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Research ArticleActive Disturbance Rejection Station-Keeping Control ofUnstable Orbits around Collinear Libration Points

Min Zhu1 Hamid Reza Karimi2 Hui Zhang1 Qing Gao13 and Yong Wang1

1 Department of Automation University of Science and Technology of China Hefei 230027 China2Department of Engineering Faculty of Engineering and Science University of Agder 4876 Grimstad Norway3 School of Engineering and Information Technology University of New South Wales at the Australian Defence Force AcademyCanberra ACT 2600 Australia

Correspondence should be addressed to Yong Wang yongwangustceducn

Received 15 January 2014 Accepted 13 March 2014 Published 10 April 2014

Academic Editor Shen Yin

Copyright copy 2014 Min Zhu et alThis 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

An active disturbance rejection station-keeping control scheme is derived and analyzed for station-keeping missions of spacecraftalong a class of unstable periodic orbits near collinear libration points of the Sun-Earth system It is an error driven rather thanmodel-based control law essentially accounting for the independence of model accuracy and linearization An extended stateobserver is designed to estimate the states in real time by setting an extended state that is the sum of unmodeled dynamic andexternal disturbance This total disturbance is compensated by a nonlinear state error feedback controller based on the extendedstate observer A nonlinear tracking differentiator is designed to obtain the velocity of the spacecraft since only position signals areavailable In addition the system contradiction between rapid response and overshoot can be effectively solved via arranging thetransient process in tracking differentiator Simulation results illustrate that the proposedmethod is adequate for station-keeping ofunstable Halo orbits in the presence of system uncertainties initial injection errors solar radiation pressure and perturbations ofthe eccentric nature of the Earthrsquos orbit It is also shown that the closed-loop control system performance is improved significantlyusing our method comparing with the general LQR method

1 Introduction

In recent years there has been an increasing interest in libra-tion pointsmissionsThe libration points which are normallycalled equilibrium points or Lagrangian points correspondto regions in space where the centrifugal forces and thegravitational forces from the Sun and the Earth cancel eachother The existences of periodic orbits and quasi-periodicorbits in the vicinity of collinear libration points have beenproved and analyzed rigorously in celestial mechanics [12] These orbits offer potentially valuable opportunities forinvestigations concerning solar and heliospheric effects onplanetary environment and highly precise visible light tele-scopes

However these libration point orbits are inherentlyunstable but controllable Thus additional control force is

needed for a spacecraft to remain close to the nominalorbit The challenges of station-keeping control emerge fromhighly accuracy low computation burden and minimal fuelcost control requirements under the condition of spacecraftdynamic uncertainties unmodeled perturbations and initialorbit injection errors [3ndash6] Hence station-keeping controlfor libration point orbit missions is of virtual importance butwith a great deal of difficulties

The study of station-keeping control on libration pointorbits has become a popular research topic ever since theproblem was firstly proposed A vast majority of the station-keeping control methods are designed based on LTI modelvia local linearization at the libration points due to thehigh nonlinearity of the dynamic equation of libration pointorbit The nonlinear dynamic equation is obtained utilizinga Clohessy-Wiltshire- (CW-) like reference frame which is

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 410989 14 pageshttpdxdoiorg1011552014410989

2 Mathematical Problems in Engineering

widely used for Earth-centered spacecraft dynamic analysis[7 8] Breakwell et al [9 10] firstly introduced classicaloptimal control strategies for Halo orbit missions Ericksonand Glass [11] specially analyzed the ISEE-3 mission to makethis approach into implementation Cielaszyk and Wie [12]employed a new LQR control method combined with a dis-turbance accommodating controller for Lissajous and Haloorbits maintenance based on LTI model Di Giamberardinoand Monaco [13] designed a nonlinear controller based onLTI model to solve the problem of tracking Halo orbit aboutthe 119871

2point However the LTI model includes only 1st

2nd or 3rd order term of the gravitational force and is onlyeffective in the neighborhood of the libration point As amatter of fact the higher the amplitude of the libration pointorbit the greater the influence of nonlinear factors

To improve the modeling accuracy LTV method isemployed instead of the LTI approach Gurfil and Kasdin [14]expended the LTI model to LTVmodel and designed a time-varying controller for formation flying in the Sun-Earth sys-tem Barcolona Group proposed the so-called Floquet ModeControl [15ndash17] which employs nonlinear techniques tocompute the invariant manifolds and determine the discreteimpulsive maneuvers to cancel deviations along the unstablecomponent based on LTV model Howell and Pernicka [18]developed a target-point approach to maintain the spacecraftwithin some torus about the nominal Halo orbit Further-more several other control strategies are developed to over-come the disadvantage of searching for an optimal controllerfor both Floquet Mode approach and target-point approachKulkarni et al [3] extended the traditional119867

infinframework to

periodic discrete LTV systems for stabilization of spacecraftflight in Halo orbits Wang et al [6] presented a nonlinearcontroller based on polynomial eigenstructure assignment ofLTV model for the control of Sun-Earth 119871

