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Advances in Space Research 52 (2013) 2067–2079

Optimal station-keeping near Earth–Moon collinear librationpoints using continuous and impulsive maneuvers

Mehrdad Ghorbani, Nima Assadian ⇑

Department of Aerospace Engineering, Sharif University of Technology, Azadi av., Tehran, Iran

Received 1 May 2013; received in revised form 17 September 2013; accepted 19 September 2013Available online 30 September 2013

Abstract

In this study the gravitational perturbations of the Sun and other planets are modeled on the dynamics near the Earth–MoonLagrange points and optimal continuous and discrete station-keeping maneuvers are found to maintain spacecraft about these points.The most critical perturbation effect near the L1 and L2 Lagrange points of the Earth–Moon is the ellipticity of the Moon’s orbit and theSun’s gravity, respectively. These perturbations deviate the spacecraft from its nominal orbit and have been modeled through a restrictedfive-body problem (R5BP) formulation compatible with circular restricted three-body problem (CR3BP). The continuous control orimpulsive maneuvers can compensate the deviation and keep the spacecraft on the closed orbit about the Lagrange point. The continuouscontrol has been computed using linear quadratic regulator (LQR) and is compared with nonlinear programming (NP). The multipleshooting (MS) has been used for the computation of impulsive maneuvers to keep the trajectory closed and subsequently an optimizedMS (OMS) method and multiple impulses optimization (MIO) method have been introduced, which minimize the summation of multipleimpulses. In these two methods the spacecraft is allowed to deviate from the nominal orbit; however, the spacecraft trajectory shouldclose itself. In this manner, some closed or nearly closed trajectories around the Earth–Moon Lagrange points are found that need almostzero station-keeping maneuver.� 2013 Published by Elsevier Ltd. on behalf of COSPAR.

Keywords: Restricted five-body problem; Optimal control; Lagrange points; Continuous and impulsive maneuvers; Multiple impulses; Linear quadraticregulator

1. Introduction

In the circular restricted three-body problem (CR3BP)five equilibrium points exist which are known as Lagrangepoints. The first three Lagrange points (L1,L2,L3), calledthe collinear Libration points, are usually unstable andthe next two points (L4,L5), known as triangular Librationpoints, are usually linearly stable (Szebehely, 1967). In thereal systems, like Earth–Moon–Spacecraft, even the trian-gular libration points may be unstable because of perturba-tions (Gomez et al., 2001b). The majority of space missionsto Lagrange points are for closed orbits around L1 and L2

0273-1177/$36.00 � 2013 Published by Elsevier Ltd. on behalf of COSPAR.

http://dx.doi.org/10.1016/j.asr.2013.09.021

⇑ Corresponding author. Tel.: +98 21 66164607; fax: +98 21 66022731.E-mail addresses: mehrdadmif@gmail.com (M. Ghorbani), assadian@

sharif.edu (N. Assadian).

(Gomez et al., 2001a). There are orbits around L1 and L2 inCR3BP about which the spacecraft can move with low fuelconsumption.

When a Lagrange point is inherently unstable or due toperturbations, station-keeping maneuvers are required tomaintain a spacecraft near it. Euler and Yu (1971) arethe first ones who defined an optimal control problem forstation-keeping of the spacecraft at the collinear points inCR3BP. Howell and Pernicka (1993) have used impulsivemaneuvers for station-keeping near L1 point of the Sun–Earth system. Dunham and Roberts (2001) have studiedthe station-keeping of ISEE-3, ACE, and SOHO missions.Williams et al. (2004) have designed the required maneu-vers for station-keeping in the halo orbit of Genesis mis-sion. Hou et al. (2006) introduced a continuous low-thrust strategy for CR3BP and then they (Hou et al.,

2068 M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079

2011) considered various perturbations like the ellipticeffect of the Moon’s orbit and the Sun’s mass as main per-turbations in the real Earth–Moon system, to adopt sta-tion-keeping approaches in small amplitude motionaround L2 point. The transfer trajectory and station-keep-ing about Lagrange points in real Earth–Moon system isalso probed in (Pavlak, 2010).

Farres and Jorba (2011) investigated solar sail station-keeping in the elliptic restricted three-body problem(ER3BP). Designing a station-keeping strategy for a solarsail in the Sun–Earth system, they have also investigatedthe solar sail dynamics and spacecraft controllabilityaround the halo orbit of the Sun–Earth system (Farresand Jorba, 2012). Low energy trajectory optimizationwas performed for real Earth–Moon system by Lei et al.(2012), and a mathematical method was proposed bySukhanov and Prado (2012) to calculate optimal station-keeping by means of electric propulsion. This methodwas based on the linearization of the satellite motion neara reference orbit.

Invariant manifolds in CR3BP are used for space mis-sion design (Gomez et al., 2001c), and also for station-keeping of halo, quasi-halo, and lissajous orbits nearLagrange points (Gomez et al., 1998a). Station-keepingcan be like mission trajectory correction maneuvers orinsertion in periodic orbits about Lagrange points (Serbanet al., 2002). Assadian and Pourtakdoust (2010a) found themulti-objective optimal trajectories of a spacecraft travel-ing from the Earth to the Moon via impulsive maneuvers.They have used the total flight time and the summation ofimpulsive maneuvers as the objective functions. Most ofthe space missions are about Sun–Earth Lagrange points,but some work used Lunar Lagrange points as well(Gomez et al., 1998b; Folta and Vaughan, 2004; Houet al., 2011; Pavlak, 2010; Pavlak and Howell, 2012).ARTEMIS mission (acceleration, reconnection, turbulenceand electrodynamics of Moon’s interaction with the Sun)was the first Earth–Moon libration orbiter. It was a suc-cessful mission that revolved about L1 and L2 Lagrangepoints of the Earth–Moon system (Folta et al., 2010,2013; Sweetser et al., 2011).

