Research ArticleLoosely Formation-Displaced Geostationary OrbitOptimization with Complex Hybrid Sail Propulsion

Yuan Liu,1 Yong Hao,1 and Weiwei Liu 2

1College of Automation, Harbin Engineering University, Harbin 150001, China2School of Economics and Management, Harbin Engineering University, Harbin 150001, China

Correspondence should be addressed to Weiwei Liu; heulww@163.com

Received 23 March 2018; Revised 16 April 2018; Accepted 8 May 2018; Published 6 June 2018

Academic Editor: Yimin Zhou

Copyright © 2018 Yuan Liu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To explore the performance of hybrid sail and overcome the congestion of geostationary orbit, this work proposes a methodintended to optimize the trajectories of the spacecraft formation and extend the concept of displaced geostationary orbit byloosening the relative distance and introducing a station-keeping box. The multispacecraft formation is a typical complex systemwith nonlinear dynamics, and the hybrid propulsion system introduces additional complexity. To solve this problem,suboptimal trajectories with constant relative distance constraints are first found with inverse methods, which were referred toas ideal displaced geostationary orbits. Then, the suboptimal trajectories are used as a first guess for a direct optimizationalgorithm based on Gauss pseudospectral algorithm, which loosens the relative distance constraints and allows the spacecraft tobe placed anywhere inside the station-keeping box. The optimization results show that the loosely formation and station-keeping box can create more flexible trajectories and achieve higher efficiency of the hybrid sail propulsion system, which cansave about 40% propellant consumption.

1. Introduction

The geostationary orbit (GEO) is a circular, equatorial orbitwhose period equals the Earth’s rotational period, enablingthe use of applications, such as telecommunications andgeoscience monitoring. The GEO allows a satellite to be sta-tionary above a certain point on the Earth’s equator. Forthis reason, it is a unique and currently congested orbit,especially at longitudes above densely populated areas [1].To ease the congestion of the GEO slot, the concept of replac-ing the GEO with a non-Keplerian orbit (NKO) has beenproposed [2].

The NKOs are spacecraft trajectories which can beachieved by a continuous control acceleration, and McKayet al. gave comprehensive introduction about NKOs in a sur-vey [3]. However, continuous propulsion is necessary forNKO maintenance, and it can be provided either by a solarelectric propulsion (SEP) thruster [4] or by a hybrid of SEPand solar sailing [5].

Solar sailing demands many high-technology require-ments on materials, control, and structures [6]. Recently,several solar sail demonstrator missions have been achievedwith Japan Aerospace Exploration Agency (JAXA), the Inter-planetary Kite-Craft Accelerated by Radiation of the Sun(IKAROS), National Aeronautics and Space Administration(NASA), NanoSail-D2, and the Planetary Society, LightSail1 [7–9]. For NKOs, there have been some solid conclusionsthat adding the propulsion-saving propellant such as solarsails is helpful to reduce the propellant consumption, whichensures a longer mission lifetime [5].

Previous works proposed that there be possibility ofcombining a relatively small sail and SEP, which will bemore feasible than equipping a big sail for a near-term space-craft [5, 10–12]. Guided by this concept, several researcheshave been done to enhance the overall performance of thehybrid propulsion system, especially to minimize the long-term propellant consumption for Earth pole-sitter [5] andother planets’ pole-sitter [13], transferring from Earth to

HindawiComplexityVolume 2018, Article ID 3576187, 9 pageshttps://doi.org/10.1155/2018/3576187

other planets [14, 15] and from north/south to south/northpole-sitter orbits [11]. According to those studies, the hybridsail is an effective solution for the long-term NKO mainte-nance, which has better performance on propellant con-sumption than pure SEP and less complexity than a large sail.