2point station-

keeping and formation flying without considering the systemuncertainties

Besides the LTI and LTV model-based method existingworks also involve methodologies directly developed fromoriginal nonlinear dynamical equation of motion Rahmaniet al [19] solved the problem of Halo orbit control usingthe optimal control theory and the variation of the extremetechnique Xin et al [4] used a suboptimal control technique(the 120579 minus 119863 technique) to complete the mission of multiplespacecraft formation flying in deep space about the 119871

2point

Marchand and Howell [20] employed the feedback lineariza-tion approach for formation flight in the vicinity of librationpoints Gurfil et al [21] presented a novel nonlinear adaptiveneural control methodology for deep-space formation flyingBai and Junkins [22] proposed a modified Chebyshev-Picardinteration method for station-keeping of 119871

2Halo orbits of

Earth-Moon system These nonlinear control methods haveimproved the control performance However the robustnessunder nonlinear system uncertainties and the request forlow computation burden are unreasonably neglected whichshould be considered before implementation

Consequently this paper derives an active disturbancerejection station-keeping control method consideringmainlythe following issues

(1) Proposing an error driven rather than model-basedcontrol law which takes into account system uncer-tainties unmodeled disturbance and orbit injectionerrors to achieve better robustness

(2) Considering output feedback from practical view-point rather than full information since only positionsignals of spacecraft can be measured

(3) Better station-keeping performance as well as simplercomputation burden

A new nonlinear station-keeping control law based onactive disturbance rejection control (ADRC) method whichrefers to the so-called active disturbance rejection station-keeping control (ADRSC) method in this paper is pro-posed and analyzed ADRC was firstly proposed by Han[23 24] and Sun [25] and has been successfully appliedin various industrial processes such as vehicle flight con-trol ship tracking control robot control and power plantcontrol

The remainder of this paper is organized as followsSection 2 presents the equations of motion and periodic ref-erence orbits about the Sun-Earth libration point Section 3presents an introduction to the theory of ADRC synthesisSection 4 presents an ADRSC method for nonlinear station-keeping of periodic orbits around collinear libration pointSection 5 carries out simulation results to validate the effec-tiveness of the new method Finally Section 6 concludes thepaper

2 Equation of Motion

In this section the dynamic models of spacecraft basedon the circular restricted three-body problem (CR3BP) areestablished Additionally the relative Halo reference orbitsderived from LP map method are presented

21 Dynamics of CR3BP The CR3BP is one of the mostcommon nonlinear models which investigate the relativemotion around the libration points As is shown in Figure 1the infinitesimal mass 119898 of spacecraft compared to the mass1198981of the Sun and the mass 119898

2of the Earth is assumed

It is further assumed that the two primary bodies rotateabout their barycenter in a circular orbit under a constantangular velocity 119908 A rotating coordinate system (119874119883 119884 119885)

is defined with the origin set at the barycenter of the Sun-Earth system The 119883-axis is directed from the Sun towardsthe EarthThe119885-axis is perpendicular to the plane of rotationand is positive when pointing upward The 119884-axis completesthe set to yield a right-hand reference frame Normalizationis performed by defining the distance 119877 between the Sunand the Earth as the unit AU of length the time of 1119908 asthe unit of time TU and the total mass of the Sun and theEarth as the unit ofmassNormalization is obtained by setting120583 = 119898

2(1198981+ 1198982) 1198982= 120583 and 119898

1= 1 minus 120583 thus 119898

1is

Mathematical Problems in Engineering 3

EarthX

Z

Y

OR

Spacecraft

j

i

km2

m1

r2r1

L1 120574

120596

Sun

Figure 1 Restricted three-body problem

located at (minus120583 0 0) and1198982is located at (1minus120583 0 0) The well-

known equations of motion for CR3BP can be written in thedimensionless form [1]

= 2 + 119883 minus(1 minus 120583) (119883 + 120583)

1199033

1

minus120583 (119883 minus (1 minus 120583))

1199033

2

= minus2 + 119884 minus(1 minus 120583)119884

1199033

1

minus120583119884

1199033

2

= minus(1 minus 120583)119885

1199033

1

minus120583119885

1199033

2

(1)

where the dot represents time derivative in the rotatingframe and the distance 119903

1= radic(119883 + 120583)

2+ 1198842 + 1198852 and 119903

2=

radic(119883 minus (1 minus 120583))2+ 1198842 + 1198852

22 Periodic Reference Orbits We can find libration pointsof the Sun-Earth system denoted by 119871

1ndash1198715 by setting all

derivatives in (1) to zero For the Sun-Earth system 1198711is

located at119883 = 0989986055 It has been proved that the threecollinear points 119871

1 1198712 and 119871

3 are unstable but controllable

[1] And a family of periodic orbits and quasi-periodic orbitsexist around the collinear points [2]