There are several studies that have incorporated the per-turbations of gravity forces to the CR3BP. The ellipticityof the major bodies orbit and the gravity of the otherbodies should be modeled. The ellipticity of the orbitscan be handled through the use of an ER3BP (Gurfil andMeltzer, 2006; Kumar Jha and Shrivastava, 2001). Gurfiland Meltzer (2006) developed a new method for generatingperiodic orbits near collinear libration points in ER3BP. Afour- and five-body problem model is required to observethe effect of the Sun and other planets. The Bi-Circularproblem (Prado, 2005), Quasi Bi-Circular problem (And-reu, 1998), and Bi-Elliptic problem (Assadian, 2009; Assa-dian and Pourtakdoust, 2010b) are among the known four-body problem (4BP) models. The Bi-Elliptic definition ofthe coordinate system is suitable for defining the equationsof motion about Lagrange points (Ghorbani, 2011).

In this study, the station-keeping of the spacecraft nearthe collinear Lagrange points of the Earth–Moon systemunder the gravitational perturbation of the Sun and otherplanets has been introduced. Most of the previous workshave used a simplified model of the system. Thus, a newformulation has been presented to take into considerationthe effect of the eccentricity of the Earth–Moon orbit,and the Sun and other planets gravitation in a similar for-mulation of the CR3BP. Using this formulation, the con-tinuous and impulsive control strategies are developedusing the classical optimization methods to keep the space-craft on a reference trajectory of the original CR3BP. Bothmethods are applicable with the current technology. Thecontinuous control can be implemented through usingmodern electric propulsion systems and impulsive maneu-vers can be implemented utilizing the classic high thrustchemical propulsions. The station-keeping on the referenceorbit is just for comparison and the final goal of thisresearch is to find closed orbits near the Lagrange pointsof the real Earth–Moon system with almost zero station-keeping maneuvers by introducing more degrees of free-dom to the station-keeping problem.

First CR3BP and Lagrange points are introduced inSection 2. Then, the elliptic effect and the Sun and the grav-ity of the other planets are modeled using a four- and five-body formulation. The linearization of the equationsemployed for the linear optimal control and introductionof control actions are discussed in this section as well.The continuous control methods including linear quadraticregulator (LQR) and nonlinear programming (NP) are uti-lized for the problem in Section 3. Afterward, the impulsivestation-keeping approached and the results are presented inSection 4 and finally the concluding remarks have beenprovided in Section 5.

2. Governing equations

In this section, first, CR3BP and Lagrange points areintroduced, and then elliptic effect of the orbit of the Moonhas been added to the equations. To demonstrate the Sunand other planets effects a restricted four-body problem(R4BP) model followed by a restricted five-body problem(R5BP) model compatible with the CR3BP are presented.Finally, the equations of motion of the spacecraft arederived and compared with the real N-body equations ofmotion to show the accuracy of the model. Then, equationsare linearized for the purpose of linear control method.

2.1. Circular restricted three-body problem

Consider two primary bodies (m1 and m2) in circularorbits around their center of mass. The study of the motionof an infinitesimal particle in the gravitational field ofprimaries is called CR3BP. The normalized equation ofmotion in the rotating frame with the primaries on the x

axis is as follows (Szebehely, 1967):

M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079 2069

€x

€y

€z

264375 ¼

2 _y þ x

�2 _xþ y

0

264

375� ð1� lÞ r1

r31

� lr2

r32

ð1Þ

The length unit in the above equation is the distance be-tween the primaries and the time unit is such that the per-iod of the primaries rotation is 2p. l is the ratio of thesmaller primary mass over the sum of the masses, r1 andr2 are the relative positions of the particle from theprimaries:

r1 ¼xþ l

y

z

264

375; r2 ¼

x� 1þ l

y

z

264

375 ð2Þ

Five equilibrium points exist in the CR3BP, three ofwhich locate in the x-axis are called collinear librationpoints and two other are called triangular points (Fig. 1).

2.2. Elliptical restricted three-body problem

Rather than assuming a circular orbit for the Moonaround the Earth, it is more close to real system if an eccen-tric approximation is utilized. To have a similar model tothat of the CR3BP, the equations of motion of the space-craft are derived in the rotating frame which rotates withthe variable speed of the Earth–Moon system and the dis-tances are scaled to the Earth–Moon distance (Assadian,2009):

€x

€y

€z

264375 ¼ �2 _hðg1 _x� _yÞ þ g3

_h2x

�2 _hðg1 _y þ _xÞ þ g3_h2y

�2 _hg1 z�g2_h2z

264

375� 1� l

r3

r1

r31

� lr3

r2

r32

ð3Þ

where r is the instantaneous distance between the Earthand the Moon divided by the semi-major axis of the orbitof the Moon, _h is the angular velocity of the Earth–Moonsystem around their center of mass, and g1, g2, and g3 aredefined as follows:

Figure 1. Schematic of the Lagrange points in the CRTBP.

g1 ¼e sin h

1þ e cos h

g2 ¼e cos h

1þ e cos h

g3 ¼ 1� g2 ¼1

1þ e cos h

ð4Þ

These equations are similar to the ones of the CR3BP,assuming the eccentricity e to be zero, and r and _h con-stants equal to 1.