The displaced GEO through solar radiation pressure wasfirst shown to be feasible by Baig and McInnes [16]. Afterthat, Heiligers et al. [1, 2] have demonstrated that the hybridsail-displaced GEO is more feasible with strict position con-straints. Then, Arnot et al. proposed that new families of rel-ative orbits for spacecraft formation flight will be generatedby applying continuous thrust with minimal intervention inthe dynamics of the problem [17].

However, we believe that loosening the formation rela-tive distance constraints is helpful to release the potentialof hybrid sail. In this paper, we extend the previous worksby keeping several spacecrafts in an assigned station-keeping box with loosely formation. The concept of astation-keeping box is found to have the drawback thatthe spacecraft does not always have constant relative posi-tion with respect to an Earth observer; it is not truly “geo-stationary”. Yet, if the station-keeping box is small enough,the negative effects of pointing problems can be restrictedto an acceptable level so that this displacement will not beperceivable from the Earth.

The structure of this paper is organized as the following.In Section 2, the system dynamics and the satellite formationmodel are described. In Section 3, mission profiles for space-craft strict formation on displaced GEOs are presented andan inverse method to minimize the SEP thrust and obtainthe corresponding solar sail control is introduced. In Section4, the optimization method used to discover the trajectoriesof loosely formation spacecraft in a station-keeping box tominimize the propellant mass is described.

2. Equations of Motion

The displaced GEO is an NKO whose period equals that ofthe Earth’s rotation [1]. Compared with the distance fromthe Sun, a spacecraft in a displaced GEO is much closer tothe Earth. Therefore, the dynamics are defined as two-body,Earth-centered dynamics, neglecting perturbations from thehigher order harmonics and from the Earth’s potential, theMoon, Sun, and so on. Figure 1 shows a rotating referenceframe A X A , Y A , Z A , which is centered at the Earth cen-ter, with the X A axis pointing towards the slot of an idealGEO and the Z A axis aligning with the angular momentum

vector of the Earth and perpendicular to the equatorial plane.Then, the Y A axis completes the right-handed Cartesian ref-erence frame.

The basic idea of displaced GEO spacecraft is that itsrelative movement with an Earth surface observer can beneglected and its period exactly equals that of the idealGEO. The equations that describe the dynamics of thespacecraft in the frame (A) are

x A = 2ωeyA + ωe

2x A −μex

A

r3+ a A

X ,

y A = −2ωexA + ωe

2y A −μey

A

r3+ a A

Y ,

z A = −μez

A

r3+ a A

Z

1

The spacecraft position vector is r A = x A , y A , z A T,

r = r A , ωe is the Earth’s constant angular velocity, andμe is the gravitational parameter of the Earth. For a hybridsail propulsion system, the thrust-induced acceleration

a A = a AX , a A

Y , a AZ

Tcan be written as

a A = a ASEP + a A

S , 2

where a ASEP is the acceleration of the SEP thruster and a A

S isthe acceleration of the solar sail.

a ASEP =

f ASEPm

,

a AS = β0

m0m

μAs

s2n A ⋅ s A 2

n A ,

3

in which f ASEP = f A

X  f AY  f A

ZTis the SEP thrust vector,

f SEP = f ASEP , and m is the spacecraft mass. Because of the

consumption of SEP propellant, m can be profiled as thefollowing:

m = −f SEPISP

g0, 4

in which ISP is the specific impulse (we adopt the value ofISP = 3200s, which can be obtained with current engine tech-nology [12]) and g0 is the standard Earth surface gravityacceleration.

Furthermore, in this paper, a perfect solar sail forcemodel is used to account for specular reflection, where μs isthe gravitational parameter of the Sun, n is the vector normalto the sail surface, m0 is the initial mass at time t = 0, s is theSun-sail vector, and s is the unit vector along s. In our work,s is approximated by a constant Sun-Earth distance of one

astronomical unit (AU). The parameter β0 indicates thelightness number, the value of β0 = 0 is plausible for a near-term mission, and a value of β0 = 0 1 for a long-term mission[18]. The unit vectors n and s can be expressed in the easiestway in an auxiliary frame B X B , Y B , Z B (see Figure 2).The X B axis coincides with the Sun-sail vector (neglecting

X(A)

Y(A)

Z(A)

E

GEO

Displaced GEO

z(A)

y(A)

x(A)

O(A)

�휔

Figure 1: Displaced GEO in the rotating reference frame (A).

