986
1.40729105
1.40253826
4.11174688
2.92880814
5.03455392
0.39404154
5.33535717
3.44750881
5.70971870
8.29639011
5.91740429
6.93924121
-0.0114
-0.0117
-0.0112
-0.0120
-0.0113
0.2929
0.4635
0.2009
0.6239
0.2510
2.3551
2.0130
2.5393
1.6918
2.4390
804.9
877.2
765.1
943.2
786.8
3941.6
4032.8
3897.1
4117.6
3924.2
5.2 The L Region
The L region encloses solutions with transfer time longer than 50 days (and shorter than 100 days; see
Figure 6). The dynamics that governs this subset of solutions is characterized by the Sun perturbation, as
the corresponding trajectories spend most of the transfer time well outside the Moon orbit (Figure 8). For
this reason, the sample solutions analyzed below are shown in both the Earth-centered and the Sun–Earth
rotating frame. The latter is a co-rotating system of reference centered at the Sun–Earth barycenter where
the Sun and the Earth are at rest. In this frame the Sun–Earth equilibrium points, L1,SE and L2,SE , are
shown, too. In normalized Sun–Earth units, these two points are located at xL1,SE = 0.9900266828 and
xL2,SE = 1.0100340264, and their energy is C1,SE = 3.0003004051 and C2,SE = 3.0003000446, respectively.
These values are obtained with the Sun-Earth mass parameter µSE = 3.0034 × 10−6 (Szebehely 1967). The
trajectory visualization in the Sun–Earth rotating frame is done to both catch the dynamics governing the
L-region transfers and check which quadrant the solutions belong to (see Section 1.1).
Beside the orbit of the Moon, the plots in the Earth-centered inertial frame report the counterclockwise
apparent orbit of the Sun (dashed, not to scale; the initial and final positions of the Sun are labelled with
bullets). In the Earth-centered inertial frame (Figures 16(b), 17(b), 19(b), 20(b), 22(b), 23(b)), two dash-dot
lines are also drawn: these connect the Earth with both the farthest point of the trajectory (Earth–apogee
line) and the Sun location when P is at apogee (Earth–Sun line). The role played by the angle formed
by these two lines in the exterior transfers dynamics is discussed in Circi and Teofilatto (2001). As many
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
x 10
1
5
3
b+
1
2
0.5
b+
2
1
y (km)
y (adim.)
21
0
b+
2
−0.5
b+
1
0
−1
−2
−1
−1.5
−3
−1
−0.5
0
0.5
x (adim.)
(a) Rotating frame.
1
−4
1.5
2
4
5
x 10
5
4
1
b+
4
b+
5
b+
3
0.5
2
b+
4
y (km)
y (adim.)
0
x (km)
(b) Earth-centered inertial frame.
x 10
0
−2
0
−0.5
−2
−1
−4
b+
5
b+
3
−1.5
−2
−1.5
−1
−0.5
0
0.5
x (adim.)
(c) Rotating frame.
1
1.5
−4
−2
0
x (km)
2
4
6
5
x 10
(d) Earth-centered inertial frame.
Fig. 13 Positive solution samples belonging to family b.
families characterize the L region, three solutions are reported for each family, and the trajectory of one
sample solution per family is illustrated for brevity’s sake.
5.2.1 Families e, f
With reference to Figure 15(a), families e, f are defined in the range ∆t ≃ 50 to 55 days. This is one of
the few cases in which there is a remarkable difference in cost between positive and negative transfers, the
latter being about 10 m/s (minimum-to-minimum) more convenient than the former. From a geometrical
point of view, families e, f give rise to short-lasting exterior transfers performing lunar gravity assist. Family
e is made up of second quadrant solutions (Figure 16), while family f contains fourth quadrant transfers
(Figure 26(a)). Unlike the families belonging to the S region, these solutions can be achieved only within
narrow ranges of the initial angular position of the Sun (see Table 8), indicating the importance of the
Sun perturbation. The peculiarity of these two families is that the Moon is approached from the interior
although most of the trajectory is defined outside the Moon orbit. No ballistic capture is performed by
these solutions (H2,f > 0 in Table 8), and the final Jacobi energy (Jf in Table 8) indicates that there are
no forbidden regions at arrival. In the literature, a solution close to e−
2 has been found in Mingotti and
Topputo (2011); see Figure 6.