2.3. Restricted four-body problem

A four-body model is used to show the effects of otherbodies in the ER3BP equations. The position vector of theinfinitesimal particle should be the same that was defined inthe ER3BP. Therefore, a R4BP model of Assadian (2009)has been used, in which the position vector q is defined fromthe Earth–Moon-barycenter (EMB) to the spacecraft. Therelative positions of the masses based on the definitions ofthe Fig. 2 (without m5) can be expressed as follows:

r12 ¼ R� lr r23 ¼ r

r13 ¼ Rþ ð1� lÞr r24 ¼ lrþ q

r14 ¼ Rþ q r34 ¼ q� ð1� lÞrð5Þ

The equation of motion for the spacecraft in the inertialframe is:

€q¼�ð1�lÞ r24

r324

�lr34

r334

�ðr� 1Þ r14

r314

�ð1�lÞ r12

r312

�lr13

r313

� �ð6Þ

where r is the total mass of the bodies in the non-dimen-sional system. Transferring the equations to the rotatingframe by the rotation velocity of the elliptic orbit of theMoon around the Earth ( _h) yields:

€x€y€z

2435 ¼ �2 _hðg1 _x� _yÞ þ g3

_h2x�2 _hðg1 _y þ _xÞ þ g3

_h2y�2 _hg1 _z� g2

_h2z

24

35� 1� l

r3

r24

r324

� lr3

r34

r334

� r� 1

r3

r14

r314

� ð1� lÞ r12

r312

� lr13

r313

� �ð7Þ

Figure 2. Definition of relative positions in R4BP and R5BP.

2070 M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079

In the Bi-Elliptic problem defined by Assadian (2009)the EMB moves around the Sun in an elliptic orbit, andthe orbit of the Moon around the Earth is also elliptic.Thus, the R and r vectors can be computed based on thetwo-body solution, the so called conic sections.

0.05

0.1

0.15

Reference TrajectoryN-Body SimulationMulti-Elliptic Simulation

L1

2.4. Generalized restricted five-body problem

Eq. (7) shows the effect of the Sun in the ER3BP. Toinclude the effect of the other planets in the equations, amore generalized equation has been derived, presented asR5BP. Fig. 2 shows the relative positions of this model.Same as the R4BP, distances between masses are expressedas relative position vectors. The new relative position vec-tors are:

r15 ¼ R0 r25 ¼ lr� Rþ R0

r35 ¼ R0 � R� ð1� lÞr r54 ¼ Rþ q� R0ð8Þ

The equations of motion of the spacecraft in the inertialframe, based on the above vectors, are derive to be:

€q ¼ �ð1� lÞ r24

r324

� lr34

r334

� ðr� 1Þ r14

r314

� ð1� lÞ r12

r312

� lr13

r313

� �

� dr54

r354

� ð1� lÞ r52

r352

� lr53

r353

� �ð9Þ

where d is the mass of the fifth body in the non-dimensionalsystem. Finally, in the rotating frame with the unit distancebe the distance from the Earth to the Moon, the equationsare derived to be:

€x

€y

€z

26643775¼

�2 _hðg1 _x� _yÞ þ g3_h2x

�2 _hðg1 _y þ _xÞ þ g3_h2y

�2 _hg1 _z� g2_h2z

2664

3775

� 1� lr3

r24

r324

� lr3

r34

r334

� r� 1

r3

r14

r314

� ð1� lÞ r12

r312

� lr13

r313

� �

� dr3

r54

r354

� ð1� lÞ r52

r352

� lr53

r353

� �ð10Þ

x

y

0.65 0.7 0.75 0.8 0.85 0.9-0.15

-0.1

-0.05

0

Figure 3. Verification of the derived formulation with the N-Bodysimulation and the escape of the spacecraft from the orbit around theEarth–Moon L1 with no station-keeping maneuver (2D projection).

2.5. Governing equations of motion

The equations of motion of the spacecraft in the samenon-dimensional system defined for the CR3BP (Eq. (1))of the Earth–Moon under the influence of the Sun andthe other planets gravitational forces has been derivedusing the generalized R5BP model. To include the effectof all planets, the relative position vectors between themasses and the planet pi are defined as:

r1pi¼ R0i r2pi

¼ lr� Rþ R0ir3pi¼ R0i � R� ð1� lÞr rpi4 ¼ Rþ q� R0i

ð11Þ

and the equation of motion in the inertial frame is:

€q¼�ð1�lÞr24

r324

�lr34

r334

�ðr�1Þ r14

r314

�ð1�lÞr12

r312

�lr13

r313

� �

�X8

i¼1i–3

dpi

rpi4

r3pi4

�ð1�lÞrpi2

r3pi2

�lrpi3

r3pi3

!ð12Þ

Finally, the equation of motion in the non-dimensionalrotating frame, called the Multi-Elliptic Problem (MEP),has been derived as follows:

