ArticlePDF Available

The cislunar low-thrust trajectories via the libration point

Authors:

Abstract and Figures

The low-thrust propulsion will be one of the most important propulsion in the future due to its large specific impulse. Different from traditional low-thrust trajectories (LTTs) yielded by some optimization algorithms, the gradient-based design methodology is investigated for LTTs in this paper with the help of invariant manifolds of \(\mathit{LL}_{1}\) point and Halo orbit near the \(\mathit{LL}_{1}\) point. Their deformations under solar gravitational perturbation are also presented to design LTTs in the restricted four-body model. The perturbed manifolds of \(\mathit{LL}_{1}\) point and its Halo orbit serve as the free-flight phase to reduce the fuel consumptions as much as possible. An open-loop control law is proposed, which is used to guide the spacecraft escaping from Earth or captured by Moon. By using a two-dimensional search strategy, the ON/OFF time of the low-thrust engine in the Earth-escaping and Moon-captured phases can be obtained. The numerical implementations show that the LTTs achieved in this paper are consistent with the one adopted by the SMART-1 mission.
Content may be subject to copyright.
Astrophys Space Sci (2017) 362:96
DOI 10.1007/s10509-017-3075-2
ORIGINAL ARTICLE
The cislunar low-thrust trajectories via the libration point
Qingyu Qu1·Ming Xu1·Kun Peng2
Received: 2 July 2016 / Accepted: 1 April 2017
©SpringerScience+BusinessMediaDordrecht2017
Abstract The low-thrust propulsion will be one of the most
important propulsion in the future due to its large spe-
cific impulse. Different from traditional low-thrust trajecto-
ries (LTTs) yielded by some optimization algorithms, the
gradient-based design methodology is investigated for LTTs
in this paper with the help of invariant manifolds of LL1
point and Halo orbit near the LL1point. Their deformations
under solar gravitational perturbation are also presented to
design LTTs in the restricted four-body model. The per-
turbed manifolds of LL1point and its Halo orbit serve as the
free-flight phase to reduce the fuel consumptions as much
as possible. An open-loop control law is proposed, which is
used to guide the spacecraft escaping from Earth or captured
by Moon. By using a two-dimensional search strategy, the
ON/OFF time of the low-thrust engine in the Earth-escaping
and Moon-captured phases can be obtained. The numerical
implementations show that the LTTs achieved in this paper
are consistent with the one adopted by the SMART-1 mis-
sion.
Keywords Earth–Moon transfer ·Low thrust ·LL1point ·
Halo orbit ·Invariant manifolds
1 Introduction
Deep Space-1 launched by NASA in 1998 and SMART-1
Lunar probe launched by ESA in 2003 prove that it is fea-
sible to select low-thrust propulsion as the main propulsion
BM. Xu
xuming@buaa.edu.cn
1School of Astronautics, Beihang University, Beijing 100191,
China
2Institute of Manned Space System Engineering, China Academy
of Space Technology, Beijing 100094, China
in deep-space exploration missions. Another recent famous
mission is the DAWN mission, launched by NASA in 2007,
in which a gravity-assist method is adopted. The space-
craft explored Ceres and Vesta located in the asteroid belt
(Polanskey et al. 2011). Considering that propellant econ-
omy is the main factor designed in deep-space exploration
missions, low-thrust propulsion will be one of the most im-
portant propulsions in the future because of its large specific
impulse.
Updates in propulsion are supposed to trigger a revo-
lution in the field of trajectory design. When low-thrust
propulsion is used in the designing of trajectories, their ad-
justment is gradual because of the long-term effects of thrust
arcs. A patched conic technique based on a two-body model
is normally used in the trans-lunar trajectory; however, when
adopting low-thrust propulsion, the large traveling time does
not support the working of the patched conic technique.
Therefore, another model needs to be proposed based on a
three- or four-body problem.
Earth–Moon low-thrust transfer trajectories normally in-
clude the Earth-escaping, free-flight, and Moon-captured
phases (Racca 2003; Racca et al. 2002; Betts and Erb 2006;
Guelman 1995; Herman and Conway 1996). In a conven-
tional LTTs design, the problem is normally transformed
into a nonlinear problem (NLP) by using the collocation
method, and then trajectories under low thrust can be ob-
tained (Betts and Erb 2006; Guelman 1995; Herman and
Conway 1996). This method only uses the optimization al-
gorithm to obtain the target orbit; the search process is
blind and requires a considerable amount of calculation.
AtypeofEarthMoontransferorbitbasedonthecircu-
lar restricted three-body problem (CR3BP) was proposed
by Conley (Conley 1968). This type of trajectory passes
the collinear libration point L1in the Earth–Moon system
to distinguish from the WSB transfer orbit that passes the
96 Page 2 of 11 Q. Qu et al.
collinear libration point L2, as proposed by Belbruno and
Miller (2015), the former is named Earth–Moon LL1trans-
fer orbit, which is being extensively studied by many schol-
ars for further development. These methods can be divided
into three classes stated as follows.
First, the direct method of optimization is used by dis-
cretizing the trajectories and control variables. Thus, high
accuracy is not required for the initial value but a large
CPU time is required for trajectory analysis and optimiza-
tion. More concretely, Howell and Ozimek obtained the so-
lution for a complete time history of the thrust magnitude
and direction by solving a calculus of variation problems to
locally maximize the final spacecraft mass (Howell and Oz-
imek 2010). Lee used a direct transcription and collocation
method to reformulate the continuous dynamic optimization
problem into a discrete optimization problem, which then
is solved using nonlinear programming software (Lee et al.
2014).
Second, an indirect method is adopted in the two fol-
lowing studies, in which boundary and first-order neces-
sary conditions for optimality are enforced for a continu-
ous system. Considering that the convergence domain of the
method is narrow, a high-precision initial value is required,
which is also computationally demanding. Specifically, Gra-
ham and Rao used a variable-order Legendre–Gauss–Radau
(LGR) quadrature orthogonal collocation method, thus ob-
taining a high-accuracy minimum-fuel Earth-orbit transfer
by using low-thrust propulsion (Graham and Rao 2014).
Furthermore, in another paper, they used the same method
to obtain a high-accuracy minimum-time Earth-orbit trans-
fer by using low-thrust propulsion (Graham and Rao 2015).
Compared with the aforementioned studies aimed to for-
mulate optimizations for LTTs, some other studies aimed to
obtain the initial value of the LTTs by using the libration
point. Some representative studies are presented as follows.
Petropoulos and Longuski proposed a type of shape-based
design (Petropoulos and Longuski 2004). They presented an
exponential sinusoid shape function to solve the dynamical
equations of a spacecraft. Vellutini and Avanzini improved
the method (Vellutini and Avanzini 2015) and proposed a
modified exponential sinusoid shape function. The two ex-
ponential sinusoid shape functions cannot satisfy the start-
ing and ending velocity boundary conditions; thus, they can
only be used to design the trajectories whose launch energy
is not zero. Furthermore, Wall and Conway proposed a sixth-
order inverse polynomial shape function, which is obtained
by fitting the optimal transfer trajectories (Wall and Con-
way 2009). However, the actual dynamic constraints were
not considered; therefore, some feasible trajectories are of-
ten omitted when adopting this method.
Unlike conventional LTTs yielded by some optimization
algorithms, the present study investigated the gradient-based
design methodology for LTTs with the help of invariant
manifolds of the LL1point and a Halo orbit near the LL1
point. In addition, their deformations under solar gravity
perturbation are represented to design LLTs in the restricted
four-body model. In this paper, the LL1point and the Halo
orbit near it serve as a free-flight phase so that the fuel con-
sumption is as low as possible (when adopting the LL1point
as the free-flight, non-thrusting phase, the fuel consumption
is decreased). Next, to design the Earth-escaping and Moon-
captured phases, an open-loop low-thrust control law used
for designing long duration spiral transfers between Earth-
and Moon-centered orbits is presented. By using the LL1
point and the Halo orbit near it as a starting point, respec-
tively, when backward and forward integrations are com-
pleted, the trajectory of Earth-escaping and Moon-captured
phases can be respectively obtained.