22
F. Topputo
Table 7 Data of transfer samples belonging to families c, d (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
c+
1
c+
2
c+
3
c+
4
c+
5
4.06378349
3.42728510
2.48070770
3.61113135
4.05803911
1.40152201
1.40356082
1.40221206
1.40253003
1.40135493
4.72945419
3.96094555
3.05881981
0.97340271
4.62727797
7.03389884
6.43288971
5.56800695
6.83505041
7.35158462
0.0112
0.0116
0.0114
0.0114
0.0112
0.2210
0.3848
0.2872
0.3034
0.2084
2.5441
2.2171
2.4118
2.3797
2.5693
773.8
844.1
802.5
809.4
768.3
3903.3
3989.4
3937.3
3946.7
3896.5
c−
1
c−
2
c−
3
c−
4
c−
5
2.75420166
3.34572601
3.64427971
4.05933148
4.31706412
1.40254918
1.40276780
1.40253854
1.40137237
1.40179564
0.14046990
3.85832529
4.21471731
1.48998604
1.62661238
5.84532752
6.51705828
6.86433002
7.33236955
7.60396758
-0.0114
-0.0114
-0.0114
-0.0112
-0.0112
0.2892
0.3017
0.2805
0.1872
0.2195
2.3623
2.3374
2.3799
2.5668
2.5021
803.3
808.7
799.6
759.0
773.1
3940.8
3947.8
3936.9
3887.3
3904.7
d−
1
d−
2
d−
3
d−
4
d−
5
3.63656515
3.83012066
3.91244561
4.08126636
4.31951500
1.40329433
1.40247571
1.40197220
1.40149990
1.40185671
4.23829875
4.59939861
4.71924905
4.71587999
4.81442874
6.62738371
6.80677187
6.88266686
7.05285562
7.30813878
-0.0115
-0.0113
-0.0113
-0.0112
-0.0112
0.3384
0.2740
0.2352
0.1971
0.2246
2.2639
2.3928
2.4706
2.5470
2.4918
824.4
796.8
780.0
763.4
775.4
3967.7
3933.7
3913.0
3892.7
3907.5
Table 8 Data of transfer samples belonging to families e, f (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
e+
1
e+
2
e+
3
4.14772378
4.19538197
4.26019251
1.40209214
1.40205106
1.40207753
2.60580334
2.61012003
2.58325006
11.92791632
11.99341945
12.18915926
0.0109
0.0109
0.0109
0.0654
0.0622
0.0665
2.8546
2.8609
2.8524
705.1
703.7
705.6
3839.0
3837.3
3839.4
e−
1
e−
2
e−
3
4.15715993
4.19883247
4.25352559
1.40207818
1.40205011
1.40207113
2.60680155
2.61552663
2.58069532
11.79207437
11.84504286
12.02828384
-0.0109
-0.0109
-0.0109
0.0444
0.0422
0.0457
2.8531
2.8575
2.8505
695.6
694.7
696.2
3829.4
3828.2
3830.0
f1+
f2+
f3+
4.15189184
4.19879952
4.26135704
1.40208600
1.40205042
1.40207872
5.75484592
5.74949182
5.72513518
11.94756971
12.02236141
12.21462196
0.0109
0.0109
0.0109
0.0665
0.0637
0.0683
2.8525
2.8580
2.8488
705.6
704.3
706.4
3839.4
3837.9
3840.2
f1−
f2−
f3−
4.16754826
4.20419774
4.25369743
1.40206691
1.40204963
1.40207123
5.76042579
5.75349438
5.71573294
11.81931468
11.88174281
12.06657536
-0.0109
-0.0109
-0.0109
0.0451
0.0436
0.0478
2.8517
2.8547
2.8462
696.0
695.3
697.2
3829.7
3828.9
3830.9
5.2.2 Families g, h, i
These three families exist in the range ∆t ≃ 60 to 72 days, where they form two minima (at ∆t ≃ 64 and
67 days, respectively; see Figure 15(b)). Only positive g and negative h, i solutions are found (Table 9).