2 Complexity

the tiny variation in the direction of the Sun-sail vector overone orbit, assuming it to be aligned with the Sun-Earth vec-tor), the Z B axis is perpendicular to the X B axis and liesin the plane spanning the Sun-sail vector and the Earth’srotation axis, and the Y B axis completes the right-handedreference frame. Using the pitch angle α and yaw angle θ asgiven in Figure 2, n can be expressed as

n B = sin α cos θ  sin α sin θ  cos α T 5

Because n B should always point away from the Sun, inthe frame (B), the following attitude constraints apply:

0 ≤ α ≤ π,

−π

2≤ θ ≤

π

26

Using ψ to describe the angle between s and the equa-torial plane, which reaches its maximum value (equals theEarth’s obliquity to the ecliptic, δ) at winter solstice andreaches its minimum value −δ at summer solstice, s B canbe expressed as

s B = cos ψ 0  sin ψ T 7

The evolution of the displaced spacecraft state X A =r A r A m

Tcan be expressed in the differential form:

X A =r A

r A

m

8

We assume that there is one leader and n n ≥ 1 followersin each formation. In this paper, we consider a test case of aformation with one leader and two followers, and the detailsare defined and shown in Table 1.

3. Strict Formation-DisplacedGeostationary Orbits

This section deals with the strict formation-displaced GEOanalyses by assuming that the problem is approximated as aformation with each spacecraft in a constant flat non-Keplerian-displaced GEO. The assumption that trajectoriesand the corresponding controls are used as the first guessfor the optimization algorithm will be presented in Section 4.

3.1. Approach. For a strict spacecraft formation, the relativepositions between each two agents are always constant (seeFigure 3). If one formation is placed on an ideal displacedGEO (which keeps still for the observers on Earth), each

spacecraft will be a fixed point in the frame (A). r AL is the

position of the leader, meanwhile, Δr AF,i is the relative posi-

tion of ith follower spacecraft with the leader, and r AL =

r AL = Δr A

F,i = Δr AF,i = 0. Once the trajectory of every space-

craft is known, the inverse method can be used to finduniquely required accelerations from (8).

For a hybrid propulsion system (both pure SEP, β0 = 0,and pure sail, f SEP = 0, are special cases), the optimal controlsof spacecraft require that the SEP consumes as little propel-lant as possible, which can be achieved by the following steps:

(1) Divide one day (Earth’s rotation period) into severalequal instants.

(2) Use MATLAB’s genetic algorithm toolbox to find theoptimal control of solar sail (including the pitch angle

α and yaw angle θ) and to minimize the a ASEP at each

instant.

(3) Assume that the a ASEP remains constant between two

adjacent instants and update the m according to theintegration of (4).

Plane spanning the sun-sail vectorand the earth’s rotation axis plane

X(B)

Z(B)

n(B)

Sail surfaceS(B)

Y(B)

X(B)

Z(B)

n(B)

�훼

�휃

Figure 2: Frame (B) with sail normal vector n, pitch angle α, andyaw angle θ.

Table 1: Spacecraft parameters.

Parameters Leader Follower 1 Follower 2

Initial mass (kg) 1000 900 800

Initial propellant mass (kg) 700 500 550

ISP (s) 3200 3200 3200

β0 0.2 0.1 0.1

X(A)

Y(A)

Z(A)

E

GEOO(A)Leader

Follower

Strict formation

rL

ΔrF,i , i = 1, 2, ..., n

�휔

(A)

(A)

Figure 3: Strict formation in the rotating reference frame (A).

3Complexity

(4) Repeat steps 1 to 3 until the orbit is complete atthe time t = t f , where t f equals one Earth’s rotationperiod.