Compared with the existing shape-based design, there is
no need to propose a kind of shape function (exponential si-
nusoids or inverse polynomial shape functions). In addition,
the situation in which feasible trajectories are omitted can
be avoided by using this method. Moreover, the Halo orbit
near the LL1point is adopted as a free-flight phase, which
cuts down the fuel consumption greatly. In this study, the in-
variant manifolds derived from the transportation tube wall
are studied as well.
The obtained cislunar LTTs fundamentally coincide with
those of SMART-1, which has similar boundary conditions.
Supposing that the obtained trajectories are used as the ini-
tial orbit to optimally design low-energy transfer trajecto-
ries to Moon, then the search scope can be narrowed and the
computation of optimal trajectories can be reduced.
2Dynamicalmodels
The Moon, Earth, and Sun constitute the general concept of
the three-body problem; in this study, the dynamic model is
simplified. In a Sun–Earth–Moon System, the accuracy of
the Spatial Bi-Circular Model (SBCM) is sufficiently high
(Koon et al. 2001). In the model, the inclination of the lunar
plane related to the ecliptic plane is considered, as shown
in Fig. 1. The Moon and Earth are regarded as a whole sys-
tem that nearly composes a two-body motion with the Sun,
and another two-body motion composed of the Earth and the
Moon is not affected by the Sun. The distance between Sun
and Earth is considerably larger than that between the Earth
and Moon; thus, the torque of the solar gravitation affecting
the Earth–Moon system can be ignored and only the force is
considered. The coordinate systems used in this model are
defined as follows.
For the geocentric inertial frame, the Earth’s center is
defined as the origin, and the intersection of the lunar and
ecliptic planes is defined as the x-axis. In addition, the nor-
mal of the lunar plane is defined as the z-axis, whose posi-
tive direction is coincident with that of spin angular velocity,
The cislunar low-thrust trajectories via the libration point Page 3 of 11 96
Fig. 1 The geometrical view of the SBCM (Xu et al. 2013)
and the y-axis is determined by the right-hand rule. There is
relative acceleration between the defined and inertial frames
but the geocentric inertial frame can be approximately re-
garded as an inertial frame relative to a low-Earth orbit. For
the selenocentric inertial frame, the Moon’s center is de-
fined as the origin, and the definitions of coordinate axes are
the same as those of the geocentric inertial frame, and the
frame can be approximately regarded as an inertial frame
relative to a low-Moon orbit. For the inertial frame in the
Sun–Earth/Moon systems denoted as ISE+M, the centroid
of the Sun–Earth–Moon system is defined as the origin, and
the definition of coordinate axes are the same as those of the
geocentric inertial frame.
For the syzygy frame in the Sun–Earth/Moon systems de-
noted as SSE+M, the centroid of the Sun–Earth–Moon sys-
tem is defined as the origin, and the direction to the centroid
of the Earth–Moon system from the Sun is defined as the x-
axis. Furthermore, the normal of the ecliptic plane is defined
as the z-axis, whose positive direction is coincident with that
of spin angular velocity, and the y-axis is determined by the
right-hand rule. For the syzygy frame in the Earth–Moon
system denoted as SEM, the centroid of the Earth–Moon
system is defined as the origin, and the direction toward the
Moon from Earth is defined as the x-axis. In addition, the
normal of the lunar plane is defined as the z-axis, whose
positive direction is coincident with that of spin angular ve-
locity, and the y-axis is determined by the right-hand rule.
In Fig. 1, the inclination of the lunar plane relative to the
ecliptic plane is considered, with an average angle of 5°9;
the lunar phasic angle βis measured as the angle between
the line from the Earth to Moon and the intersecting line
of the ecliptic and lunar planes. Moreover, the solar phasic
angle θsis the angle between the line from the Sun to the
barycenter of the Earth–Moon system and the intersecting
line of the ecliptic and lunar planes; the ecliptic plane is de-
picted in yellow, while the lunar plane is depicted in green
(Xu et al. 2013).
To improve efficiency and accuracy, a normalized unit
based on CR3BP is adopted throughout this investigation
(Szebehely 1967). The normalized unit is defined as follows.
When the spacecraft is located in the gravitational influence
region of two primary bodies (m1and m2; Sun and Earth,
or Earth and Moon in this paper), the unit of account is nor-
mally taken as
[M]=mE+mM
[L]=LEM
[T]=$L3
EM/G(mE+mM)%1/2(1)
where LEMis the distance between the two primary bod-
ies, mEis the mass of Earth, and mMis the mass of Moon.
According to the new units defined earlier, the gravitational
constant is G=1.
For SBCM, the dynamical equation is written as
¨x
¨y
¨z
=2
˙y
˙x
0
+
x
y
0
(1µ) rrE
3
µrrM
3+ω2
sRz(β)Rx(i)As
Rz(β)Rx(i)Rz(θs)ms
RasRs
′′ 3(2)
where ri(i =1,2,3)is the distance between the space-
craft and Earth, Moon, and Sun; ωsis the angular veloc-
ity of the Earth relative to the Sun; iis the angle between
the lunar plane and the ecliptic plane; µis proportion of
the mass of Moon in the Earth–Moon system; µsis the
mass of Sun; asis the average distance between the Sun
and the centroid of the Earth–Moon system; θsis the so-
lar phasic angle; and βis the lunar phasic angle. In addi-
tion, As=[ascosθs,a
ssin θs,0]Tis the position vector of
Sun in ISE+M, and r=[x,y,z]Tis the position vector
of the spacecraft in SEM,rE=[µ00]Tis the posi-
tion vector of Earth in SEM,rM=[1µ00]Tis the
position vector of Moon in SEM,ms=328900.54 is the
ratio of the mass of Sun to that of the Earth–Moon system,
Ris the position vector of the spacecraft in the SSE+M,
and Rs=[µS00]Tis the position vector of Sun in
SSE+M. The symbol represents the term rrE,rep-
resents rrM,and′′ represents the term RasRs.
The initial values t0=0°, θs0=0°, and β0is undeter-
mined. Therefore, the geometrical relationship between the
Sun, Earth, and Moon is only determined by β0.
3 Cislunar transfers via LL1point and its Halo
orbits
3.1 Cislunar transfer via LL1point with lowest energy
In the CR3BP frame, the invariant manifolds of LL1point
can be adopted to design the cislunar transfer. In this case,
96 Page 4 of 11 Q. Qu et al.
the spacecraft can fly with the stable manifolds from the in-
terior area affected by the gravity of Earth to the LL1point,
Fig. 2 Cislunar transfer opportunities measured using the lunar phasic
angle β: in this study, β=286°
Fig. 3 Cislunar low-thrust transfer trajectories in the syzygy frame
and then fly from the LL1point to the exterior area affected
by the Moon’s gravity with the unstable manifolds. How-
ever, considering that the transfer time will approach infin-
ity, this trajectory is unpractical.
For SBCM, under the perturbation of solar gravity, the
infinite transfer time is cut down to a finite time; this is quite
significant for the Earth–Moon transfers. Considering that
a particle at LL1point has the lowest energy that can form
a “neck” near the LL1point in the Hill’s open region, the
transfer via LL1point will obviously have the lowest energy.
In Fig. 2, only the interval of βcorresponding to the dot-
ted line can drive the trajectories from Earth to the LL1point.
Further, only the interval of βcorresponding to the solid
line can drive the trajectories from the LL1point to Moon.