These solutions show unique features: the transfer orbit foresees a lunar gravity assist that places P in a
close Earth encounter orbit, followed by an exterior-like transfer (see Figures 17 and 26(b)). Families g , i
generate second quadrant transfers, whereas family h contains fourth quadrant solutions. The function H2,f
changes sign along these families, and therefore ballistic capture appears according to Definition 1. This
occurs even though Jf < 3, which means that there are no forbidden regions for P upon Moon arrival. With
reference to Figure 6, no reference solutions having these features are found in the literature (to the best of
the author’s knowledge).
5.2.3 Families j, k
Families j , k are defined within ∆t ≃ 72 and 82 days (see Figure 18(a)). These present two minima at
∆t ≃ 75 and 80, respectively. Positive j and negative k solutions are found. Together with families g , h, i,
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
c+
1
0.6
3
0.4
2
0.2
1
y (km)
y (adim.)
0.8
4
0
−0.2
5
c+
5
c+
3
0
c+
1
−1
c+
5
−0.4
x 10
−2
−0.6
c+
3
−0.8
−1
−3
−0.5
0
0.5
x (adim.)
(a) Rotating frame.
−4
1
−2
0
x (km)
2
4
5
x 10
(b) Earth-centered inertial frame.
x 10
5
3
c+
2
0.5
2
c+
2
1
y (km)
y (adim.)
23
0
c+
4
−1
c+
4
−0.5
0
−2
−3
−1
−1
−0.5
0
0.5
x (adim.)
(c) Rotating frame.
1
−4
−2
0
x (km)
2
4
5
x 10
(d) Earth-centered inertial frame.
Fig. 14 Positive solution samples belonging to family c.
these solutions show a transition between ballistic capture and escape upon Moon arrival (see the sign of
∼ 3, meaning that the energy level at Moon arrival is
H2,f in Table 10). However, these solutions have Jf =
close to C4,5 . Families j , k are made up by second, fourth quadrant transfers, respectively. The geometry of
the trajectories recalls that of an exterior transfer with lunar gravity assist, although from the Sun–Earth
rotating frames it can be inferred that the dynamics of L1,SE and L2,SE is not exploited (i.e. the trajectories
are neither close to possible L1/2,SE periodic orbit nor their manifolds; see Figure 19). No reference solutions
are found in the region families j , k belong to (Figure 6).
5.2.4 Families l, m, n
These three families are defined within ∆t ≃ 78 and 90 days. Each of the three families presents a local
minimum, with the one of family l being the global minimum of the whole ∆v (∆t) function (see Figure
18(b)). This is located at ∆t = 82.6 days, ∆v = 3769.3 m/s. From the solution parameters summarized in
Table 11 it can be inferred that: 1) only positive transfers are found; 2) all of the solutions perform ballistic
capture at Moon arrival (H2,f < 0); 3) the final value of the Jacobi energy at arrival satisfies C3 ≤ Jf < C2
(see Table 2). The latter condition corresponds to the case b) discussed in Section 2.1 and is shown in Figures
2 and 10(b). From Figures 20 (solution l2+ , the global minimum) and 27(a) (solution m+
2 ) it can be seen that
all of the transfers are second quadrant solutions performing a lunar gravity assist. In particular, it can be
seen that the post-swing-by orbit in m+
2 is in 3:1 resonance with the Moon orbit. It is important to notice
that these transfers (the best in terms of total cost) do not exploit the L1/2,SE invariant manifolds structure.
24
F. Topputo
3855
3830
3845
3825
f 1+
3840
e+
1
3835
f 2+
∆v (m/s)
∆v (m/s)
3835
3850
f 3+
e+
3
e+
2
f 3−
f 1− f −
3830
2
−
e1 e−
2
e−
3
g 1−
h−
1
3820
3815
g 2−
h−
2
3810
h−
3
3805
g 3−
i−
1
i−
2
i−
3
3800
52
53
54
∆t (days)
(a) Families e, f .
55
62
63
64
65
66
67
∆t (days)
(b) Families g, h, i.
68
69
Fig. 15 Families e, f , g, h, i in the ∆t, ∆v plane.
5
−3
x 10
x 10
8
6
6
4
2
2
0
L1
L2
y (km)
y (SErf)
4
0
−2
−2
−4
−4
−6
−6
0.995
1
x (SErf)
1.005
(a) Sun-Earth rotating frame.