3.2. Results. In the test, the spacecraft’s positions are

defined as r AL = 40 km + rGEO0 km0 km T , Δr A

F,1 = 0 km5 km 5 km T , Δr A

F,2 = 0 km −5 km 5 km T , where r AL

= r AF,i + Δr A

F,i i = 1, 2,… , n . In the pure SEP case, theSEP provides the whole acceleration for orbital mainte-nance. In a relative short period like one day, ignoringthe impact of orbit perturbation, the accelerations pro-vided by SEP are constant, which lies at a separation of6.375× 10−4m/s2 for leader and 6.381× 10−4m/s2 for bothfollowers 1 and 2. As to the hybrid sail case, because theSun-Earth direction changes with the period of one year,

the accelerations of SEP and solar sail are no longer con-stant and show symmetrical characteristic on differentdays with different Sun-Earth directions (see Figures 4and 5). Obviously, with the help of the solar sail, theSEP provided less acceleration than the pure SEP case atmost of the time. This means that a part of propellant issaved, and the spacecraft formation will be maintainedfor a longer time.

Figure 6 shows the mass of the simulation result. Tomake the comparison clear, we assume that the spacecraft’sparameters are as described in Table 1. In the pure SEP case,the mass of the spacecraft decreased at a constant rate. Afterone-day formation maintenance, the leader’s mass dropsfrom 1000 kg to 998.252 kg, the follower 1’s mass drops from900 kg to 898.425 kg, and the follower 2’s mass drops from800 kg to 798.600 kg. For the hybrid sail case, compared with

0

2

4

6

8

SEP

(m/s

2 )

0 5 10 15 20 25 Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

× 10−4

(a)

0 5 10 15 20 25Time (h)

0

2

4

6

8

SEP

(m/s

2 )Winter solstice

Springequinox

Summersolstice

Autumnequinox

× 10−4

(b)

0 5 10 15 20 25Time (h)

0

2

4

6

8

SEP

(m/s

2 )

Winter solstice

Springequinox

Summersolstice

Autumnequinox

× 10−4

(c)

Figure 4: Strict formation on ideal displaced GEO for the hybrid sail case: acceleration provided by SEP (a) leader, (b) follower 1, and (c)follower 2 in the frame (A).

0

2

4

6

8 × 10−4

Sola

r sai

l (m

/s2 )

0 5 10 15 20 25 Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(a)

× 10−4

0 5 10 15 20 25Time (h)

0

2

4

6

8

Sola

r sai

l (m

/s2 )

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(b)

× 10−4

0 5 10 15 20 25Time (h)

0

2

4

6

8

Sola

r sai

l (m

/s2 )

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(c)

Figure 5: Strict formation on ideal displaced GEO for the hybrid sail case: acceleration provided by solar sail (a) leader, (b) follower 1, and (c)follower 2 in the frame (A).

4 Complexity

the pure SEP case, significant mass saving is achieved, whichproves the effectiveness of hybrid sail on the propellant sav-ing. In the next section, we try to get more mass saving byintroducing a station-keeping box and loosening the relativedistance constraints between each two spacecrafts.

4. Optimal Loosely Formation-DisplacedGeostationary Orbits

For a loosely spacecraft formation, the relative distancesbetween any two agents are variable and are limited to a min-imum value of dmin (to avoid collision dmin > 0) and to a

maximum value of dmax. The value of dmax − dmin is definedas the loosely distance. In addition, to ensure the loosely for-mation to be nearly geostationary (not truly geostationary)for an Earth observer, the concept of a station-keeping boxplaced around a real GEO is introduced and all the space-crafts are required to be in the station-keeping box all thetime (see Figure 7). For a loosely spacecraft formation, withloosely relative distance and the station-keeping box, it ispossible for the spacecraft to enable propellant optimal tra-jectories to save more mass. However, because the trajecto-ries of spacecraft are functions of time and the expressionof them is unknown, the optimal trajectories cannot be sim-ply obtained with the inverse method. Hence, a method isused to find both optimal trajectories and controls, whichminimizes the whole formation’s propellant usage and allowsthe formation to be kept in a displaced GEO.