The intersection of these two intervals, that is, [77°, 109°]
[285°, 342°], can be considered as the cislunar transfer op-
portunity bounded by the vertical dashed lines in Fig. 2.In
this paper, it is chosen as β=286°.
When considering LL1point as the trajectory in the free-
flight phase, the procedure used to search cislunar LTTs
is described as follows. In the syzygy frame, for a spe-
cific value of β,substituteitintoEq.(2), and integrate the
SBCM dynamics equations. When backward and forward
integrations are finished, the trajectories of Earth-escaping
and Moon-captured phases can be obtained, respectively.
The initial conditions are β=286° and θs=0°. The integral
initial point is [xLL100000]T. Figure 3presents the
cislunar low-thrust transfer trajectories in the syzygy frame.
Figure 4presents the history of eccentricity and semi-major
axis in free flying without control. The moment when the
initial integration is defined is defined as the epoch time
t=0.
The motion before the Earth-escaping phase and after the
Moon-captured phase can be regarded as a two-body prob-
lem under perturbation. Commonly, the radius of the Earth’s
sphere-of-influence (SOI) is approximately 924647 km, and
Fig. 4 History of orbit elements in the free-flying phase without control: (a) the history of eccentricity; (b) the history of semi-major axis
The cislunar low-thrust trajectories via the libration point Page 5 of 11 96
Fig. 5 Invariant manifolds of the Halo orbit near the LL1point: (a) the manifolds in the CR3BP frame; and (b)themanifoldsintheSBCMframe
the radius of the Moon’s SOI is approximately 66190 km.
Figure 3shows that the altitude of the trajectories of the
Earth-escaping phase is no more than 4 ×105km, while
the maximum distance from the node on the trajectories
of the Moon-captured phase to the Moon is approximately
5.5×104km. In addition, the trajectories of the Earth-
escaping phase are observed to be located entirely in the
Earth’s SOI; therefore, when a two-body model is adopted
to solve this problem, the accuracy meets the requirement,
and the orbit is stable. On the other hand, some parts of the
trajectories of the Moon-captured phase are located at the
boundary of the Moon’s SOI, and therefore will be unstable;
this is proved by the result in Fig. 4.Moreover,inFig.4,the
eccentricity and semi-major axis are defined with respect to
two central bodies depending on time. That is, for t<0, the
Earth is the center and for t<0, the Moon is the center.
By considering the instability of the transfer trajectories
in Moon captured phase, the trajectory with a lower altitude
of periapsis is more appropriate to be adopted in the free-
flight phase. If the altitude of periapsis is very high, it will
increase the risk of the spacecraft escaping again before be-
ing captured.
3.2 Cislunar transfer through the Halo orbit near LL1
point with more opportunities
Compared with the cislunar transfers via LL1point, those
via the Halo orbit near the LL1point provide more trans-
fer opportunities because another variable is introduced (i.e.,
the phase of the Halo orbit near the LL1point). This part pro-
vides some results about the cislunar transfer via the Halo
orbit near the LL1point are obtained, which are similar to
those obtained in Sect. 3.1.
The Halo orbit is the periodic solution around the li-
bration point in CR3BP, and results from orbit bifurcations
(Barden et al. 1996). It is symmetrical with the xzplane in
the syzygy frame. Poincaré mapping P(z)is defined as
P(z)=φT(z), zΓ(θ). (3)
Based on the Hamiltonian system’s theory, the derivative of
P(z), that is, Φ=DzP(z)is a symplectic matrix. The com-
plex eigenvalues of the matrix are |λi|=1, i=1,2,3,4,
and the real eigenvalues are λ5=λ1
6>1. These eigenval-
ues of Φare named characteristic exponents of P(z).The
real eigenvalues reflect the stability of the Halo orbit.
In the symplectic matrix, λ5>1andλ6<1; therefore,
the Halo orbit has both stable and unstable manifolds. The
invariant manifolds of the Halo orbit are globally repre-
sented as two-dimensional compact manifolds in the phase
space, and their representation in the position space are
shown in Fig. 5(a). For the Earth–Moon system, the invari-
ant manifolds of the Halo orbit in CR3BP can be divided
into Ws
E,Wu
E,Ws
M,andWu
Mbased on stability and ori-
entation; the subscripts “E”and“M” represent the Earth
and Moon orientations, respectively, and the superscripts
s”and“u” represent the stable and unstable manifolds, re-
spectively. In Fig. 5(a), the green and red areas represent
the stable and unstable manifolds, respectively. Moreover,
Ws
Eand Wu
Mconsist of the transportation tube wall from
Earth to Moon, and Ws
Mand Wu
Econsist of the transporta-
tion tube wall from Moon to Earth. The transportation tube
wall is still represented as a two-dimensional submanifold
in the phase space, while all the transfer orbits between the
Earth and Moon with the same energy are comprised only
inside the transportation tube (Koon et al. 2000). However,
in SBCM, considering that the Halo orbit is affected by
the solar gravitational perturbations, the closed orbit is no
longer maintained. In addition, the solar gravity can cause
the transportation tube to be deformed or broken (Yamato
and Spencer 2003,2004).
96 Page 6 of 11 Q. Qu et al.
Fig. 6 Contour map of transfer opportunities: (a) the opportunities for the Earth-escaping phase; the blue areas represent the feasible opportunities.
(b) The opportunities for the Moon-captured phase; the blue areas represent the feasible opportunities. In this study, (β,τ)=(150°, 0.5)
In invariant manifolds, the spacecraft travels multiple
loops in the elliptical orbit of the Earth or Moon, and can
be captured by the Moon or Earth’s gravitational force. It
then travels multiple loops in the elliptical orbit of the Moon
or Earth after being captured. During the whole process, the
spacecraft makes a free flight. Ws
Eand Wu
Mconsist of the
transportation tube wall from Earth to Moon, represented by
the dark area in Fig. 5(b), and Ws
Mand Wu
Econsist of the
transportation tube wall from Moon to Earth, represented by
the light area in Fig. 5(b). Hill’s forbidden area in SBCM is
time-variant but the variation is minute. Therefore, this area
can be approximately derived based on CR3BP. Moreover,
in Fig. 5(b), the solar and lunar phases in the SBCM are 0°
and 60°, respectively.
The spacecraft has enough time to realize the Earth-
escaping and Moon-captured phases in the invariant mani-
folds through the cumulative effects of low thrust. It is not
necessary to consider the chaotic behavior that a tiny change
in velocity can allow the spacecraft escape from the grav-
itational field of Earth–Moon system, which exists in the
N-body problem.
Although the transportation tube wall based on SBCM is
deformed, the invariant manifolds derived from it also dif-
fer. The long-term effects of solar gravity limit the trans-
fer time instead of supporting limitlessness; this is signifi-
cant for the Earth–Moon transfer. Some pairs (β,τ), rang-
ing over βin [0,2π]andτin [0,1], do not represent true
transfer opportunities through the LL1Halo orbit as pertur-
bations to the manifolds cause crashes or escapes over short
time scales. Only the pairs that enable the transfer from the
Earth to the Halo orbit near the LL1point and from the Halo
orbit near the LL1point to the Moon can be adopted to de-
sign the Earth–Moon transfer; these are depicted in Fig. 6.
In this study, the transfer opportunities make up 12.96%
of all possible (β,τ)combinations,thussatisfyingboththe
contour-maps shown in Fig. 6. In addition, the satisfactory
(β,τ)pairsareasfollows:([95°,150°][262°, 345°]) ×
[0,1] and ([100°, 200°] [270°, 30°]) ×([0.16,0.26]
[0.47,0.58]). In this paper, (β,τ)=(150°, 0.5), as marked
in Fig. 6.