Periodic solutions of the nonlinear equations of motioncan be constructed using the method of successive approx-imations in conjunction with a technique similar to theLindstedt-Poincaremethod [26]These periodic solutions arefirstly presented by Richardson [27 28] and have been widelyquoted and used To obtain the periodic solutions around 119871

1

a new rotating coordinate system (1198711 119894 119895 119896) is defined with

the origin set at 1198711and the distance 120574 between the Earth and

1198711as the new unit of length which is also shown in Figure 1

The relationship between the (119874119883 119884 119885) reference frame andthe (119871

1 119894 119895 119896) reference frame is as follows

119894 =119883 minus (1 minus 120583 minus 120574)

120574 119895 =

119884

120574 119896 =

119885

120574 (2)

where 120574 = 0010010905 [27]

The equations of theHalo orbit to the third order are givenby

119909119903(119905) = 119886

211198602

119909+ 119886221198602

119911minus 119860119909cos (120582119905)

+ (119886231198602

119909minus 119886241198602

119911) cos (2120582119905)

+ (119886311198603minus 119886321198601199091198602

119911) cos (3120582119905)

119910119903(119905) = (119860

119910+ 119887331198603

119909+ (11988734

minus 11988735) 1198601199091198602

119911) sin (120582119905)

+ (119887211198602

119909minus 119887221198602

119911) sin (2120582119905)

+ (119887311198603

119909minus 119887321198601199091198602

119911) sin (3120582119905)

119911119903(119905) = minus 3119889

21119860119909119860119911+ 119860119911cos (120582119905)

+ 11988921119860119909119860119911cos (2120582119905)

+ (119889321198601199111198602

119909minus 119889311198603

119911) cos (3120582119905)

(3)

where 119909119903(119905) 119910119903(119905) and 119911

119903(119905) are the coordinates along 119894 119895 and

119896 axes The Halo orbit designed for ISEE-3 mission [27] isselected as one of the target station-keeping orbits The otherreference orbit is selected with higher vertical amplitudeThevalues of the various constants in (2) of both Halo orbits aregiven in Table 1

3 Introduction of ADRC

Due to complexity of modern systems more attention hasbeen paid on data-driven control scheme recently [29 30]The ADRC is a typical data-driven control It inherits fromPID using the error driven rather than model-based controllaw to eliminate errors

Hanrsquos ADRC consists of three parts a nonlinear trackingdifferentiator (TD) [31] which is used to arrange the idealtransient process of the system an extended state observer(ESO) [32] which could estimate the disturbance fromthe system output and then the ADRC compensates thedisturbances according to estimated values and a nonlinearstate error feedback (NLSEF) [24] which is used to get thecontrol input of the system

Consider the following system

1199091= 119910

1= 1199092

2= 119891 (119909

1 1199092 119908 (119905) 119905) + 119887119906

(4)

where 119910 is the output variable 119906 is the control input 119887 ismagnification factor and 120596(119905) is the external disturbance119891(1199091 1199092 119908(119905) 119905) includes three parts modeling dynamics

uncertain dynamics and disturbance The ADRC approachmakes an effort to compensate for the unknown dynamicsand the external disturbances in real time without an explicitmathematical expression

4 Mathematical Problems in Engineering

Table 1The values of the various constants in (2) for ISEE-3 (119885 = 110000 km) orbit and Halo orbit (119885 = 800000 km) near 1198711libration point

of the Sun-Earth system For ISEE-3 (119885 = 110000 km) orbit and for Halo orbit (119885 = 800000 km)

Parameters Value Parameters Value Parameters Value120582 208645 119896 322927 119886

21209270

11988622

248298 times 10minus1

11988623

minus905965 times 10minus1

11988624

minus104464 times 10minus1

11988631

793820 times 10minus1

11988632

826854 times 10minus2

11988721

minus492446 times 10minus1

11988722

607465 times 10minus2

11988731

885701 times 10minus1

11988732

230198 times 10minus2

11988733

minus284508 11988734

minus230206 11988735

minus18703711988921

minus346865 times 10minus1

11988931

190439 times 10minus2

11988932

398095 times 10minus1

31 TD The TD has the ability to track the given inputreference signalwith quick response andnoovershoot by pro-viding transition process for expected input V and differentialtrajectory of set value that is V

1 and its differential V

2

One feasible second-order TD can be designed as [31]

V1= V2

V2= fhan (V

1minus V (119905) V

2 119903 ℎ0)

(5)

where V(119905) denotes the control objective 119903 is speed factor anddecides tracking speed ℎ

0is filtering factor and fhan(V

1minus

V(119905) V2 119903 ℎ0) is as follows

119889 = 119903ℎ2

0

1198860= ℎ0V2

119910 = (V1minus V (119905)) + 119886

0

1198861= radic119889 (119889 + 8 |119889|)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119904119910=sign (119910 + 119889) minus sign (119910 minus 119889)