€x

€y

€z

264375¼ �2 _hðg1 _x� _yÞþg3

_h2x

�2 _hðg1 _yþ _xÞþg3_h2y

�2 _hg1 _z�g2_h2z

264

375

�1�lr3

r24

r324

� lr3

r34

r334

�r�1

r3

r14

r314

�ð1�lÞr12

r312

�lr13

r313

� �

�X8

i¼1i–3

dpi

r3

rpi4

r3pi4

�ð1�lÞrpi2

r3pi2

�lrpi3

r3pi3

!ð13Þ

The trajectories computed by using the CR3BP, N-body, and MEP models and the initial conditions of a typ-ical Lyapunov reference orbit around the L1 of the Earth–Moon CR3BP are plotted in Fig. 3. The Moon, the EMB,and the planets orbits in the simulation are assumed to beelliptic with fixed parameters obtained from the JPL

Ephemeris data on April 10, 2000 (Table 1). The trajectorywith CR3BP model is clearly closed. However, due to per-turbations in N-body and MEP models, the trajectory is nolonger closed without using any control. The results of theMEP simulation for one period of the reference orbit iscompared with that of the N-Body model and Fig. 3 shows

Table 1Specifications of major body masses and their orbits (for April 10, 2000).

l (km3/s2) a (km) e X (deg) i (deg) x (deg) h0 (deg)

Sun 132,712,440,018 – – – – – –Moon 4902.798 381,055.426 0.033544 119.9538 5.04246 314.4519 15.34489Mercury 22,032.09 57,909,001.7 0.205634 48.33183 7.005227 29.12456 208.8803Venus 324,858.63 108,208,151 0.006764 76.68055 3.394693 55.15912 209.1832Earth 398,600.44 149,646,179 0.016293 25.76081 0.001819 78.17091 96.50889EMB 149,598,120 0.016702 147.7248 0.000117 315.2441 97.46413Mars 42,828.3 227,952,576 0.09341 49.56443 1.849832 286.5379 81.97815Jupiter 126,686,511 778,594,604 0.048899 100.4872 1.304793 274.9365 29.91779Saturn 37,931,207.8 1,433,841,956 0.056201 113.6901 2.484398 336.1696 319.4688Uranus 5,793,966 2,875,034,755 0.044861 74.03684 0.772878 96.01097 147.464Neptune 6,835,107 4,499,107,778 0.011198 131.777 1.773145 270.6264 262.1344

t (day)

Perturbation(m/s^2)

240 2600

5E-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.00035

L1L2L3L4L5

Elliptical Effect

Figure 4. The perturbation effect of the eccentricity of the orbit of theMoon on all Earth–Moon Lagrange points.

t (day)

Perturbation(m/s^2)

230 240 250 260 270

1.5E-05

2E-05

2.5E-05

3E-05

3.5E-05

L1L2L3

Sun Effect

Figure 5. The perturbation acceleration due to the gravity field of the Sunon the Earth–Moon collinear Lagrange points.

M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079 2071

the accuracy of this model near the L1 of Earth–Moon sys-tem. Although the model can be verified in other condi-tions, the validity of the MEP model near the Lagrangepoints of the Earth–Moon system is sufficient for the caseof this study.

One of the advantages of the MEP model is that theeffect of the eccentricity of the orbit of the Moon andthe effect of the Sun and the other planets gravity fielddefined separately in the equations of motion. Figs. 4and 5 show the two main perturbations on the Earth–Moon CR3BP around the Lagrange points; the eccentric-ity of the orbit of the Moon and the gravity effect of theSun. In Fig. 6 the Venus and Jupiter perturbed accelera-tions on a Lyapunov orbit around L1 has been shown.The perturbations are evaluated for almost three years,considering the fact that the synodic period of the EMB

with other planets is less than 2.14 years (with Mars).Then, the graphs are plotted for the time period of thepeak values of each perturbation.

It can be seen from these figures that the largest planetseffect is of four orders of magnitude smaller than the eccen-tricity of the orbit of the Moon and the gravity effect of theSun.

The equations of motion of the spacecraft includingthree terms that you can be categorized is shown below:

€x ¼ a ¼�2 _hðg1 _x� _yÞ�2 _hðg1 _y þ _xÞ�2 _hg1 _z

264

375þF ðtÞ þ Xxðt; xÞ ð14Þ

in which a is the acceleration of the spacecraft and F ðtÞand X are defined as:

F ðtÞ ¼ r� 1

r3ð1� lÞ r12

r312

þ lr13

r313

� �

þX8

i¼1i–3

dpi

r3ð1� lÞ rpi2

r3pi2

þ lrpi3

r3pi3

!ð15Þ

t (day)

Pert

urba

tion(

m/s

^2)

250 300 350 400 450 5000

5E-10

1E-09

1.5E-09

2E-09

2.5E-09Venus EffectJupiter Effect

Figure 6. The largest perturbation acceleration of the planets on an orbitabout Earth–Moon L1.

2072 M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079

Xðt; xÞ ¼_h2

2ðg3x2 þ g3y2 � g2z2Þ

þ 1

r3

1� lrþ l

r0þ r� 1

RþX9

i¼1i–3

di

R0i

0@

1A ð16Þ

where r, r0, R and R0 are the spacecraft positions from theEarth, Moon, Sun and planets, respectively. For simplicity,D is defined as a generic spacecraft position from eachmass as follows:

D ¼ Dðt; qÞ ¼ ½D1 D2 D3 �T,r; r0;R or R0 ð17Þ

The position of the spacecraft from each major body is de-fined as:

r ¼ lrþ q ¼ ½ r1 r2 r3 �T

r0 ¼ q� ð1� lÞr ¼ ½ r01 r02 r03 �T

R ¼ Rþ q ¼ ½R1 R2 R3 �T

R0i ¼ Rþ q� R0i ¼ ½R

0i1 R0i2 R0i3 �

T

ð18Þ

The dimensionless masses and the spacecraft distancesfrom them are given in Table 2. The orbital elements ofthe planets around the Sun, which is used in this paperfor the simulations, are given in Table 1.