The procedure used to search for the transfer opportu-
nities is described as follows. (i) The lunar phasic angle β
changes in the range [0°, 360°], and the phase of serial points
on the Halo orbit τchanges in the range [0,360]/360. By
substituting these values in Eq. (3), SBCM dynamics equa-
tions can be integrated. After obtaining a backward inte-
gration, the trajectory of the Earth-escaping phase can be
obtained. When a forward integration is derived, the trajec-
tory of the Moon-captured phase can be obtained. (ii) When
reaching the first periapsis in the Earth-escaping or Moon-
captured phases, record the altitude of the periapsis and draw
the contour map, which can be adopted to find the trans-
fer opportunities. (iii) The position and velocity vectors of
serial points on the Halo orbit are at the same initial condi-
tions as those of the two types of integrations (backward and
forward). Furthermore, the initial lunar phasic angle β=
and solar phasic angle θschange in the range [0°, 360°],
the phase of serial points on the Halo orbit τchanges in the
range [0,1], and the amplitude in the ydirection of the Halo
orbit is 40142.16 km.
The most important aspect in the designing of LTTs is to
determine invariant manifolds to extend the low-thrust arcs.
In a time sequence, the spacecraft travels multiple loops in
the elliptic orbit of Earth, and then travels in the Halo or-
bit near the LL1point. It is then captured by the Moon and
travels in the elliptic orbit of the Moon. There is no con-
trol during the whole process. The initial lunar phasic angle
β0=150°, and the phase of the Halo orbit is 180°, which
has been mentioned earlier.
The cislunar low-thrust trajectories via the libration point Page 7 of 11 96
Similar to the procedure in Sect. 3.1, when considering
the Halo orbit near the LL1point as the trajectory in the
free-flight phase, the procedure used to search for cislunar
LTTs is described as follows. In the syzygy frame, sub-
stitute values of a specific pair of (β,τ)intoEq.(2), and
integrate the SBCM dynamics equations. When backward
and forward integrations are completed, the trajectory of the
Earth-escaping and Moon-captured phases can be obtained,
respectively. The initial lunar phasic angle β0=150°, phase
of Halo orbit is 180°, as mentioned earlier. Figure 7presents
the cislunar low-thrust transfer trajectories in the syzygy
frame. Figure 8shows the history of eccentricity and semi-
major axis in the free-flying phase without control. The mo-
ment of initializing integration is defined as the epoch time
t=0.
The motion before the Earth-escaping phase and after the
Moon-captured phase can be regarded as a two-body prob-
Fig. 7 Cislunar low-thrust transfer trajectories in the syzygy frame
lem under perturbation. Commonly, the radius of Earth’s
SOI is approximately 924647 km, and the radius Moon’s
SOI is approximately 66190 km. Figure 7shows that the
altitude of the trajectories of the Earth-escaping phase is no
more than 4×105km, while the maximum distance from the
cislunar trajectories to Moon is approximately 5.5×104km.
The trajectories of the Earth-escaping phase are observed to
be located entirely in the Earth’s SOI, indicating that the tra-
jectories are affected mainly by the Earth and will be stable
enough around the Earth with a slight change in their ec-
centricities. Therefore, when a two-body model is adopted
to solve this problem, the accuracy meets the requirement,
with the orbit being stable in this model. Some parts of the
trajectories of the Moon-captured phase are located at the
boundary of the Moon’s SOI, and they are therefore unsta-
ble. This is proved in the results showed in Fig. 8.
Considering the instability of the transfer trajectories in
the Moon-captured phase, the trajectory with a lower alti-
tude of periapsis is more appropriate to be adopted in the
free-flight phase. If the altitude of periapsis is very high, it
will increase the risk of the spacecraft escaping again before
being captured.
4 Low-thrust control strategy for cislunar
trajectories
This section shows the low-thrust transfer by transiting the
Halo orbit near the LL1point as an instance to design the
low-thrust control strategy.
When initial and terminal conditions are limited, a two-
point boundary problem must be solved to obtain the
ON/OFF time and control law. In this study, the feasibility
of the low-thrust transfer based on the transportation tube
is the main point of our study. Therefore, the satisfaction of
specific phasing requirements of particular missions is not
Fig. 8 History of orbit elements in the free-flying phase without control: (a) the history of eccentricity and (b) of semi-major axis
96 Page 8 of 11 Q. Qu et al.
Fig. 9 History of eccentricity in the escaping earth phase under the subcontrol law I:(a) the history of the semi-major axis, and (b)ofthe
eccentricity
required. In addition, only the semi-major axis (ae,a
m)and
eccentricity (ee,e
m) are limited in the initial and terminal
states, where aeand eeare defined as the semi-major axis
and eccentricity of the orbit in the geocentric inertial frame.
The parameters amand emare defined as the semi-major
axis and eccentricity of the orbit in the selenocentric inertial
frame.
4.1 Design of subcontrol law for low thrust
Without considering the effects of solar gravity perturbation,
when the spacecraft travels from the Earth to Moon, the Ja-
cobi energy integral Jchanges from small to large and fi-
nally to small under the effects of low thrust. In this process,
the function of the subcontrol law Iis to change the direc-
tion of energy along with the maximum gradient direction.
In CR3BP, the maximum energy gradient direction is
in the direction of the velocity (Petropoulos 2003). J=
1
2(x2+y2+z2)+¯
Uis a constant under the condition of
no thrust. When the effects of low thrust are considered, the
Hamilton system is no longer conservative. Let the control
acceleration f=[fx,f
y,f
z]T, then dJ
dt =[˙x, ˙y, ˙z]T·f.Un-
der the control law I,thethrustangleis0rad.
For example, consider the Earth-escaping phase. Under
the subcontrol law provided in this paper, the change rule of
the orbital elements is shown in Fig. 9.
Under the subcontrol law I, the variation of eccentricity
will exceed the allowable range ([0,0.7927]). However, the
semi-major axis is generally small; therefore, based on the
two-body model, another subcontrol law, that is, subcontrol
law II is provided to adjust the eccentricity. To avoid the
additional change in the semi-major axis, the control law II
is made to align with the normal of the inertia velocity, and
the thrust angle is π/2 rad. To demonstrate the foundation
of the optimal controls, the governing differential equation
for eccentricity is presented as follows:
˙e=1
V*2(e +cosθ)ft1
arsin θfn+(4)
where Vis the inertia velocity, θis the true anomaly, trep-
resents the tangential direction of the inertia velocity, and n
represents the normal direction of the inertia velocity. In the
two-body model, control law II does not result in an addi-
tional change of energy.
By applying control law II to Eq. (4),
˙e=1
V*1
ar|sin θfn+.(5)
To obtain more degrees of freedom to develop optimiza-
tions, another transitional subcontrol law is used between
the two given ones. Instead, a linear subcontrol law is
adopted to simplify the problem. The unified control law is
presented as follows:
0|sin θ|<κ1α=0
κ1|sin θ|<κ2α=π
2-|sin θ|κ1
κ2κ1.
κ2|sin θ|1α=π
2.
(6)
In the law, two thresholds are adopted so that there are two
degrees of freedom to develop optimizations. The optimized
object is the thrust angle required to achieve the following
target: the eccentricity and semi-major axis can reach the
final value simultaneously. The procedure used to describe
the control law is as follows.
The termination condition requires both the semi-major
axis and eccentricity to reach their target values simultane-
ously. Thus, the values of κ1and κ2can be obtained as fol-
lows. First, utilize κ1in the range of (0, 1) and utilize κ2in
The cislunar low-thrust trajectories via the libration point Page 9 of 11 96
the range of (κ1,1). Second, for each κ1,acorresponding
κ2can be obtained according to the termination condition.
In this study, the pair (κ1,κ2)=(0.2,0.7)is achieved nu-
merically to meet the constraints of both flight time and fuel
consumption.