1.01
−5
0
5
x (km)
10
5
x 10
(b) Earth-centered inertial frame.
Fig. 16 Solution e−
2 belonging to family e.
Reference solutions matching any of these three families are not found, though examples in Yamakawa et al.
(1992, 1993); Mingotti and Topputo (2011) are defined in the same region of families l, m, n (Figure 6).
5.2.5 Families o, p
Families o, p span a wide range of transfer times, from ∆t ≃ 70 to 100 days, with a transfer cost that drops
50 m/s along the families. Both positive and negative solutions are found for the two families, the former
being cheaper than the latter for ∆t > 75 (see Figure 21(a)). The dynamics of these solutions recall that of
a classic exterior transfer (see Figure 22, solution o+
5 , second quadrant transfer, and Figure 27(b), solution
,
fourth
quadrant
transfer).
No
lunar
gravity
assist
p−
exists, and this explains the variability of α in Table
5
12 (departure may occur in any direction, if the Moon has not to be encountered, provided that a suitable
value of θi is considered). From Table 12, a direct relation between transfer cost and final Kepler energy at
Moon arrival can be observed. When H2,f decreases, ∆v decreases accordingly. These families are close to
the reference solutions in Mengali and Quarta (2005); Da Silva Fernandes and Marinho (2011); Assadian
and Pourtakdoust (2010) and adhere to the trajectories used in GRAIL mission (Hoffman 2009; Roncoli
and Fujii 2010; Chung et al. 2010; Hatch et al. 2010); see Figure 6.
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
25
Table 9 Data of transfer samples belonging to families g, h, i (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
g1+
g2+
g3+
4.18237304
4.19369433
4.21070714
1.40268471
1.40265901
1.40262644
0.36355032
0.34820064
0.34162206
14.26905351
14.48063955
14.77502012
-0.0108
-0.0108
-0.0107
0.0282
0.0029
-0.0156
2.8854
2.9362
2.9733
688.4
676.9
668.5
3826.9
3815.2
3806.6
h−
1
h−
2
h−
3
4.14682062
4.15885546
4.15635259
1.40281171
1.40276662
1.40277811
3.48471172
3.46143639
3.44611191
14.29339564
14.55097864
14.79936225
-0.0108
-0.0108
-0.0107
0.0237
-0.0044
-0.0160
2.8945
2.9508
2.9741
686.3
673.6
668.3
3825.9
3812.8
3807.6
i−
1
i−
2
i−
3
4.14224224
4.27648072
4.51532179
1.40286493
1.40258858
1.40284772
0.14736246
0.26443401
0.58605743
14.99949586
15.32147461
15.73544729
-0.0107
-0.0107
-0.0107
-0.0139
-0.0262
-0.0231
2.9699
2.9946
2.9884
669.3
663.7
665.1
3809.2
3801.4
3804.9
5
−3
x 10
x 10
8
10
6
8
6
2
0
L1
L2
y (km)
y (SErf)
4
4
2
−2
0
−4
−2
−6
0.995
1
x (SErf)
1.005
−5
1.01
0
5
10
x (km)
(a) Sun-Earth rotating frame.
5
x 10
(b) Earth-centered inertial frame.
Fig. 17 Solution i−
2 belonging to family i.
Table 10 Data of transfer samples belonging to families j, k (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
j1+
j2+
j3+
3.96454437
4.22463033
4.37950445
1.40367734
1.40207788
1.40226193
2.56638459
3.12430176
3.26548328
16.72624653
17.34374511
17.77648273
0.0108
0.0107
0.0108
-0.0123
-0.0155
0.0160
3.0096
3.0161
2.9531
670.0
668.5
682.9
3816.3
3802.3
3818.1
k1−
k2−
k3−
3.99864546
4.18510811
4.41582183
1.40294205
1.40206304
1.40234774
5.08214538
5.42476134
5.60169530
17.75063250
18.34078867
18.69107060
-0.0107
-0.0107
-0.0107
-0.0233
-0.0248
-0.0153
2.9886
2.9916
2.9726
665.0
664.3
668.6
3805.5
3798.0
3804.5
5.2.6 Families q, r, s, t
The branches defining these families are reported in Figure 21(b) and their parameters are summarized
in Table 13. Solutions s, t perform a lunar gravity assist similar to families g , h, i, although no ballistic
capture is accomplished (see Figure 23, solution s−
3 , second quadrant transfer). Families q , r exploit instead
the dynamics of the Sun–Earth Lagrange points and their transfer structure recalls that of families o, p (see
Figure 28, solution q2+ , second quadrant transfer). With reference to Figure 6, no reference solutions close
to these families are found.