By defining the position vector of the spacecraft withx , q, the potential function can be written as:

Table 2Definition of the major body masses and position vectors of the spacecraftfrom them in non-dimensional system.

mj D

Earth 1 � l r

Moon l r0

Sun r� 1 R

Planet di R0i

Xðt; xÞ ¼_h2

2ðg3x2 þ g3y2 � g2z2Þ þ 1

r3

Xj

mj

Dð19Þ

2.6. Linearization of the equations

In optimal control approaches, a linear point massmodel of the spacecraft is needed. Thus, the Jacobian ofthe acceleration vector a (Eq. (14)) should be derived:

a _x ¼@a

@ _x¼ �2 _h

g1 �1 0

1 g1 0

0 0 g1

264

375 ð20aÞ

ax ¼@a

@x¼ Xxx ¼ _h2

g3 0 0

0 g3 0

0 0 �g2

264

375þ 1

r3

Xj

@2

@x2

mj

D

� �

ð20bÞ

where:

@2

@x2

mj

D

� �¼ � mj

D5

D2 � 3D21 �3D1D2 �3D1D3

�3D1D2 D2 � 3D22 �3D2D3

�3D1D3 �3D2D3 D2 � 3D23

264

375ð21Þ

Before the linearization of the equations on a referencetrajectory, the equations are re-written in the followingform:

_X ¼ f ðX ; tÞ ð22Þ

in which X is the state vector that consists of the positionand velocity of the spacecraft:

X ¼x

_x

� �¼

x

v

� �ð23Þ

Linearization on a reference trajectory leads to the follow-ing perturbed equation:

D _X ¼ AðX ; tÞDX ð24Þ

where DX is the deviation from the reference trajectoryðX rÞ:DX ¼ X � X rðtÞ ð25Þ

Finally, the coefficient matrix is obtained:

A ¼ @f

@X¼

03�3 I3�3

ax a _x

� �ð26Þ

2.7. Control action in the equations of motion

The previous sections showed the governing equationsof an uncontrolled spacecraft. In this work, two types ofmaneuvers are used; continuous and impulsive maneuvers.When using continuous maneuvers, the engines are opera-tive for all flight times. On the other hand, when using animpulsive control, the engines burn time is very small in

x

y

0.65 0.7 0.75 0.8 0.85 0.9 0.95-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Reference TrajectoryQ, H, R = IQ = 10IH = 10IR = 10I

L1

Figure 7. The controlled trajectories around Earth–Moon L1 orbit using

M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079 2073

comparison to the total mission flight time and can beregarded as impulsive maneuvers.

The main purpose of this study is to keep the spacecraftin a near closed orbit around the Lagrange points. To thisend, a closed reference orbit, like periodic orbits in CR3BPis an appropriate alternative. A typical Lyapunov orbitnear L1 and L2 is taken as a reference trajectory in thispaper, as the uncontrolled motion of the spacecraft withan initial condition on L1 orbit can be seen in Fig. 3. Nev-ertheless, this orbit is arbitrary and the proposed methodsare suitable for all periodic orbits in the CR3BP. For con-tinuous control action, the nonlinear equation of motion istransformed to:

_X ¼ f ðX ; tÞ þ Bu ð27Þ

where u is the control acceleration vector in the non-dimen-sional rotating coordinate system, and B is defined as:

B ¼03�3

I3�3

� �ð28Þ

LQR method.

t (day)

ControlMagnitude(m/s^2)

0 2 4 6 8 10 120

2E-05

4E-05

6E-05

8E-05

0.0001

0.00012R, H, Q = IQ = 10IH = 10IR = 10I

Figure 8. Time history of the control magnitude for station-keeping of theorbit around Earth–Moon L1 using LQR method.

3. Continuous control

In this section, the system governing Eq. (27) is utilizedand the control vector is supposed to be continuous. Twoapproaches are investigated; LQR and NP. The followingquadratic cost function is used (Kirk, 2004):

J ¼ 1

2DXT

f HDX f þ1

2

Z tf

0

ðuT ðtÞRuðtÞ

þ DXT ðtÞQDXðtÞÞdt ð29Þ

The weight matrices H, R, and Q are for final tracking er-ror, control effort and trajectory error penalty factors,respectively.

3.1. Linear quadratic regulator (LQR)

The solution of the LQR yields to find the control as afeedback of the states as shown below.

u� ¼ �R�1BT KDX ð30Þ

in which K is obtained from the Riccati differential equa-tion (Kirk, 2004):

_K ¼ �KA� AT K � Qþ KBR�1BT K; Kðtf Þ ¼ H ð31Þ

The K differential equations should be solved in back-ward time with final condition H. The A matrix is evalu-ated in time along the reference trajectory. The LQR

controlled trajectory of the spacecraft around L1 and thecontrol magnitude for different weight matrices are shownin Figs 7 and 8, respectively. It should be noted that the tra-jectories and the positions of the Sun and the other planetsare three-dimensional and out of the reference orbit. How-ever, the two-dimensional trajectory of the original orbit isof interest in this study and, therefore, only the planar tra-jectory of the spacecraft is plotted. Moreover, the graphs

are for a specific start time, which is January 1, 2000.The results can be a little different for different startingtimes, but it does not change the analysis and thearguments.