4.1. Approach. The problem is to find optimal periodic orbitswith minimum propellant consumption. Therefore, the costfunction can be simply defined as

J = − mL t f + 〠n

i=1mF,i t f , 9

998.0

998.5

999.0

999.5

1000.0 M

ass (

kg)

0 5 10 15 20 25Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(a)

898.0

898.5

899.0

899.5

900.0

Mas

s (kg

)

0 5 10 15 20 25Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(b)

798.5

799.0

799.5

800.0

Mas

s (kg

)

0 5 10 15 20 25Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(c)

998.5

999.0

999.5

1000.0

Mas

s (kg

)

0 5 10 15 20 25Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(d)

898.5

899.0

899.5

900.0

Mas

s (kg

)

0 5 10 15 20 25Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(e)

799.0

799.2

799.4

799.6

799.8

800.0

Mas

s (kg

)

0 5 10 15 20 25Time (h)

Winter solstice

Springequinox

Summersolstice

Autumnequinox

(f)

Figure 6: Strict formation on ideal displaced GEO: mass of pure SEP (a) leader, (b) follower 1, and (c) follower 2 and mass of hybrid sail (d)leader, (e) follower 1, and (f) follower 2 in the frame (A).

Y(A)

X(A)

Z(A)�휔E

GEOO(A)LeaderFollower

Loosely formation

Station-keeping box

Figure 7: Loosely formation in the rotating reference frame (A).

5Complexity

that is, to maximize the whole final mass of all spacecraftsafter one period (for displaced GOE t f equals one day).In the frame (A), the leader spacecraft’s dynamics can bedescribed as (1), the followers’ relative dynamics (with the

leader) Δr AF,i = Δx A

F,i , ΔyAF,i , Δz

AF,i

Ti = 1, 2, 3,… , n can

be written as

Δx AF,i = x A

L − x AF,i = 2ωe y A

L − y AF,i

+ ωe2 x A

L − x AF,i − μe

x ALr3L

−x AF,ir3F,i

+ a AX,L − a A

X,F,i,

Δy AF,i = y A

L − y AF,i = −2ωe x A

L − x AF,i + ωe

2 y AL − y A

F,i

− μey ALr3L

−y AF,ir3F,i

+ a AY,L − a A

Y,F,i,

Δz AF,i = z A

L − z AF,i = −μe

z ALr3L

−z AF,ir3F,i

+ a AZ,L − a A

Z,F,i

10

When the spacecrafts in it are close to each other, theformation is rL ≈ rF,i. Therefore, (9) can be simplified as

Δx AF,i = 2ωeΔy

AF,i + ωe

2Δx AF,i − μe

Δx AF,ir3L

+ Δa AX,F,i,

Δy AF,i = −2ωeΔx

AF,i + ωe

2Δy AF,i − μe

Δy Ai

r3L+ Δa A

Y,F,i,

Δz AF,i = −μe

Δz AF,ir3L

+ Δa AZ,F,i

11

By introducing the follower’s relative state vector

ΔX AF,i =

Δr AF,i

Δr AF,i

mF,i

, 12

the state of the formation can be written as X A = X AL ,

ΔX AF,1 ,… , ΔX A

F,n .The center of the station-keeping box is located at an

ideal displaced GEO, which is a fixed point in the frame

(A). Its position is defined as r ASKB = rGEO + dX, dY, dZ

T ,where the constants dX, dY, and dZ are position correc-tion values and the rGEO is the GEO orbit radius. Toallow the spacecraft to move freely within the box, thestation-keeping box is defined as K = x, y, z ∣ x ∈ rGEO +dX − ρX, rGEO + dX + ρX , y ∈ dY − ρY, dY + ρY , z ∈ dZ − ρZ,dZ + ρZ ρX, ρY, ρZ ≥ 0 , where ρX, ρY, and ρZ are relax-ing parameters.