26
F. Topputo
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(a) Families j, k.
(b) Families l, m, n.
Fig. 18 Families j, k, l, m, n in the ∆t, ∆v plane.
−3
6
x 10
1
x 10
10
8
0.5
y (km)
y (SErf)
6
4
2
0
L1
L2
0
−0.5
−2
−1
−4
0.995
1
x (SErf)
1.005
(a) Sun-Earth rotating frame.
1.01
−1
−0.5
0
0.5
x (km)
1
1.5
6
x 10
(b) Earth-centered inertial frame.
Fig. 19 Solution j2+ belonging to family j.
6 Conclusions
Since the discovery of low-energy transfers, the problem of finding alternative, efficient Earth–Moon transfers
has been approached with a focus on a small class of solutions, if not on just one kind of solutions. Typically,
when a novel solution is found, one compares it to the Hohmann transfer, leaving unanswered the question
on the existence of other solutions, possibly having intermediate features between the solution at hand and
the Hohmann transfer. This modus operandi prevents the formulation of a complete trade-off between cost
and transfer time.
Motivated by this need, the present paper reconstructs the global set of solutions for the two-impulse
Earth–Moon transfers. The idea is to view the Hohmann, the interior, and the exterior transfers as special
solutions of the same problem. This means that known cases may be thought of as belonging to a more
general picture. The framework used in this paper is the (∆t, ∆v ) plane, which allows direct assessment of
solutions for practical purposes.
The global set of optimal two-impulse Earth–Moon transfers presented in this paper demonstrates that
the intuition on the existence of intermediate solutions was valid. Known solutions reduce to points belonging
to different families of transfers. The families deemed of more interest have been characterized and studied
in detail, and their main features have been discussed. The parameters defining all the solutions presented
are given to allow the reproduction of these transfers for possible further independent investigations.
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
27
Table 11 Data of transfer samples belonging to families l, m, n (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
l1+
l2+
l3+
3.93000540
4.22063005
4.35687235
1.40447433
1.40207720
1.40222223
2.32536708
3.07943365
3.19400904
18.01942133
18.99518723
19.60848008
0.0106
0.0106
0.0106
-0.0811
-0.0875
-0.0830
3.1470
3.1597
3.1508
638.5
635.5
637.6
3791.0
3769.3
3772.5
m+
1
m+
2
m+
3
4.10457544
4.22350820
4.25793279
1.40221448
1.40206865
1.40208973
2.32660368
2.48849928
2.51902225
19.30732266
19.98666422
20.50160843
0.0106
0.0106
0.0106
-0.0731
-0.0799
-0.0786
3.1309
3.1445
3.1420
642.2
639.1
639.6
3777.0
3798.0
3773.5
n+
1
n+
2
n+
3
4.17979291
4.22841073
4.29017247
1.40207328
1.40206951
1.40212432
2.41578647
2.46855661
2.51636438
19.27387149
19.54086245
19.81143467
0.0106
0.0106
0.0106
-0.0676
-0.0685
-0.0648
3.1200
3.1219
3.1146
644.7
644.3
646.0
3778.5
3778.0
3780.1
−0.5
0
6
−3
x 10
x 10
1
10
0.5
y (km)
y (SErf)
5
0
0
L2
L1
−0.5
−1
−5
0.995
1
x (SErf)
1.005
(a) Sun-Earth rotating frame.
1.01
−1
0.5
x (km)
1
1.5
6
x 10
(b) Earth-centered inertial frame.
Fig. 20 Solution l2+ belonging to family l.
7 Acknowledgments
The author is grateful to Pierluigi Di Lizia, Alexander Wittig, and Koen Geurts for having proof-read the
paper, to Mauro Massari and Franco Bernelli-Zazzera for having shared their workstations, and to Roberto
Armellin for the computation of the Pareto-efficient solutions.