2

119886 = (1198860+ 119910 minus 119886

2) 119904119910+ 1198862

119904119886=sign (119886 + 119889) minus sign (119886 minus 119889)

2

fhan (V1minus V (119905) V

2 119903 ℎ0)

= minus119903 (119886

119889minus sign (119886)) 119904

119886minus 119903 sign (119886)

(6)

32 ESO ESO is used to estimate 119891(1199091 1199092 119908(119905) 119905) in real

time and tomake adjustments at each sampling point in a dig-ital controller An augmented variable 119909

3= 119891(119909

1 1199092 119908(119905) 119905)

is introduced in (4) Using 1199111 1199112 and 119911

3to estimate 119909

1 1199092

and 1199093 respectively a nonlinear observer is designed as [32]

119890 = 1199111minus 119910

1= 1199112minus 1205731119890

2= 1199113minus 1205732fal (119890 120572

1 120575) + 119887119906

3= minus1205733fal (119890 120572

2 120575)

(7)

where 1205731 1205732 and 120573

3are observer gains 119890 is the error and

fal(119909 120572 120575) is as follows

fal (119909 120572 120575) =

119909

1205751minus120572 |119909| le 120575

sign (119909) |119909|120572 |119909| gt 120575

(8)

33 NLSEF NLSEF generates control voltage 119906 for system byusing the errors between the output of ESO and TD

1198901= V1minus 1199111

1198902= V2minus 1199112

(9)

A nonlinear combination of errors signal can be con-structed as [24]

1199060= minusfhan (119890

1 1198881198902 120572 ℎ1) (10)

where the nonlinear coefficient 120572 is selected as 0 lt 120572 lt 1 and119888 is the proportional coefficients

The controller is designed as

119906 = 1199060minus1199113

119887 (11)

4 ADRSC for Station-Keeping

As is mentioned in Section 2 the reference Halo orbitaround collinear libration orbit is inherently unstable andstation-keeping control approach is designed under modeluncertainties and disturbances In this paper we extend theADRC for the ADRSCThe structure of ADRSC algorithm ispresented in Figure 2 ADRSC will be proposed and analyzedin this section from the following three aspects discrete TDdiscrete ESO and discrete NLSEF

41 Discrete TD for the Tracking of the Reference Orbit andthe SpacecraftTrajectory Assuming that only position signalsof spacecraft can be detected and used for feedback thereare two different kinds of discrete TDs designed for station-keeping control

411 TD I It is designed for reference Halo orbit tracking

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 119903119894 ℎ119894)

(12)

Mathematical Problems in Engineering 5

ESO

u

w

minus

minusminus

+Xr(t)r1

r

12

2 e2

e1

z1z2z3

u0 X(t)TD I

TD II

NLSEFSpacecraftnonlineardynamic

Disturbingforces

xr(t) =[xr(t) yr(t) zr(t)]T

Figure 2 The structure of ADRSC algorithm

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively V

119894(119896) denotes the reference orbit TD I provides

transition process for nominal orbit V119894and differential trajec-

tory of set position value that is V1198941and its differential V

1198942

412 TD II It is designed for flight position tracking andestimating of the velocity of the spacecraft

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 1199030119894 ℎ0119894)

(13)

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively For instance state vector [V11 V

12]119879 denotes the

estimation of [119883 ]119879 which is the position and velocity

vector of spacecraft trajectory along119883 axes V119894(119896) denotes the

flight trajectory of spacecraft And 1199030119894is the tracking speed

parameter and ℎ0119894is the filter factor which makes an effort

of filtering While the integration step is fixed increasingthe filtering factor will make the effort of filter better Theoutputs of TD II which are the spacecraft flight position andvelocity provide the inputs of ESO for feedback as describedin Figure 2

42 Discrete ESO for Estimating the Nonlinear DynamicsModel Uncertainties and Unmodeled Disturbance The equa-tions of station-keeping of Halo orbits using ADRSC can begenerally defined as

1198941= 1199091198942

1198942= 119892119894(X X) + 119908

119894+ 119906119894

119910 = 1199091198941

(14)

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectivelyX = [11990911 119909

2111990931]119879 denotes the position vector

of the spacecraft w = [1199081 1199082

1199083]119879 denotes the disturbance

vector and u = [1199061 1199062

1199063]119879 denotes the vector of station-

keeping control input g(XX) = [1198921 1198922

1198923]119879 represents

the uncertain dynamics vector respectively Note that (1)will be simulated as the real flight dynamic mode Hence

robust performance will be presented under such formationof dynamic model uncertainties

It is necessary to point out that a distinct improvementcan be obtained using ADRSC for station-keeping since thedynamic model does not need to be expressively knownwhich is different with the aforementioned studies DefineF(X X) = g(X X) + w In fact in the context of feedbackcontrol F(X X) is something to be overcome by the controlsignal and it is therefore denoted as the ldquototal disturbancerdquo Atthis point F(X X) is extended as an additional state variablethat is