The problem is solved with a numerical direct pseudos-pectral method implemented in the software tool PSOPT[19], which makes use of the automatic differentiation by

overloading in C++ (ADOL-C) library for the automatic dif-ferentiation of the objective, dynamics, constraint functionsand the initial guess for the problem. We use the optimal tra-jectories and controls of strict formation-displaced GEO(results of Section 3) as the suboptimal initial guess forPSOPT, which allows the Gauss-pseudospectral algorithmto converge quickly and smoothly.

4.2. Results. In the test, the spacecraft’s parameters are listedin Table 1 and the station-keeping box is defined in Table 2.According to Table 2, dmin = 1 km, dmax = 10 km, and theloosely degree is 9 km. The efficiency of solar sail is greatlyaffected by the sunlight direction which is the same as thatof the Sun sail. Ignoring the minor differences betweenSun-sail vector and the Sun-Earth vector in the simulation,we use the Sun-Earth vector rather than the Sun-sail vectorto describe the sunlight direction. Considering the universalsituations, we select the Sun-Earth vectors of winter solstice,spring equinox, summer solstice, and autumn equinox as thetypical values. Unlike the strict formation, with the optimaltrajectories of one orbital period (one day, see Figure 8), therelative distances between each two agents are not constant,which is always larger than dmin and smaller than dmax.Furthermore, the trajectories of the leader, follower 1, andfollower 2 in a day are shown in Figure 9. It is obvious thatthe positions of all spacecrafts are restricted in the station-keeping box at any time. Because of the symmetry character-istic of the Sun-sail direction (see Figure 9), the shapes ofspacecraft trajectories at winter solstice and summer solsticeare symmetric (the same as the shapes at spring equinox andautumn equinox).

The mass consumption of each spacecraft is listed inTable 3, and the overall mass consumption (according to(9)) is listed in Table 4. With the station-keeping boxand loosely formation, the mass consumption is greatlyreduced both in the whole formation and each spacecraft.In the whole formation, the 2.805 kg, 3.022 kg, 2.808 kg, and3.021 kg propellants are needed to maintain the strict forma-tion at the ideal displaced GEOs for the winter solstice, springequinox, summer solstice, and autumn equinox, respectively.By loosening the relative distance constraints and applyingthe station-keeping box, propellant mass is saved corre-sponding to the four seasons, 39.89% for winter solstice,42.19% for spring equinox, 41.67% for summer solstice, and42.04% for autumn equinox.

5. Conclusions

In this paper, with a hybrid (solar sail and solar electricpropulsion, SEP) propulsion system, by inviting two con-cepts, the loosely relative distance constraints and thestation-keeping box, a method to find the optimal trajectoriesof spacecraft formation for displaced geostationary orbits

Table 2: Station-keeping box and relative distance parameters.

dmin(km)

dmax(km)

dX(km)

dY(km)

dZ(km)

ρX(km)

ρY(km)

ρZ(km)

1 10 40 0 0 10 10 10

6 Complexity

02468

10

Dist

ance

(km

)

0 5 10 15 20 25Time (h)

Leader-follower 1Leader-follower 2Follower 2-follower 3

UpperboundaryLowerboundary

(a)

02468

10

Dist

ance

(km

)

0 5 10 15 20 25Time (h)

Leader-follower 1Leader-follower 2Follower 2-follower 3

UpperboundaryLowerboundary

(b)

02468

10

Dist

ance

(km

)

0 5 10 15 20 25Time (h)

Leader-follower 1Leader-follower 2Follower 2-follower 3

UpperboundaryLowerboundary

(c)

02468

10

Dist

ance

(km

)0 5 10 15 20 25

Time (h)

Leader-follower 1Leader-follower 2Follower 2-follower 3

UpperboundaryLowerboundary

(d)

Figure 8: Loosely formation with station-keeping box: the relative distance of hybrid sail on (a) winter solstice, (b) spring equinox, (c)summer solstice, and (d) autumn equinox in the frame (A).