4.2 Design of ON/OFF time
The spacecraft starts up in the low orbit of Earth and acceler-
ates to the OFF time TE
off,travelsinthefree-ightphase,and
then travels in the Moon-captured phase. At the time of TL
on,
the spacecraft starts up again and slows down to the Moon’s
low orbit.
In the low-thrust transfer by the transiting of LL1point,
the trajectory is stable before the spacecraft escapes from
Earth; therefore, the choice of TE
off can be loose. Figure 4
shows that there is an inflection point (near 66.9th day) in
the trend curve of both the eccentricity and semi-major axis
in the Earth-escaping phase. Furthermore, any time before
the inflection point can be adopted as TE
off.Inthisstudy,itis
66.9th day. After the spacecraft is captured by the Moon’s
gravitational force, the trajectory is unstable and the eccen-
tricity increases under the control law; as the eccentricity
starts decreasing, the time is chosen as TL
on.Inthisstudy,it
is 9.609th day.
In the low-thrust transfer by the transiting of the Halo
orbit near the LL1point, the trajectory is stable before the
spacecraft escapes from Earth; therefore, the choice of TE
off
is relaxed. Figure 8shows that there is an inflection point
(near 14.4th day) in the trend curve of both the eccentric-
ity and semi-major axis in the Earth-escaping phase. Further,
any time before the inflection point can be adopted as TE
off.In
this study, it is 14.4th day. After the spacecraft is captured
by the Moon, the trajectory is unstable and the eccentricity
increases under the control law. As the eccentricity starts de-
creasing, the time is chosen as TL
on. In this study, it is 7.075th
day.
To satisfy the requirement that m0=350 kg (the mass of
SMART-1), an iterative search must be performed. mfree is
the parameter of search (mfree is the mass of the spacecraft
in the free-flight phase). ,mfree is the step size of the search.
The diagram is shown in Fig. 10.
5 Numerical implementations
PPS-1350 Hall ion engine used in SMART-1 is considered
in this example. The thrust is 73.19 mN, and the velocity of
the fuel gas is 16.434 km/s (Betts and Erb 2006). The initial
time is recorded as 0, during which the lunar phase angle
is 291.1°, and the mass of the spacecraft is 349.2678 kg. In
addition, the semi-major axis of the initial orbit measures
24661.11 km, and the eccentricity is 0.7157.
Fig. 10 Diagram for iterative search to satisfy the requirement of ini-
tial mass
In the low-thrust transfer by the transiting of the LL1
point, the control law functions from the initial time to
the 292.2668th day. The free-flight phase must be instanta-
neous. Then the control law functions from the 292.2668th
to the 331.5628th day. At this time, the spacecraft arrives in
the terminal orbit and the low-thrust transfer is completed.
The semi-major axis of the terminal orbit is 7238.244 km
and the eccentricity is 0.5882. The final mass of the space-
craft is 280.7816 kg. Figure 11 depicts the trajectories
in geocentric inertial and syzygy frames. The total trans-
fer time is 331.5628 days, and the fuel consumption is
69.2184 kg.
In the low-thrust transfer by the transiting of the Halo
orbit near the LL1point, the control law works from the ini-
tial time to the 191.5556th day. The free-flight phase ranges
from the 191.5556th to the 241.5380th day. Then the con-
trol law works from the 241.5380th to the 260.2504th day.
At this time, the spacecraft arrives in the terminal orbit and
the low-thrust transfer is completed. The semi-major axis
of the terminal orbit is 7238.244 km, and the eccentricity
is 0.5882. The final mass of the spacecraft is 269.0420 kg.
Figure 12 depicts the trajectories in geocentric inertial and
syzygy frames. The total transfer time is 260.2504 days, and
the fuel consumption is 80.9580 kg, including 18.0242 kg
used to control the eccentricity under the subcontrol law II.
Betts and Erb (2006) used the direct collocation method
to optimize the SMART-1 orbit with similar boundary con-
ditions; however, the total transfer time was 201.7267 days
and the fuel consumption was 74.994 kg. The performance
comparison of the three trajectories is shown in Table 1.
96 Page 10 of 11 Q. Qu et al.
Fig. 11 Trajectory from the Earth to Moon: (a) the trajectory in geocentric inertial frame; (b) the trajectory in syzygy frame
Fig. 12 Trajectory from the Earth to Moon: (a) the trajectory in geocentric inertial frame; (b) the trajectory in syzygy frame
Table 1 Performance
comparison of the three LTTs LL1point Halo orbit near LL1point SMART-1
Transfer time 331.5628 days 260.2504 days 201.7267 days
Fuel consumption 69.2184 kg 80.9580 kg 74.994 kg
The table shows that the fuel consumption of the LTTs
when transiting the LL1point is 5.7756 kg less than that of
SMART-1; however, the diminution of fuel consumption is
at the expense of the transfer time. When making further
optimizations to the LLTs when transiting the LL1point, the
obtained LLTs may have the lowest fuel consumption. How-
ever, compared with the LTTs when transiting the Halo orbit
near the LL1point, the window of the passing of LTTs when
transiting the LL1point is narrower. In other words, the cis-
lunar transfer opportunities when transiting the LL1point is
[77°, 109°] [285°, 342°], while the ones when transiting
the Halo orbit near the LL1point is [0°, 30°] [96°, 207°]
[269°, 360°].
The fuel consumption of the LTTs when transiting the
Halo orbit near the LL1point is 5.964 kg more than that
of SMART-1, the main reason of which is described as fol-
lows. The following example is based on the deformation in
SBCM of invariant manifolds derived from the transporta-
tion tube. By considering the dependence on initial values of
the solution, the invariant manifolds can still remain a part of
the transportation tube. However, LLTs of SMART-1 must
be in the transportation tube. Therefore, further optimiza-
tion is required in the future. This paper provides the bound-
ary of trajectory optimization (i.e., the transportation tube).
However, when the trajectory optimization is performed in
SMART-1, there is no boundary and a large amount of com-
putation is needed.
6 Conclusions
In this paper, the low-thrust transfer orbit was analyzed by
studying the invariant manifolds derived from the trans-
portation tube. The deformation of transportation tube in
SBCM was examined thoroughly. Next, a gradient-based de-
sign of the LTTs was represented by analyzing the invariant
manifolds of the LL1point and the Halo orbit near it. Its
deformation under solar gravity perturbation was also pre-
sented; this is significant for designing LTTs. In other words,
The cislunar low-thrust trajectories via the libration point Page 11 of 11 96
the LL1point and the Halo orbit near it will serve in the free-
flight phase such that the fuel consumption can be as low as
possible. In addition, the control law that changes energy
most rapidly was designed based on CR3BP. Furthermore,
a certain ON/OFF time is provided with a two-dimensional
search method. The numerical results fundamentally coin-
cide with the SMART-1 orbit, which has similar boundary
conditions.
The paper focuses on the feasibility of low-thrust trans-
fer based on the transportation tube. However, the results
obtained in this paper have not been optimized. Therefore,
the result can be used as an initial low-energy transfer orbit
between the Earth and Moon. If further optimization can be
conducted, another orbit that can save more energy can be
obtained.
Acknowledgements The research is supported by the National Nat-
ural Science Foundation of China (11172020 and 11432001), Beijing
Natural Science Foundation (4153060), and the Fundamental Research
Funds for the Central Universities.
References
Barden, B.T., Howell, K.C., et al.: Application of dynamical systems
theory to trajectory design for a libration point mission. J. Astro-
naut. Sci. 45(2), 161–178 (1996)
Belbruno, E.A., Miller, J.K.: Sun-perturbated Earth-to-Moon trans-
fers with ballistic capture. J. Guid. Control Dyn. 16(4), 770–775
(2015)
Betts, J.T., Erb, S.O.: Optimal low thrust trajectories to the Moon.