The integral of the control accelerations, Dv, as a mea-surement of the mass consumption and the integral oftracking error are computed in Table 3. All weight matricesare assumed to be diagonal matrices. As it can be seen, byincreasing the Q and H gains, the control effort isincreased, but the trajectory error is decreased. The Q

matrix is the penalty of the trajectory error and guaranteesthe closeness of the controlled trajectory to the referencetrajectory. Therefore, not only the large value of Q holdsthe spacecraft on the desired orbit during the trajectory,

Table 3Effect of weight matices on fuel consumption and trajectory error for station-keeping around Earth–Moon L1 using LQR method.

Fuel Consumption (m/s) Integral of the error along the trajectory (dimensionless)

Q, H, R = I 56.95 5.83E�04R, H = I, Q = 10I 72.74 1.88E�04R, H = I,Q = 100I 81.55 2.49E�05Q, R = I, H = 10I 74.76 3.82E�04Q, R = I, H = 100I 84.64 2.60E�04Q, H = I, R = 10I 32.93 1.10E�03Q, H = I, R = 100I 23.66 1.66E�03

2074 M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079

but it is also useful for keeping the spacecraft on its desiredfinal conditions. Nevertheless, if the accuracy in the endingcondition is needed, the usage of both gains is recom-mended. Unlike these two gains, when R increases, the con-trol effort decreases and the trajectory error increases.

Figs. 9 and 10 show the trajectory and the control effortfor different gains about L2 orbit, respectively. The numer-ical results are shown in Table 4. As it can be seen, theresults are similar to L1. However, due to the larger pertur-bations effects near the L2 point (see Figs. 4 and 5), themagnitudes of fuel consumption and trajectory error islarger.

3.2. Nonlinear programming

A NP approach for keeping the spacecraft in its orbitaround the collinear libration points is presented in thissection. To this end, the control acceleration is discretizedin time and the optimum value of them is computed usinga zero initial guess and BFGS (named for Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton gradient-basedoptimization method. The control action in between thediscretized time intervals are computed using a cubic inter-polation algorithm. The quadratic interpolation method isalso used for line search in the BFGS.

x

y

1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32 1.34

-0.1

-0.05

0

0.05

0.1 Reference TrajectoryR, Q, H = IQ = 10IH = 10IR = 10I

L2

Figure 9. The controlled trajectories around Earth–Moon L2 orbit usingLQR method.

The BFGS needs the gradient of the cost function.Based on calculus of variation, the Hamiltonian is definedas (Kirk, 2004):

H ¼ 1

2ðuT Ruþ ðX � X rÞT QðX � X rÞÞ þ kT ðf þ BuÞ ð32Þ

and the states and co-states equations are:

_X ¼ @H@k¼ f þ Bu

_k ¼ � @H@X¼ �ðQðX � X rÞ þ AT kÞ

(ð33Þ

with the boundary conditions:

Xð0Þ ¼ X rð0Þ; kðtf Þ ¼ HDX f ¼ HðXðtf Þ � X rðtf ÞÞ ð34Þ

By solving Eq. (33) with boundary conditions (34), theHamiltonian and its gradient as a function of time can becomputed. The gradient of the cost function (29) withrespect to control actions is computed using the followingapproximation (Agrawal and Fabien, 1999):

@J@uðtÞ ¼

@H

@uðtÞ � Dt ð35Þ

in which the Dt is the discretization time step.The results of the optimal station-keeping on a reference

trajectory around L1 using NP is shown in Fig. 11. The

t (day)

ControlMagnitude(m/s^2)

0 2 4 6 8 10 12 140

2E-05

4E-05

6E-05

8E-05

0.0001

0.00012R, H, Q = IQ = 10IH = 10IR = 10I

Figure 10. Time history of the control magnitude for station-keeping onorbit around Earth–Moon L2 using LQR method.

Table 4Effect of weight matices on fuel consumption and trajectory error for station-keeping around Earth–Moon L2 using the LQR method.

DVtotal (m/s) Integral of trajectory error (dimensionless)

Q, H, R = I 79.31 6.64E�04R, H = I, Q = 10I 89.42 1.62E�04R, H = I,Q = 100I 95.05 2.29E�05Q, R = I, H = 10I 93.62 3.31E�04Q, R = I, H = 100I 106.59 1.89E�04Q, H = I, R = 10I 66.95 1.85E�03Q, H = I, R = 100I 57.16 5.42E�03

x

y

0.78 0.8 0.820.840.860.88 0.9 0.920.940.960.98 1 1.021.041.06

-0.1

-0.05

0

0.05

0.1

Reference TrajectoryNonlinear Programming

Q = 100IR = IH = I

L1

Figure 11. The controlled trajectory around Earth–Moon L1 orbit usingNP method.

t (day)

ControlMagnitude(m/s^2)

0 2 4 6 8 10 120

2E-05

4E-05

6E-05

8E-05

0.0001

0.00012Q = 100IR = IH = I

Figure 12. Time history of the control magnitude for station-keeping onorbit around Earth–Moon L1 using NP method.

M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079 2075

magnitude of the control time history is also plotted inFig. 12. Total impulse used for this trajectory is 81.52 m/s and the dimensionless integral of error from the referencetrajectory is 2.333 � 10�5. It can be seen that the NP canfind better tracking solution than LQR. The reason is obvi-ously that the system is highly nonlinear.