1199091198943= 119865119894(X X) (15)

where 1198651 1198652 and 119865

3denote the corresponding axis com-

ponents of F(X X) respectively and let F(X X) = a(X X)where a(X X) is unknown

One can rewrite (14) as follows

1198941= 1199091198942

1198942= 1199091198943+ 119906119894

1198943= 119886119894

119910 = 1199091198941

(16)

where 1199091198941 1199091198942 and 119909

1198943represent the position velocity and

119865119894(X X) respectively 119894 = 1 2 3 Then one can use the

following discrete nonlinear observer 1199111198941 1199111198942 1199111198943 to estimate

state vector 1199091198941 1199091198942 1199091198943

119890119894= 119911119894minus 119910119894

1199111198941(119896 + 1) = 119911

1198941(119896) + ℎ times (119911

1198942(119896) minus 120573

1198941119890)

1199111198942(119896 + 1)

= 1199111198942(119896) + ℎ times (119891

0119894+ 1199111198943(119896) minus 120573

1198942times fal (119890 05 ℎ))

+ 119906 (119894)

1199111198943(119896 + 1) = 119911

1198943(119896) minus ℎ times 120573

1198943times fal (119890 025 ℎ)

(17)

43 Discrete NLSEF for Station-Keeping Since the statesunmodeled dynamics and disturbances have been estimatedby ESO presented in Section 42 state errors between the

6 Mathematical Problems in Engineering

output of ESO and TD I are combined for feedback controllaw large errors corresponding to low gains and small errorscorresponding to high gains

119906119894(119896 + 1) = minusfhan (119890

1198941 1198881198941198901198942 120572119894 ℎ119894) minus 1199111198943(119896) (18)

where 119888119894 120572119894 ℎ119894 are the NLSEF parameters described in (10)

1198901198941and 119890

1198942are the state errors described in (9) 119911

1198943denotes

unmodeled dynamics and disturbances 119865119894(X X) which is

estimated from (17)The parameter ℎ

119894of NLSEF which is also named filter

factor of fhan makes great contribution in NLSEF controlperformance With the integration step fixed ℎ

119894must be

selected properly It is noted that larger errors correspondingto larger ℎ

119894and smaller errors corresponding to smaller ℎ

119894will

make a better performanceIn this paper a parameter self-turning approach is firstly

proposed for ℎ119894selection as follows

ℎ119894=

119896log10radic

3

sum

119894=1

1198902

1198941+ ℎ0 radic

3

sum

119894=1

1198902

1198941gt 120585

119896log10120585 + ℎ0 radic

3

sum

119894=1

1198902

1198941le 120585

(19)

where the parameters 119896 ℎ0 and 120585 of self-turning ℎ

119894can

be easily chosen in practice as will be presented in thesimulation

44 Disturbing Forces In any actual mission the pertur-bation factors and the injection errors coupled with theinherent unstable nature of the Halo orbits around thecollinear libration points will cause a spacecraft to drift fromthe periodic reference orbit [33] A robust performance inthe presence of disturbance can be obtained by ADRSC andthe model of disturbance is not necessary for ADRSC as longas the disturbance is bounded Physically speaking deep-space disturbances are always bounded [21] To illustrate thecapability of ADRSC to reject unknowndisturbances hereweintroduce the perturbative forces due to the eccentric natureof the Earthrsquos orbit and the solar radiation pressure (SRP)disturbance

441 The Perturbative Forces due to the Eccentric Nature ofthe Earthrsquos Orbit The most important perturbative effects inthe CR3BP are the eccentric nature of the Earthrsquos orbit andthe gravitational force of the moon [3 28] In this paper thelargest perturbative force per unit mass dENE of spacecraftdue to the eccentric nature of the Earthrsquos orbit is taken intoaccount which has been given approximately in [3]

1003816100381610038161003816dENE1003816100381610038161003816 = 119866119872Earth

100381610038161003816100381610038161003816100381610038161003816

1

1199032119888

minus1

1199032119890

100381610038161003816100381610038161003816100381610038161003816

(20)

where 119903119890is the position vector of the Earth at the pericenter

of the Earthrsquos elliptical orbit from the libration point 119903119888is the

position vector similar to a circular orbit

442 SRP Disturbance In the deep-space mission SRP isanother disturbance to account for Here we adopt a widelyused model [34] According to this model the disturbanceacceleration to SRP is

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

2(_r1sdot n)2

n (21)

where 120573 is a parameter that depends on the coefficient ofreflectivity the area andmass of the spacecraft the solar fluxand the speed of light

_r1= r1r1 and n is the attitude

vector of the spacecraft in the rotating frame To simplify (21)we assume that n =

_r1 which yields

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

3r1 (22)

5 Simulation Results and Analysis

51 Simulation Scenario and Numerical Values To comparethe performance of ADRSC with linearization method thesimulation of an LQR controller [1] based on LTI systemfrom local linearization around 119871