References
Armellin, R., P. Di Lizia, F. Topputo, M. Lavagna, F. Bernelli-Zazzera, and M. Berz. 2010. Gravity Assist Space Pruning
Based on Differential Algebra. Celestial Mechanics and Dynamical Astronomy 106 (1): 1–24. doi:10.1007/s10569-0099235-0.
Assadian, N., and S. H. Pourtakdoust. 2010. Multiobjective Genetic Optimization of Earth–Moon Trajectories in the
Restricted Four-Body Problem. Advances in Space Research 45 (3): 398–409.
Battin, R. H. 1987. An Introduction to the Mathematics and Methods of Astrodynamics. AIAA, New York.
Belbruno, E. 1987. Lunar Capture Orbits, a Method of Constructing Earth–Moon Trajectories and the Lunar GAS Mission.
In Aiaa paper 97-1054, proceedings of the aiaa/dglr/jsass international electric propulsion conference.
Belbruno, E. 1994. The Dynamical Mechanism of Ballistic Lunar Capture Transfers in the Four-Body Problem from the
Perspective of Invariant Manifolds and Hill’s Regions. In Technical report, centre de recerca matematica, barcelona,
spain.
Belbruno, E. 2004. Capture Dynamics and Chaotic Motions in Celestial Mechanics: With Applications to the Construction
of Low Energy Transfers. Princeton University Press.
Belbruno, E., and J. Carrico. 2000. Calculation of Weak Stability Boundary Ballistic Lunar Transfer Trajectories. In
Aiaa/aas astrodynamics specialist conference, paper aiaa 2000-4142.
28
F. Topputo
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(a) Families o, p.
(b) Families q, r, s, t.
Fig. 21 Families o, p, q, r, s, t in the ∆t, ∆v plane.
Table 12 Data of transfer samples belonging to families o, p (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
o−
1
o−
2
o−
3
o−
4
o−
5
4.87924505
5.18402876
6.01081082
0.93009504
2.63901403
1.41088384
1.41078261
1.41029724
1.40999519
1.41009626
2.16174675
2.45241547
3.09897520
4.16463357
5.70421254
16.36363453
17.07355978
18.13605289
19.50110233
21.24636234
-0.0109
-0.0108
-0.0108
-0.0107
-0.0107
0.0445
-0.0019
-0.0111
-0.0142
-0.0149
2.8529
2.9459
2.9643
2.9705
2.9719
695.7
674.7
670.5
669.1
668.8
3898.1
3876.3
3868.4
3864.6
3865.1
o+
1
o+
2
o+
3
o+
4
o+
5
5.15585335
5.58251216
0.20095333
1.58324803
2.99264083
1.41084129
1.41078410
1.41024314
1.41004949
1.41013880
2.45717541
2.90767751
3.61092707
4.84316834
6.12422036
16.73769746
17.64177466
18.94414311
20.56435823
22.10419030
0.0109
0.0107
0.0107
0.0107
0.0107
0.0337
-0.0269
-0.0430
-0.0484
-0.0502
2.9179
3.0389
3.0710
3.0818
3.0853
690.8
663.3
656.0
653.5
652.7
3892.9
3865.0
3853.4
3849.4
3849.3
p−
1
p−
2
p−
3
p−
4
p−
5
5.02222917
5.49763936
0.02048615
1.51967236
3.05842328
1.41085460
1.41058926
1.41022252
1.41000824
1.41011989
5.45004921
5.83714409
0.22874828
1.56238694
2.95811499
16.55725651
17.51562844
18.43533710
20.04968749
21.63452349
-0.0109
-0.0108
-0.0108
-0.0108
-0.0107
0.0373
-0.0068
-0.0110
-0.0134
-0.0139
2.8672
2.9556
2.9640
2.9688
2.9699
692.5
672.5
670.6
669.5
669.3
3894.7
3872.6
3867.9
3865.1
3865.7
p+
1
p+
2
p+
3
p+
4
p+
5
5.29149080
5.70978560
0.44578644
2.49730625
3.34228464
1.41082818
1.41072464
1.41019712
1.41015273
1.41005213
5.74447801
6.16102181
0.69440616
2.52992889
3.32154711
16.91153673
17.85988596
19.24256202
21.52688793
22.47702442
0.0108
0.0107
0.0107
0.0107
0.0107
0.0280
-0.0310
-0.0427
-0.0477
-0.0491
2.9292
3.0469
3.0703
3.0802
3.0830
688.3
661.5
656.1
653.9
653.2
3890.3
3862.7
3853.2
3850.6
3849.2
Belbruno, E., and J. Miller. 1993. Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture. Journal of Guidance,
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Betts, J. T. 1998. Survey of Numerical Methods for Trajectory Optimization. Journal of Guidance, Control and Dynamics
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On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
29
5
−3
x 10
x 10
8
10
6
8
4
0
L1
L2
y (km)
y (SErf)
6
2
4
2
−2
0
−4
−2
−6
0.995
1
x (SErf)
1.005
−5
1.01
(a) Sun-Earth rotating frame.