4. Discrete control

When the spacecraft uses impulsive maneuvers, it isassumed that the engines are only active for specific timeintervals which results in velocity change. Thus, the equa-tions of motion of the spacecraft without control (Eq.(14), (22)) is used in this section. Three methods are intro-duced for discrete control.

Figure 13. Schematic of the simple shooting method.

4.1. Multiple shooting method

A simple way to have closed orbits around Lagrangepoints is to select some points on the reference trajectoryand force the spacecraft to pass by them. According toFig. 13, using the simple shooting method between twopoints, the velocity v0 should be found in each point x0

such that the spacecraft pass by the next point x1. In each

point, the impulse (Dv) is the difference between the initialvelocity in the next trajectory and final velocity in the pre-vious trajectory. In this study, the reference orbit period isequally divided by the number of reference points and amultiple shooting (MS) method is used.

Because the shooting method guarantees that the trajec-tory is continuous and closed, only the total velocitychange is used as the cost function for comparison of theresults with the continuous control:

J ¼ DV total ¼Xn

i¼1

jDvij ð36Þ

Figs. 14 and 15 show the trajectories that result for 4 ref-erence points on the original orbit around the Earth–MoonL1 and L2, respectively. The trajectory between each twopoints is continuous and smooth (with continuous differen-tial). However, there is a corner at each point between two

x

y

0.8 0.85 0.9 0.95 1 1.05 1.1

-0.1

-0.05

0

0.05

0.1 MS Method (4 Points)Reference Trajectory

L1

Figure 14. The controlled trajectory around Earth–Moon L1 orbit usingMS method.

x

y

1.08 1.1 1.121.141.161.18 1.2 1.221.241.261.28 1.3 1.321.34

-0.1

-0.05

0

0.05

0.1 MS Method (4 Points)Reference Trajectory

L2

Figure 15. The controlled trajectory around Earth–Moon L2 orbit usingMS method.

2076 M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079

trajectories, which is due to the impulsive maneuvers Dvi.Tables 5 and 6 present the results for different numbersof reference points. As it can be seen from these tables,increasing the number of reference points, which meansthat the spacecraft trajectory is close to the reference trajec-tory, results in more fuel consumption. This is similar toLQR with high Q gains. The maximum value of Dv for thismethod approaches 83.36 m/s and 89.86 m/s for L1 and L2,

Table 5Effect of the number of reference points on the total impulse of station-keepin

Number of reference points n = 4 n = 5Total impulse (m/s) 80.32 81.15

respectively, which is still better than LQR in term of fuelconsumption. It can be compared with the caseQ = R = I, H = 100I in Tables 3 and 4.

4.2. Optimized multiple shooting method

The problem with the MS method is that the referencepoints are fixed and may force the spacecraft to stay onthem with high cost of fuel consumption. Thus, in this sec-tion, the reference points are free in such a way to minimizethe total impulse of Eq. (36). In this method, called opti-mized MS (OMS) method, the locations of the referencepoints are the optimization variables:

Xopt ¼

x0

x1

..

.

xn�1

266664

377775 ð37Þ

It should be noted that the reference points are three-dimensional vectors. For each set of reference points, ashooting method (discussed in the previous section) is ap-plied to find the required velocity change in each point.The last trajectory segment starts from xn�1 and ends atxn ¼ x0.

A BFGS gradient-based method and quadratic interpo-lation line-search approach is used to find the optimal solu-tion. The gradient of the cost function is computed usingnumerical finite-difference methods.

The planar trajectory of the spacecraft using the OMS

method with 4 impulses is shown in Figs. 16 and 17 for tra-jectories about L1 and L2, respectively. The trajectories areshifted in such a way that the required DV total is much lessthan the one required by the MS method that keeps thespacecraft on the CR3BP reference trajectory. The fourimpulses of the trajectories shown in Figs. 16 and 17 arepresented in Tables 7 and 8, respectively. As it can be seen,the total impulse is less than 1 m/s for both cases, which ismuch smaller than other methods. The main disadvantageof this approach is its long running time, and also the factthat sometimes it converges to a local minimum.

It should be noted that the shift in the orbits are due tothe position of the Sun and the planet at the starting timeand results may be different for other start times. As it canbe seen in the figures, the controlled trajectories are closedbut they are not smooth. The velocity of the spacecraft isdifferent for the initial and final times, which results innon-repetitive orbits. This is because they should be com-pared with the previous results which are for one orbitand the trajectory is not repeated. Obviously, a smoothnesscriterion should be included in the cost function for longperiod (more than one period) station-keeping.

g around Earth–Moon L1 using the MS method.

n = 10 n = 100 n = 20082.25 83.25 83.36

Table 6Effect of the number of reference points on the total impulse of station-keeping around Earth–Moon L2 using the MS method.

Number of reference points n = 4 n = 5 n = 10 n = 100 n = 200Total impulse (m/s) 78.25 82.10 84.75 89.55 89.86

x

y

0.8 0.820.840.860.88 0.9 0.920.940.960.98 1 1.021.04

-0.1

-0.05

0

0.05

0.1

Reference TrajectoryOMS Method (4 points)

L1Delta_V = 0.55 m/s

Figure 16. The controlled trajectory around Earth–Moon L1 orbit usingOMS method.

x

y

1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32

-0.1

-0.05

0

0.05

0.1 Reference TrajectoryOMS method (4 points)

L2

Delta_V = 0.4 m/s

Figure 17. The controlled trajectory around Earth–Moon L2 orbit usingOMS method.