1point of nonlinear system

equation (1) is also carried out In order to compare theperformance of ADRSC between low and large amplitudeorbits we select two orbits for simulation one orbit for theISEE-3 mission with vertical displacement 110000 km andthe other of 800000 km

The control forces for station-keeping are provided byan ionic engine with the maximum thrust 60mN whichis currently commercially available [35] The mass of thespacecraft is designed as119898 = 500 kg

The disturbing forces described in 44 are considered inthe simulation The eccentricity of Earth orbit 119890 = 0016675And the SRP parameter is calculated assuming that the solarflux is 119865

119904= 1358Wm2 the speed of light is 119888 = 3 times 10

8msthe cross-sectional area is 119878 = 35m2 and the coefficient ofreflectivity is 119902 = 06 which results in 120573 = 119878119865

119904(1 + 119902)(119898119888) =

5 times 10minus8ms2 [21] It is also assumed that the initial position

injection errors along the 119909 119910 and 119911 directions are 1000 kmand the velocity injection errors are 1ms The MATLABSimulink software is used for simulation The time step ℎ isset as 00001 TU for better accuracy of the spacecraft Thesimulation time is chosen as 1800 days almost 10 periods ofHalo motion about 119871

1libration point

In the simulation considering the system is not full statefeedback the TD is designed to track the spacecraft flightposition and obtain the velocity as well Figure 3 illustratesthe TD II tracking result of the spacecraft flight In Figure 3FP and FV denote the spacecraft flight position and velocityTP and TV denote the TD tracking position and velocityrespectively Figure 3 presents the accurately tracking ofposition signal and velocity signal along the 119909 119910 and 119911 axeswhich will be used for feedback control

52 ISEE-3 Halo Orbit (119885 = 110 000 km)

521 Related Parameters In dimensionless coordinates theinitial orbit parameters without injection errors of ISEE-3 are

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

ISEE-3 (110000 km) 0988872932669558 0 8109335378305802119890 minus 04 0 0008853797264729 0Halo (800000 km) 0989390221855232 0 0006248517078177 0 0012547011257083 0

given in Table 2 (in unit of AU for position and in unit ofAUTU for velocity)The parameters of ADRSC of ISEE-3 aregiven as follows 119903

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

522 Simulation Results Figures 4 and 5 illustrate thestation-keeping control results for ISEE-3 with orbit injectionerrors as well as disturbing forces which show that preferablestation-keeping accuracy can be obtained using ADRSC

through 60mNengine FromFigures 5(a) and 5(b) the errorsof relative position and relative velocity drop down veryquickly and steadily to zero within 5 days compared withLQR controller 14 days in Figures 6(a) and 6(b) The positionerrors along 119909 119910 and 119911 directions can be kept within 1 kmwhich is very small compared with the orbit altitude of theISEE-3 mission (110000 km) Also as shown in Figure 5(c)the control forces begin at 60mN to drive the relative positionerrors to zero but quickly reduce to the steady progressin 5 days Hence these results illustrate the effectiveness ofthe ADRSC of unstable Halo orbits near collinear librationpoints

In order to describe the results of ADRSCmore accuratelywhen the spacecraft moves steady on the Halo orbit here

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

we introduce the popular evaluation parameters for station-keeping [3 12 33] the velocity incrementsΔ119881

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

10038161003816100381610038161199061198941003816100381610038161003816 d119905 (119894 = 119909 119910 119911)

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

The velocity increment and themean absolute value of theposition errors of ISEE-3 station-keeping with orbit injectionerrors and disturbance are given in Table 3 It can be seenthat themean absolute position error is 0389 kmwhich illus-trates the higher precision characteristic for the spacecraftstation-keeping mission with ADRSC comparing with thatof LQR controller 2965 km shown in Table 3 Meanwhilethe velocity increment of ADRSC under disturbing forceand injection error is 36567msT which is approximatelyequal to that of LQR controller 36705msT Note thatthere is a trade-off between accuracy of the spacecraft tofollow the Halo orbit and the thrust usage [3] The earlierapproaches which use impulsive maneuvers are designed

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

for station-keeping missions and do not have stringent orbit-tracking requirements While high accuracy orbit station-keeping is definitely required for current missions which usecontinuous low-thrust propulsion

53 Halo Orbit (119885 = 800 000 km)

531 Related Parameters The parameters of ADRSC of Haloorbit (119885 = 800 000 km) are the same as ADRSC of ISEE-3mission except the parameters of self-turning ℎ

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

532 Simulation Results Figures 7 and 8 illustrate theADRSC results for Halo orbit (119885 = 800 000 km) with orbitinjection errors as well as disturbing force From Figures