0
5
x (km)
10
5
x 10
(b) Earth-centered inertial frame.
Fig. 22 Solution o+
5 belonging to family o.
Table 13 Data of transfer samples belonging to families q, r, s, t (∆v are in m/s).
Sol
α
β
θi
δ
h2,f
H2,f
Jf
∆vf
∆v
q1+
q2+
q3+
0.03381194
0.10879184
0.49122400
1.41065912
1.41056240
1.41031915
3.67413052
3.71022277
3.99219506
19.74181230
19.96824167
20.65689374
0.0106
0.0106
0.0106
-0.0797
-0.0842
-0.0844
3.1442
3.1532
3.1535
639.1
637.0
637.0
3839.8
3837.0
3835.0
r1+
r2+
r3+
0.13483581
0.22346700
0.43650553
1.41065431
1.41053662
1.41039878
0.63885104
0.67779827
0.83354004
19.84872767
20.06106177
20.58467910
0.0106
0.0106
0.0106
-0.0799
-0.0837
-0.0849
3.1446
3.1522
3.1546
639.1
637.3
636.7
3839.7
3837.0
3835.4
s−
1
s−
2
s−
3
4.17015923
4.19420474
4.25274883
1.40272187
1.40265697
1.40257540
6.23928017
6.24424980
0.00339720
20.12726592
20.38791538
20.54123860
-0.0109
-0.0108
-0.0108
0.0606
0.0258
0.0207
2.8205
2.8903
2.9006
703.0
687.3
685.0
3841.8
3825.6
3822.6
t−
1
t−
2
t−
3
4.39366912
4.40612382
4.41997347
1.40263808
1.40265462
1.40267407
3.34123544
3.35659232
3.37438159
20.34934312
20.54099714
20.73265115
-0.0109
-0.0109
-0.0108
0.0671
0.0375
0.0223
2.8076
2.8668
2.8972
705.8
692.6
685.7
3844.0
3830.8
3824.2
Springer.
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1
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8
0.5
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4
0
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0
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Fig. 23 Solution s−
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Appendix 1
An orbit in the rotating frame, x(t) = {x(t), y (t), ẋ(t), ẏ(t)}, is converted to an orbit expressed in the P1 centered (or Earth-centered), inertial frame, X 1 (t) = {X1 (t), Y1 (t), Ẋ1 (t), Ẏ1 (t)}, through
X1 (t) = (x(t) + µ) cos t − y (t) sin t
Y1 (t) = (x(t) + µ) sin t + y (t) cos t
Ẋ1 (t) = (ẋ(t) − y (t)) cos t − (ẏ(t) + x(t) + µ) sin t
Ẏ1 (t) = (ẋ(t) − y (t)) sin t + (ẏ(t) + x(t) + µ) cos t
(34)
where t is the present, scaled time. The transformation into the P2 -centered (i.e. Moon-centered), inertial
frame is obtained from (34) by replacing ‘µ’ with ‘µ − 1’. The initial conditions of the points highlighted
in Figure 7 are reported in Table 14 with high accuracy. This allows to reproduce the optimal solutions
by simply integrating the initial conditions under the PBRFBP dynamics; i.e. the i-th transfer orbit is
γ (i) = {ϕ(xi , ti ; t), t ≤ tf } where ϕ is the flow of (5).