Table 7Station-keeping impulse maneuvers for station-keeping around Earth–Moon L1 using optimization of MS method.

Dv1 Dv2 Dv3 Dv4

Dvx 8.73E�03 1.17E�01 8.05E�03 �9.71E�03Dvy �5.15E�03 6.93E�02 �3.86E�02 �9.80E�02Dvz 7.76E�03 3.32E�01 5.33E�04 �1.23E�01

Table 8Station-keeping impulse maneuvers for station-keeping around Earth–Moon L2 using optimization of MS method.

Dv1 Dv2 Dv3 Dv4

Dvx �1.28E�03 �7.13E�02 1.23E�01 �9.43E�02Dvy �2.53E�03 5.18E�02 �4.91E�02 �4.35E�02Dvz �9.41E�03 1.67E�02 5.27E�02 1.18E�01

M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079 2077

4.3. Multiple impulses optimization

The final formulation that has been used for the station-keeping on the orbits around L1 and L2 of the Earth–Moon

is parameterized based on the impulse vectors, called multi-ple impulses optimization (MIO) method. In this method,the total flight time is divided into some specified intervals.An impulsive maneuver is performed between each inter-val. The structure of this method is similar to that of theprevious discrete controls. However, the trajectory hasbeen controlled through its initial position and velocity aswell as some impulsive maneuvers. Thus, the optimizationvariables are:

Xopt ¼

x0

v0

Dv1

..

.

Dvn�1

26666666664

37777777775

ð38Þ

In this method, the spacecraft starts with initial positionx0 and velocity v0 and then, after each time interval animpulse vector equal to Dvi is applied to the spacecraft.To enforce the trajectory to be closed, the final positionerror with respect to the initial position ef is taken intoconsideration in the cost function:

J ¼Xn�1

i¼1

ðjDvijÞ þ1

2eT

f Hef ð39Þ

The first term is the total impulse or, equivalently, thefuel consumption measurement, and the second term isthe final error penalty in the quadratic form.

Figs. 18 and 19 show the trajectories found by thismethod about L1 and L2, respectively. The initial condi-tions for these solutions are computed as:

x

y

0.8 0.820.840.860.88 0.9 0.920.940.960.98 1 1.021.04

-0.1

-0.05

0

0.05

0.1

Reference TrajectoryMultiple Impulse Optimization

L1

Delta_V = 0.4 m/s

Figure 18. The controlled trajectory around Earth–Moon L1 orbit usingMIO method.

x

y

1.08 1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 1.32

-0.1

-0.05

0

0.05

0.1 Reference TrajectoryMIO method (4 points)

L2

Delta_V = 0

Figure 19. The controlled trajectory around Earth–Moon L2 orbit usingMIO method.

Table 9Station-keeping impulse maneuvers for station-keeping around Earth–Moon L1 using MIO method.

Dv1 Dv2 Dv3 Dv4

Dvx 1.91E�05 2.35E�02 1.35E�04 �1.15E�04Dvy 1.03E�05 1.62E�02 1.63E�05 �1.25E�05Dvz �2.42E�06 �3.95E�01 �5.56E�04 6.98E�03

2078 M. Ghorbani, N. Assadian / Advances in Space Research 52 (2013) 2067–2079

x0jL1¼

0:801093

0:00664

�0:000098

0B@

1CA; v0jL1

¼�0:01102

0:288221

�0:00092

0B@

1CA

x0jL2¼

1:101615

�0:00428

0:008271

0B@

1CA; v0jL2

¼0:00425

0:24151

�0:01423

0B@

1CA

The required impulses of the trajectory around L1 arepresented in Table 9, which are much smaller than thatof OMS method (Table 7). No impulse is required forL2. This results show that, using this method, a closed orbitaround L2 is found that does not require station-keepingmaneuvers for orbits around L2 of the Earth–Moon sys-tem. The station-keeping maneuvers around L1 are alsosmall enough that makes it almost possible to revolve itwith no propulsion system.

5. Concluding remarks

The optimal station-keeping of spacecraft in orbitsaround the Earth–Moon collinear libration points in thepresence of gravitational perturbations has been investi-gated. In this regard, a general R5BP model has been intro-duced, which is suitable for studies near the Lagrangepoints of the original CR3BP. The continuous control ofthe spacecraft has been implemented through the LQR

and NP and it has been revealed that the NP approach ismore powerful than LQR for the station-keeping. Themajor disadvantage of NP is its more required computa-tion time, which is not very important in the real applica-tions considering the fact that the mission time spans arehighly long.

It has been shown that the discontinuous controlsthrough the impulsive maneuvers are, to a great extent, bet-ter in the sense of the required total impulse. The MS

method has been applied in such a manner that it guaran-tees that the spacecraft stays on the reference trajectorywith less fuel needed than the continuous control. How-ever, when the reference points are not restricted to the ori-ginal CR3BP reference orbit, the promising results withvery small required impulse have been obtained. It hasbeen proved in this study that the OMS method and theMIO approach can find fuel efficient trajectories aroundL1 and L2 of the Earth–Moon system by letting the refer-ence orbit move to a new orbit that the required totalimpulse for station-keeping is much smaller (almost zero)than the original orbit. If the multiple revolutions aroundthe Lagrange points are used, these approaches can beemployed for finding the quasi-periodic closed orbitsaround the L1 and L2 with no fuel consumption in the pres-ence of gravitational perturbations.

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