8(a) and 8(b) one can find that ADRSC controller can drivethe relative position errors and velocity errors along 119909 119910and 119911 steadily to zero within 75 days compared with LQRcontroller 25 days in Figures 9(a) and 9(b) During the steadyperiodic the position errors are kept within 2 km whichillustrate high accurately maintained ability of ADRSC ofhigh Halo orbit

The velocity increment and the mean absolute value ofthe position errors of both ADRSC and LQR are presentedin Table 3 The position error of ADRSC is 1146 km which isgreatly less than LQR controller 17084 km Meanwhile thecontrol consumption of ADRSC is 195248msT which isalmost equal to that of LQR controller 195394msT

54 Summary From the above analyses and control resultsbetween ADRSC and LQR we can find the following

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Table 3 The velocity increments and the mean absolute value of the position errors of ADRSC and LQR control

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

ADRSC ISEE-3 (110000 km) 24872 25948 6721 36567 0135 0343 0119 0389LQR ISEE-3 (110000 km) 25030 25991 6727 36705 1991 2126 0555 2965ADRSC Halo (800000 km) 160770 81501 75049 195248 0380 0607 0895 1146LQR Halo (800000 km) 160815 81703 75115 195394 14325 6516 6647 17084

(1) TheADRSCwhich is an error driven controlmethodis adequate for the station-keeping control of unstableorbits without any knowledge about the spacecraftdynamic model

(2) The ADRSC presents praiseworthy station-keepingperformance with orbit injection errors as well as

unmodeled disturbances such as the SRP and theperturbative forces due to the eccentric nature of theEarthrsquos orbit using an ionic engine with maximumthrust 60mN

(3) The ADRSC approach has better station-keepingcontrol ability and higher orbit maintenance accuracy

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

(4) TheADRSC has better robust performance comparedwith the general station-keeping method dependingon spacecraft dynamic model

There has been a great deal of commendable research onunstable libration point orbits station-keeping as mentionedbefore A major point of departure between ADRSC methodand earlier approaches to station-keeping is that ADRSC is anerror driven rather than model-based control law which caninherently get across the dependency on model accuracy as

well as the drawbacks of linearization With the combinationof TD ESO and NLSEF the unmodeled disturbances as wellas the unmodeled system dynamic can be compensated inreal time Thus fast response-time requirement and highaccuracy of orbit maintenance requirement can be satisfiedby ADRSC

It is important to keep in mind not only the trackingaccuracy but also the robustness of the station-keepingcontroller as the space environment cannot be accuratelymodeled as well as the internal and external disturbanceADRSC extends the unmodeled spacecraft dynamic andthe disturbance as a state which can be estimated from

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

Figure 8 The relative position errors velocity errors and control input of ADRSC control of Halo orbit

ESO and then ldquorejectedrdquo from nonlinear feedback controlThus no matter the spacecraft system is model known orunknown linear or nonlinear time invariant or time variantwith disturbance or without ADRSC is able to show desiredperformance

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

output of nonlinear spacecraft dynamic system rather thanprecise nonlinear dynamic model or linearization The non-linear simulation results indicate that ADRSC is adequate forthe station-keeping of unstable orbits that take into account

the system uncertainties initial injection errors SRP andperturbations of the eccentric nature of the Earthrsquos orbitWith this ADRSC method the system control performanceis improved significantly by comparing with a general LQRmethod in the simulation The thrust required by the controllaw is also reasonable and can be implemented using acontinuous low-thrust propulsion device such as an ionengine ADRSC can be qualified for future missions whichrequire better performance more robustness and higherstation-keeping accuracy It is anticipated that this methodwill work for other unstable orbits such as Lissajous andHalo orbits near other collinear points as well as the for-mation flying on these orbits which are the focus for futurestudy

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Nature Science Founda-tion of China under Grant no 61004017 and by the Polish-Norwegian Research Programme in the frame of ProjectContract no Pol-Nor200957472013 The authors wouldlike to thank the editors and the anonymous reviewers fortheir keen and insightful comments which greatly improvethe paper

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

[2] K C Howell ldquoThree-dimensional periodic halo orbitsrdquo Celes-tial Mechanics vol 32 no 1 pp 53ndash71 1984

[3] J E Kulkarni M E Campbell and G E Dullerud ldquoStabiliza-tion of spacecraft flight in halo orbits an 119867

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

[4] M Xin S N Balakrishnan and H J Pernicka ldquoMultiplespacecraft formation control with 120579-D methodrdquo IET ControlTheory and Applications vol 1 no 2 pp 485ndash493 2007

[5] D Sheng X Yang and H R Karimi ldquoRobust control forautonomous spacecraft evacuation with model uncertainty and

14 Mathematical Problems in Engineering

upper bound of performance with constraintsrdquo MathematicalProblems in Engineering vol 2013 Article ID 589381 2013

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

[7] Z Li M Liu H R Karimi and X Cao ldquoObserver-based sta-bilization of spacecraft rendezvous with variable sampling andsensor nonlinearityrdquoMathematical Problems in Engineering vol2013 Article ID 902452 11 pages 2013