SIAM J. Appl. Dyn. Syst. 2(2), 144–170 (2006)
Conley, C.C.: Low energy transit orbits in the restricted three-body
problem. SIAM J. Appl. Math. 97(4), 732–746 (1968)
Graham, K.F., Rao, A.V.: Minimum-fuel trajectory optimization of
many revolution low-thrust Earth-orbit transfers. Preprint. Acta
Astronaut. (2014, submitted)
Graham, K.F., Rao, A.V.: Minimum-time trajectory optimization of
many revolution low-thrust Earth-orbit transfers. J. Spacecr.
Rockets 52(3), 1–17 (2015)
Guelman, M.: Earth-to-Moon transfer with a limited power engine.
J. Guid. Control Dyn. 18(18), 1133–1138 (1995)
Herman, A.L., Conway, B.A.: Direct optimization using collocation
based on high-order Gauss-Lobatto quadrature rules. J. Guid.
Control Dyn. 19(3), 592–599 (1996)
Howell, K.C., Ozimek, M.T.: Low-thrust transfers in the Earth–Moon
system, including applications to libration point orbits. J. Guid.
Control Dyn. 33(2), 533–549 (2010)
Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Dynamical Systems,
the Three-Body Problem and Space Mission Design. World Sci-
entific, Singapore (2000)
Koon, W.S., Lo, M.W., Marsden, J.E., Ross, S.D.: Low energy transfer
to the Moon. Celest. Mech. Dyn. Astron. 9,6373(2001)
Lee, D., Butcher, E.A., Sanyal, A.K.: Optimal interior Earth–Moon
Lagrange point transfer trajectories using mixed impulsive and
continuous thrust. Aerosp. Sci. Technol. 39,281292(2014)
Petropoulos, A.E.: Simple control laws for low-thrust orbit transfers.
In: AAS/AIAA Astrodynamics Specialists Conference (2003).
Paper AAS 03-630
Petropoulos, A.E., Longuski, J.M.: Shape-based algorithm for the auto-
mated design of low-thrust, gravity assist trajectories. J. Spacecr.
Rockets 41(5), 787–796 (2004)
Polanskey, C.A., Joy, S.P., Raymond, C.A.: DAWN science planning,
operations and archiving. Space Sci. Rev. 163(1–4), 511–543
(2011)
Racca, G.D.: New challenges to trajectory design by the use of electric
propulsion and other new means of wandering in the solar system.
Celest. Mech. Dyn. Astron. 85(1), 1–24 (2003)
Racca, G.D., Marini, A., et al.: SMART-1 mission description and de-
velopment status. Planet. Space Sci. 50(2), 1323–1337 (2002)
Szebehely, V.: Theory of Orbits. Academic Press, New York (1967)
Vellutini, E., Avanzini, G.: Shape-based design of LTTs to cislunar La-
grangian point. J. Guid. Control Dyn. 37(4), 1329–1335 (2015)
Wall, B.J., Conway, B.A.: Shape-based approach to low-thrust ren-
dezvous trajectory design. J. Guid. Control Dyn. 32(1), 95–101
(2009)
Xu, M., Yan, W., Xu, S.: On the construction of low-energy cislunar
and translunar transfers based on the libration points. Astrophys.
Space Sci. 348(1), 65–68 (2013)
Yamato , H. , Sp encer , D. B. : Orbit t ra nsfer via t ub e jumpi ng o n pla-
nar restricted problems of four bodies. J. Spacecr. Rockets 42(2),
321–328 (2003)
Yamato, H., Spencer, D.B.: Transit-orbit search for planar restricted
three-body problems with perturbations. J. Guid. Control Dyn.
27(6), 1035–1045 (2004)
... In both strategies of Earth-Moon transfers, the Earth-Moon libration points L 1 and L 2 always serve as transfer stations for Moon missions due to its accessibility to the Moon. Therefore, some useful mathematical tools such as invariant manifolds, lunar flyby and low thrust have been utilized to design low energy transfers to the Moon and Earth-Moon L 1 or L 2 (Gao et al. 2019;Lei and Xu 2018;Lian et al. 2015;Qi and de Ruiter 2019;Qu et al. 2017). Moreover, invariant manifolds also have been regarded as an effective tool to design low-cost Earth-Moon transfer trajectories (Davis et al. 2011;Howell and Kakoi 2006;Koon et al. 2000Koon et al. , 2011Lei and Xu 2016;Mingotti et al. 2012a;Xu et al. 2012). ...
... More specifically, interior Earth-Moon transfers are always defined in the Earth-Moon CR3BP model without considering any perturbation. Since invariant manifolds associated with the Earth-Moon L 1 point cannot naturally approach the neighborhood of the Earth, additional maneuvers are usually required to connect the invariant manifold and the transfer trajectory starting from low Earth orbit (LEO) (Marson et al. 2010;Mingotti and Topputo 2011;Qu et al. 2017). Then bi-impulsive Earth-Moon transfers have been investigated by using the patched-conic method and differential correction to avoid the mid-course maneuver (da Silva Fernandes and Maranhão Porto Marinho 2012;Lv et al. 2017;Yagasaki 2004a). ...
Article
Full-text available
Low-energy bi-impulsive Earth-Moon transfers are investigated by using periodic orbits. Two Earth-Moon transfer design strategies in the CR3BP are proposed, termed direct and indirect design strategy. In the direct design strategy, periodic orbits which approach both vicinities of the Earth and Moon are selected as candidate periodic orbits, which can provide an initial guess of bi-impulsive Earth-Moon transfers. In the indirect design strategy, new bi-impulsive Earth-Moon transfers can be designed by patching together a bi-impulsive Earth-Moon transfer and a candidate periodic orbit which can approach the vicinity of the Moon. Optimizations in the CR3BP are undertaken based on the Gradient Descent method. Finally, bi-impulsive Earth-Moon transfer design and optimizations in the Sun-Earth-Moon bi-circular model (BCM) are carried out, using bi-impulsive Earth-Moon transfers in the CR3BP as initial guesses. Results show that the bi-impulsive Earth-Moon transfer in the CR3BP can serve as a good approximation for the BCM. Moreover, numerical results indicate that the optimal transfers in the BCM have the potential to be of lower cost in terms of velocity impulse than optimal transfers in the CR3BP.
... Similar approaches to low-thrust trajectory design also appear in Refs. [8][9][10]. Additionally, transfers in the lunar domain, particularly those involving periodic orbits such as NRHOs and distant retrograde orbits have also been studied extensively by various researchers [11][12][13]. ...
Article
In this study, a supervised machine learning approach called Gaussian process regression (GPR) was applied to approximate optimal bi-impulse rendezvous maneuvers in the cis-lunar space. We demonstrate the use of the GPR approximation of the optimal bi-impulse transfer to patch points associated with various invariant manifolds in the cis-lunar space. The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solutions of the optimal bi-impulsive Lambert transfer because the learned map is computationally efficient. This approach promises to be useful for aiding in preliminary mission design. The use of invariant manifolds as part of the transfer trajectory design offers unique features for reducing propellant consumption while facilitating the solution of trajectory optimization problems. Long ballistic capture coasts are also very attractive for mission guidance, navigation, and control robustness. A multi-input single-output GPR model is presented to represent the fuel costs (in terms of the ΔV magnitude) associated with the class of orbital transfers of interest efficiently. The developed model is also proven to provide efficient approximations. The multi-resolution use of local GPRs over smaller sub-domains and their use for constructing a global GPR model are also demonstrated. One of the unique features of GPRs is that they provide an estimate of the quality of approximations in the form of covariance, which is proven to provide statistical consistency with the optimal trajectories generated through the approximation process. The numerical results demonstrate our basis for optimism for the utility of the proposed method.