Appendix 2
The figures corresponding to the solutions characterized in Section 5 are reported below.
32
F. Topputo
Table 14 Initial condition xi = (xi , yi , ẋi , ẏi ) and initial, final time, ti , tf of solutions (i)–(x) in Figure 7, 8 and Table 4.
(i)
(ii)
(iii)
(iv)
(v)
xi
yi
ẋi
ẏi
ti
tf
-0.019710793706630
-0.015255813698632
9.554404402969761
-4.734752068884238
-1.803851247295145
-0.745915731505332
0.004430665316507
0.003867133727872
-2.421772563013686
10.383974704885071
-4.242568609277442
-0.928062659449309
-0.022223338375692
-0.013727223510871
8.593048071999794
-6.305349228506921
28.854324263774792
35.904880040219062
-0.022469248813271
-0.013543346852322
8.477163558651014
-6.458691168284453
5.183780363655846
12.518449758919608
-0.020509279478864
-0.014833373417121
9.289091958818513
-5.234406611705439
-2.783372600934104
7.256777745724602
(vi)
(vii)
(viii)
(ix)
(x)
xi
yi
ẋi
ẏi
ti
tf
-0.020796119659883
-0.014668044128481
9.185639864842763
-5.414082611512081
-2.825197110547974
8.989837035669957
-0.021300676720177
-0.014358714903565
8.996820692867965
-5.733172198727265
3.020458855979923
17.277056927521521
-0.019048265996746
-0.015566582598150
9.751858086858906
-4.321076476149917
2.932062269968396
17.819572739337175
-0.019022730972677
-0.015577872203673
9.759101553325923
-4.305155198773667
6.480457482011631
21.838729667546875
-0.020288471040515
-0.014955651584857
9.365880586172640
-5.096246610790654
0.068219944002269
19.078513034970545
5
5
x 10
x 10
6
4
a−
4
2
a−
1
a−
2
2
y (km)
0
y (km)
b−
5
4
−2
b−
3
b−
2
0
a−
3
−4
b−
1
−2
a−
5
−6
−4
−8
−6
−4
−2
0
x (km)
2
4
6
5
x 10
b−
4
−8
−6
(a) Family a (negative).
−4
−2
0
x (km)
2
4
6
5
x 10
(b) Family b (negative).
Fig. 24 Negative solution samples belonging to family a and b (Earth-centered inertial frame).
5
5
x 10
x 10
3
3
2
2
c−
2
c−
4
1
y (km)
y (km)
1
0
−1
c−
3
0
−1
−2
c−
5
−2
c−
1
−3
−3
−4
−2
0
x (km)
2
(a) Family c (negative).
4
5
x 10
−4
−2
0
x (km)
2
(b) Family c (negative).
Fig. 25 Negative solution samples belonging to family c (Earth-centered inertial frame).
4
5
x 10
On Optimal Two-Impulse Earth–Moon Transfers in a Four-Body Model
−3
33
−3
x 10
x 10
6
4
4
2
0
L2
L1
−2
y (SErf)
y (SErf)
2
0
L1
L2
−2
−4
−4
−6
−6
−8
−8
0.995
1
1.005
x (SErf)
(a) Solution f1+ , SE rotating frame.
1.01
0.995
1
1.005
x (SErf)
(b) Solution h−
3 , SE rotating frame.
1.01
Fig. 26 Solutions f1+ and h−
3 .
−3
5
x 10
10
6
8
4
6
2
4
y (SErf)
y (km)
x 10
2
0
L1
L2
−2
0
−4
−2
−6
−4
−8
−6
−5
0
5
10
x (km)
0.995
1
1.005
x (SErf)
(b) Solution p−
5 , SE rotating frame.
5
x 10
(a) Solution m+
2 , Earth-centered frame.
1.01
−
Fig. 27 Solution m+
2 and p5 .
−3
5
x 10
8
10
6
8
4
6
y (km)
y (SErf)
x 10
2
0
L1
L2
4
2
−2
0
−4
−2
0.995
1
1.005
x (SErf)
(a) Sun-Earth rotating frame.
Fig. 28 Solution q2+ belonging to family q.
1.01
−5
0
5
x (km)
10
(b) Earth-centered inertial frame.
5
x 10