4.219 4.220 4.221

LeaderFollower 1

Follower 2Station-keeping box

×104 X (km)−10 0 10

Y (km)

−100

10

Z (k

m)

(a)

4.219 4.220 4.221×104 X (km)−10 0 10

Y (km)

−100

10

Z (k

m)

LeaderFollower 1

Follower 2Station-keeping box

(b)

4.219 4.220 4.221×104 X (km)−10 0 10

Y (km)

−100

10

Z (k

m)

LeaderFollower 1

Follower 2Station-keeping box

(c)

4.219 4.220 4.221×104 X (km)−10 0 10

Y (km)

−100

10

Z (k

m)

LeaderFollower 1

Follower 2Station-keeping box

(d)

Figure 9: Loosely formation with station-keeping box: the trajectories of hybrid sail on (a) winter solstice, (b) spring equinox, (c) summersolstice, and (d) autumn equinox in the inertial frame.

7Complexity

(GEOs) is introduced, which can be used as a solution to thecongestion of the GEO. The hybrid sail spacecraft will con-sume less propellant to maintain the formation. For a10× 10× 10 km3 station-keeping box, which centers 40 kmoutside a real GEO and a 9 km loosely degree, propellant con-sumption saved 39.89%, 42.19%, 41.67%, and 42.04% for thewinter solstice, spring equinox, summer solstice, and autumnequinox, respectively. The obvious mass saving shows thatthe station-keeping box and loosely relative distance con-straints can improve the efficiency of the hybrid propellantsystem more efficiently than the ideal displaced GEO andstrict relative distance formation. Besides, the optimizationtrajectories show that the station-keeping box can success-fully restrict all the spacecrafts of the formation inside it. Fur-thermore, their relative distance is always larger than thelower boundary and smaller than the upper boundary.

Data Availability

All data in this manuscript are available for noncommercialapplications upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was jointly funded by China Natural ScienceFoundation (no. 61633008); Nature Science Foundation ofHeilongjiang Province of China (no. F2017005); Fundamen-tal Research Funds for the Central Universities, China (nos.HEUCFP201768 and HEUCFP201770); National NaturalScience Fund (no. 11772185); Harbin Science and Technol-ogy Innovation Talent Youth Fund (no. 2015RQQXJ003);and Postdoctoral Science-Research Foundation (no. LBH-Q13042).

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Table 3: Mass consumption for each spacecraft after one day.

Time Roles

Strictformationwith idealdisplacedGEO (kg)

Looselyformation

with a station-keeping box (kg)

Masssaving(%)

Wintersolstice

Leader 0.9155 0.4940 46.04%

Follower 1 1.000 0.6220 37.80%

Follower 2 0.8899 0.5700 31.99%

Springequinox

Leader 1.077 0.5850 45.68%

Follower 1 1.030 0.6160 40.19%

Follower 2 0.9153 0.5460 40.34%

Summersolstice

Leader 0.9183 0.4550 50.45%

Follower 1 1.000 0.6290 37.10%

Follower 2 0.8894 0.5540 37.71%

Autumnequinox

Leader 1.076 0.5870 45.45%

Follower 1 1.030 0.6180 40.00%

Follower 2 0.9152 0.5460 40.34%

Table 4: Mass consumption for all spacecrafts after one day.

Time

Strict formationwith idealdisplacedGEO (kg)

Looselyformation

with a station-keeping box (kg)

Masssaving(%)

Winter solstice 2.805 1.686 39.89%

Spring equinox 3.022 1.747 42.19%

Summer solstice 2.808 1.638 41.67%

Autumn equinox 3.021 1.751 42.04%

8 Complexity

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9Complexity

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