... Rectilinear Halo Orbits (NRHOs) in a multi-body system for a more accurate representation of the manifolds, and using them as terminal coast arcs for trajectory design [7]. Similar approaches to low-thrust trajectory design also appear in [8][9][10]. Additionally, transfers in the lunar domain especially involving periodic orbits like NRHOs and Distant Retrograde Orbits (DROs) have also been studied extensively by various researchers [11][12][13]. ...
Preprint
Full-text available
A supervised machine learning approach called the Gaussian Process Regression (GPR) is applied to approximate the optimal bi-impulse rendezvous maneuvers in cis-lunar space. The use of GPR approximation of the optimal bi-impulse transfer to patch-points associated with various invariant manifolds in the cis-lunar space is demonstrated. The proposed method advances preliminary mission design operations by avoiding the computational costs associated with repeated solution of the optimal bi-impulsive Lambert transfer because the learned map is efficient to compute. This approach promises to be useful for aiding preliminary mission design. The use of invariant manifolds as part of the transfer trajectory design offers unique features in reducing propellant consumption while facilitating the solution of the trajectory optimization problems. Long ballistic capture coasts are also very attractive for mission guidance, navigation and control robustness. A multi-input single-output GPR model is shown to efficiently represent the fuel costs (in terms of the $\Delta$V magnitude) associated with the class of orbital transfers of interest. A multi-input multi-output GPR model is developed and shown to provide efficient approximations. Multi-resolution use of local GPRs over smaller sub domains, and their use to construct a global GPR model is also demonstrated. One of the unique features of GPRs is to provide an estimate on the quality of the approximations in the form of covariance, which is shown to provide statistical consistency to the optimal trajectories generated from the approximation process. Numerical results demonstrate a basis for optimism for the utility of the proposed method.
... They also extended their work to study the behavior of the invariant manifolds of Southern L 2 Near Rectilinear Halo Orbits (NRHOs) in a multi-body system for a more accurate representation of the manifolds, and using them as terminal coast arcs for trajectory design [19]. Similar approaches to low-thrust trajectory design also appear in [20,21]. ...
Conference Paper
Full-text available
A novel indirect-based trajectory optimization framework is proposed that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs while incorporating eclipses and perturbations during the optimization process in an ephemeris model; a feature lacking in state of the art software like MYSTIC and Copernicus. The end-to-end trajectories are generated by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of the proposed trajectory optimization framework using advanced indirect methods and by leveraging a Composite Smooth Control (CSC) construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). The results quantify accurate ∆V costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to be used for analogous un-crewed lunar missions.
... Transfers in the low-thrust domain exploiting manifolds of periodic orbits as well as multi-body equilibria in the cislunar space as a research topic have seen an exciting impetus in the recent past [5,29,37]. Although the NRHOs and associated transfers have been studied in some detail, an end-to-end low-thrust transfer from a geocentric orbit to a NRHO, especially one which leverages manifolds is important from the point of view of the Lunar Gateway mission. ...
Article
Full-text available
In this paper, we investigate the manifolds of three Near-Rectilinear Halo Orbits (NRHOs) and optimal low-thrust transfer trajectories using a high-fidelity dynamical model. Time- and fuel-optimal low-thrust transfers to (and from) these NRHOs are generated leveraging their ‘invariant’ manifolds, which serve as long terminal coast arcs. Analyses are performed to identify suitable manifold entry/exit conditions based on inclination and minimum distance from the Earth. The relative merits of the stable/unstable manifolds are studied with regard to time- and fuel-optimality criteria, for a set of representative low-thrust family of transfers.
... They also extended this to study the behavior of the invariant manifolds of Southern 2 Near Rectilinear Halo Orbits (NRHOs) in a multi-body system for a more accurate representation of the manifolds, and using them as terminal coast arcs for trajectory design [21]. Similar approaches to low-thrust trajectory design also appear in [22][23][24]. Additionally, transfers in the lunar domain especially involving periodic orbits like NRHOs and Distant Retrograde Orbits (DROs) have also been studied extensively by various researchers [25][26][27][28]. ...
Article
Full-text available
A novel methodology is proposed for designing low-thrust trajectories to quasi-periodic, near rectilinear Halo orbits that leverages ephemeris-driven, "invariant manifold analogues" as long-duration asymptotic terminal coast arcs. The proposed methodology generates end-to-end, eclipse-conscious, fuel-optimal transfers in an ephemeris model using an indirect formulation of optimal control theory. The end-to-end trajectories are achieved by patching Earth-escape spirals to a judiciously chosen set of states on pre-computed manifolds. The results elucidate the efficacy of employing such a hybrid optimization algorithm for solving end-to-end analogous fuel-optimal problems using indirect methods and leveraging a composite smooth control construct. Multiple representative cargo re-supply trajectories are generated for the Lunar Orbital Platform-Gateway (LOP-G). A novel process is introduced to incorporate eclipse-induced coast arcs and their impact within optimization. The results quantify accurate Δ costs required for achieving efficient eclipse-conscious transfers for several launch opportunities in 2025 and are anticipated to find applications for analogous uncrewed missions.
... The features of the circular restricted three-body problem (CR3BP) [1,2] allow transfer trajectories with small propellant consumption to be designed by exploiting the existence of invariant manifolds, as is discussed in Refs. [3,4,5,6] in the case of the Earth-Moon CR3BP. A potential application of the results of the CR3BP analysis is constituted by space missions orbiting around equilibrium points. ...
Article
Full-text available
A solar sail generates thrust without consuming any propellant, so it constitutes a promising option for mission scenarios requiring a continuous propulsive acceleration, such as the maintenance of a (collinear) L1-type artificial equilibrium point in the Sun-[Earth+Moon] circular restricted three-body problem. The usefulness of a spacecraft placed at such an artificial equilibrium point is in its capabilities of solar observation, as it guarantees a continuous monitoring of solar activity and is able to give an early warning in case of catastrophic solar flares. Because those vantage points are known to be intrinsically unstable, a suitable control system is necessary for station keeping purposes. This work discusses on how to stabilize an L1-type artificial equilibrium point with a solar sail by suitably adjusting its lightness number and thrust vector orientation. A full-state feedback control law is assumed, where the control gains are chosen with a linear-quadratic regulator approach. In particular, the numerical simulation results show that an L1-type artificial equilibrium point can be maintained with small required control torques, by using a set of reflectivity control devices.
... Tang (2013a, 2013b) and Lian et al. (2012) studied the problem of libration point orbit rendezvous using terminal sliding mode control. Qu et al. (2017) investigated a gradient-based design methodology for low-thrust trajectories with the help of invariant manifolds and halo orbit of LL1 point. ...
Article
Full-text available
A methodology is proposed to design optimal time-fixed impulsive transfers in the vicinity of the L2 libration point of the Earth-Moon system, taking the construction of a space station around the collinear libration points as the background. The approximate analytical expression of motions around the L2 point in the CRTBP is given, and the expression in the ERTBP is derived by linearizing the dynamical equations for the purpose of expanding the methodology from the CRTBP to the ERTBP. Thus, the approximate analytical solution of the transfer between two points is obtained by substituting the position vectors of the two points into the expression, which solves the Lambert problem in the three-body system. Furthermore, the transfer between different orbits is constructed by parameterization of the position vectors with the amplitudes and phases of the initial orbit and the final orbit. The transfers are optimized such that the total velocity increment required to implement the transfer exhibits a global minimum. The values of variables involved in the optimal transfers are determined by the unconstrained minimization of a function of one or nine variables using a multivariable search technique. To numerically ensure that the transfers are accurate and to eliminate the linearization bias, the differential correction and SQP method are employed. The optimality of the transfers is determined lastly by the primer vector theory. Simulations of point-to-point transfers, Lissajous-to-Lissajous transfers, halo-to-halo transfers and Lissajous-to-halo transfers are made. The results of this study indicate that the approximate analytical solutions, as well as the differential correction and SQP method, are valid in the design of the optimal transfers around the libration points of the restricted three-body problem.
