Content uploaded by Xu Ming

Author content

All content in this area was uploaded by Xu Ming on Oct 10, 2017

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.

AtypeofEarth–Moontransferorbitbasedonthecircu-

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 IS−E+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 SS−E+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 SE−M, 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]=LE−M

[T]=$L3

E−M/G(mE+mM)%1/2(1)

where LE−Mis 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⎤

⎦−(1−µ) r−rE

∥•∥3

−µr−rM

∥•′∥3+ω2

sRz(β)Rx(i)As

−Rz(β)Rx(i)Rz(−θs)ms

R−asRs

∥•′′ ∥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 IS−E+M, and r=[x,y,z]Tis the position vector

of the spacecraft in SE−M,rE=[−µ00]Tis the posi-

tion vector of Earth in SE−M,rM=[1−µ00]Tis the

position vector of Moon in SE−M,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 SS−E+M,

and Rs=[−µS00]Tis the position vector of Sun in

SS−E+M. The symbol •represents the term r−rE,•′rep-

resents r−rM,and•′′ represents the term R−asRs.

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 x–zplane in

the syzygy frame. Poincaré mapping P(z)is deﬁned as

P(z)=φT(z), ∀z∈Γ(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=λ−1

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=0°

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)ft−1

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 θ|·fn+.(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

κ2−κ1.

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-ﬂightphase,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. 9,63–73(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. 39,281–292(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)

- A preview of this full-text is provided by Springer Nature.
- Learn more

Preview content only

Content available from Astrophysics and Space Science

This content is subject to copyright. Terms and conditions apply.