Article
This paper investigates the energy change of the two-impulse Earth-Moon trajectory in the Sun-Earth-Moon bicircular model analytically. An analytic expression is first derived to describe the change of the Jacobi value of the two-impulse Earth-Moon trajectory in the Sun-Earth-Moon system. Then the analytical results of the total cost of the Earth-Moon transfer can be obtained by solving the equations related to the change in the Jacobi value of the transfer trajectory. Numerical results show that the analytic expressions can provide good descriptions of the energy changes of the two-impulse Earth-Moon transfer. Moreover, it indicates that the total cost mainly depends on a critical angle, termed the perilune angle. Accordingly, we explore the correlation of the total cost and the perilune angle. Finally, optimizations of two-impulse Earth-Moon transfers are further carried out.
Book
Full-text available
Space missions which reach destinations such as the moon, asteroids, or the satellites of Jupiter are complex and challenging to design, requiring new and unusual kinds of orbits to meet their goals, orbits that cannot be found by classical approaches to the problem. In addition, libration point orbits are seeing greater use. This book guides the reader through the trajectory design of both libration point missions and a new class of low energy trajectories which have recently been discovered and make possible missions which classical trajectories (conic sections) could not. Low energy trajectories are achieved by making use of gravity as much as possible, using the natural dynamics arising from the presence of a third body (or more bodies). The term "low-energy" is used to refer to the low fuel and therefore low energy required to control the trajectory from a given starting condition to a targeted final condition. Low energy trajectory technology allows space agencies and aerospace companies to envision missions in the near future involving long duration observations and/or constellations of spacecraft using little fuel. A proper understanding of low energy trajectory technology begins with a study of the restricted three-body problem, a classic problem of astrodynamics, which we approach from a rigorous and geometric point of view. We develop systematic and computationally efficient methods for the design of both libration point orbits and low energy trajectories based on fundamental ideas of invariant manifold theory. Furthermore, we develop the computational techniques needed to design trajectories for a spacecraft in the field of N bodies by patching together solutions of the 3 body problem. These computational methods are key for the development of some NASA and ESA mission trajectories, such as low energy libration point orbit missions (e.g., the Genesis Discovery Mission and Terrestrial Planet Finder, to mention only two), low energy lunar missions and low energy tours of outer planet moon systems, such as the a mission to tour and explore in detail the icy moons of Jupiter. This book can serve as a valuable resource for graduate students and advanced undergraduates in aerospace engineering, as well as a manual for practitioners who work on libration point and deep space missions in industry and at government laboratories. The book contains a wealth of background material, but also brings the reader up to a portion of the research frontier. Furthermore, the book will also have appeal to students of applied mathematics, especially those with an interest in dynamical systems and real world applications.
Conference Paper
Full-text available
Recently, there has been accelerated interest in missions utilizing trajectories near libration points. The trajectory design issues involved in missions of such complexity go beyond the lack of preliminary baseline trajectories (since conic analysis fails in this region of the solution space). Successful and efficient design of mission options will require new persepectives; a more complete understanding of the solution space is imperative. In this investigation, dynamical systems theory is applied to better understand the geometry of the phase space in the three-body problem via stable and unstable mamlolds. Then, the manifolds are used to generate various solution arcs and establish trajectory options that are then utilized in preliminary design for the proposed Suess-Urey mission.
Article
Full-text available
There exist cislunar and translunar libration points near the Moon, which are referred to as the LL 1 and LL 2 points, respectively. They can generate the different types of low-energy trajectories transferring from Earth to Moon. The time-dependent analytic model including the gravitational forces from the Sun, Earth, and Moon is employed to investigate the energy-minimal and practical transfer trajectories. However, different from the circular restricted three-body problem, the equivalent gravitational equilibria are defined according to the geometry of the instantaneous Hill boundary due to the gravitational perturbation from the Sun. The relationship between the altitudes of periapsis and eccentricities is achieved from the Poincaré mapping for all the captured lunar trajectories, which presents the statistical feature of the fuel cost and captured orbital elements rather than generating a specified Moon-captured segment. The minimum energy required by the captured trajectory on a lunar circular orbit is deduced in the spatial bi-circular model. The idea is presented that the asymptotical behaviors of invariant manifolds approaching to/traveling from the libration points or halo orbits are destroyed by the solar perturbation. In fact, the energy-minimal cislunar transfer trajectory is acquired by transiting the LL 1 point, while the energy-minimal translunar transfer trajectory is obtained by transiting the LL 2 point. Finally, the transfer opportunities for the practical trajectories that have escaped from the Earth and have been captured by the Moon are yielded by the transiting halo orbits near the LL 1 and LL 2 points, which can be used to generate the whole of the trajectories.
Article
Recently, there has been accelerated interest in missions utilizing trajectories near libration points. The trajectory design issues involved in missions of such complexity go beyond the lack of preliminary baseline trajectories (since conic analysis fails in this region of the solution space). Successful and efficient design of mission options will require new persepectives; a more complete understanding of the solution space is imperative. In this investigation, dynamical systems theory is applied to better understand the geometry of the phase space in the three-body problem via stable and unstable manifolds. Then, the manifolds are used to generate various solution arcs and establish trajectory options that are then utilized in preliminary design for the proposed Suess-Urey mission.
Article
The problem of determining high-accuracy minimum-time Earth-orbit transfers using low-thrust propulsion is considered. The optimal orbital transfer problem is posed as a constrained nonlinear optimal control problem and is solved using a variable-order Legendre-Gauss-Radau quadrature orthogonal collocation method. Initial guesses for the optimal control problem are obtained by solving a sequence of modified optimal control problems where the final true longitude is constrained and the mean square difference between the specified terminal boundary conditions and the computed terminal conditions is minimized. It is found that solutions to the minimumtime low-thrust optimal control problem are only locally optimal, in that the solution has essentially the same number of orbital revolutions as that of the initial guess. A search method is then devised that enables computation of solutions with an even lower cost where the final true longitude is constrained to be different from that obtained in the original locally optimal solution. A numerical optimization study is then performed to determine optimal trajectories and control inputs for a range of initial thrust accelerations and constant specific impulses. The key features of the solutions are then determined, and relationships are obtained between the optimal transfer time and the optimal final true longitude as a function of the initial thrust acceleration and specific impulse. Finally, a detailed postoptimality analysis is performed to verify the close proximity of the numerical solutions to the true optimal solution. Copyright © 2014 by Kathryn F. Graham and Anil V. Rao. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
Article
Given the benefits of coupling low-thrust propulsion with gravity assists, techniques for easily identifying candi- date trajectories would be extremely useful to mission designers. The computational implementation of an analytic, shape-based method for the design of low-thrust, gravity-assist trajectories is described. Two-body motion (cen- tral body and spacecraft) is assumed between the flybys, and the gravity-assists are modeled as discontinuities in velocity arising from an instantaneous turning of the spacecraft's hyperbolic excess velocity vector with respect to the flyby body. The method is augmented by allowing coast arcs to be patched with thrust arcs on the transfers between bodies. The shape-based approach permits not only rapid, broad searches over the design space, but also provides initial estimates for use in trajectory optimization. Numerical examples computed with the shape-based method, using an exponential sinusoid shape, are presented for an Earth-Mars-Ceres rendezvous trajectory and an Earth-Venus-Earth-Mars-Jupiter flyby trajectory. Selected trajectories from the shape-based method are successfully used as initial estimates in an optimization program employing direct methods.