[8] Z LiM LiuH RKarimi andXCao ldquoSampled-data control ofspacecraft rendezvous with discontinuous lyapunov approachrdquoMathematical Problems in Engineering vol 2013 Article ID814271 10 pages 2013

[9] J V Breakwell ldquoInvestigation of halo satellite orbit controlrdquoTech Rep CR-132 858 NASA 1973

[10] J V Breakwell A A Kamel and M J Ratner ldquoStation-keepingfor a translunar communication stationrdquo Celestial Mechanicsvol 10 no 3 pp 357ndash373 1974

[11] J Erickson and A Glass ldquoImplementation of ISEE-3 trajectorycontrolrdquo in American Astronautical Society and American Insti-tute of Aeronautics and Astronautics Astrodynamics SpecialistConference Provincetown UK 1979

[12] D Cielaszyk and B Wie ldquoNew approach to halo orbit determi-nation and controlrdquo Journal of Guidance Control and Dynam-ics vol 19 no 2 pp 266ndash273 1996

[13] P Di Giamberardino and S Monaco ldquoOn halo orbits spacecraftstabilizationrdquo Acta Astronautica vol 38 no 12 pp 903ndash9251996

[14] P Gurfil and N J Kasdin ldquoStability and control of spacecraftformation flying in trajectories of the restricted three-bodyproblemrdquo Acta Astronautica vol 54 no 6 pp 433ndash453 2004

[15] C Simo G Gomez J Llibre R Martınez and J RodrıguezldquoOn the optimal station keeping control of halo orbitsrdquo ActaAstronautica vol 15 no 6 pp 391ndash397 1987

[16] G G Gomez J Llibre R Martınez and C Simo Dynamicsand Mission Design Near Libration Point Orbits vol 1 ofFundamentals The Case of Collinear Libration Points WorldScientific Singapore 2001

[17] G G Gomez Dynamics and Mission Design Near LibrationPoints vol 3 of Advanced Methods for Collinear Points WorldScientific 2001

[18] K C Howell and H J Pernicka ldquoStationkeeping method forlibration point trajectoriesrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 151ndash159 1993

[19] A Rahmani M-A Jalali and S Pourtakdoust ldquoOptimalapproach to halo orbit controlrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and Exhibit pp11ndash14 Austin Tex USA 2003

[20] B G Marchand and K C Howell ldquoControl strategies forformation flight in the vicinity of the libration pointsrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1210ndash12192005

[21] P Gurfil M Idan and N J Kasdin ldquoAdaptive neural control ofdeep-space formation flyingrdquo Journal of Guidance Control andDynamics vol 26 no 3 pp 491ndash501 2003

[22] X Bai and J L Junkins ldquoModified Chebyshev-Picard iterationmethods for station-keeping of translunar halo orbitsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 926158 18pages 2012

[23] J Han ldquoControl theory is it a model analysis or direct controlapproachrdquo Journal of Systems Science andMathematical Sciencevol 9 no 4 pp 328ndash335 1989

[24] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[25] D Sun ldquoComments on active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 54 no 6 pp3428ndash3429 2007

[26] G E Dullerud and F Paganini A Course in Robust ControlTheory vol 6 Springer New York NY USA 2000

[27] D L Richardson ldquoHalo orbit formulation for the isee-3 mis-sionrdquo Journal of Guidance Control and Dynamics vol 3 no 6pp 543ndash548 1980

[28] D L Richardson ldquoAnalytic construction of periodic orbitsabout the collinear pointsrdquo Celestial Mechanics vol 22 no 3pp 241ndash253 1980

[29] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark tennesseeeastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

[30] S Yin S XDingAHAbandan Sari andHHao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[31] J Han and W Wang ldquoNonlinear tracking-differentiatorrdquo Jour-nal of Systems Science and Mathematical Science vol 14 no 2pp 177ndash183 1994

[32] J Han ldquoThe ldquoextended state observerrdquo of a class of uncertainsystemsrdquo Control and Decision vol 10 no 1 pp 85ndash88 1995

[33] H Peng J Zhao Z Wu and W Zhong ldquoOptimal periodiccontroller for formation flying on libration point orbitsrdquo ActaAstronautica vol 69 no 7-8 pp 537ndash550 2011

[34] P Gurfil ldquoControl-theoretic analysis of low-thrust orbital trans-fer using orbital elementsrdquo Journal of Guidance Control andDynamics vol 26 no 6 pp 979ndash983 2003

[35] M Martinez-Sanchez and J E Pollard ldquoSpacecraft electricpropulsion-an overviewrdquo Journal of Propulsion and Power vol14 no 5 pp 688ndash699 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

widely used for Earth-centered spacecraft dynamic analysis[7 8] Breakwell et al [9 10] firstly introduced classicaloptimal control strategies for Halo orbit missions Ericksonand Glass [11] specially analyzed the ISEE-3 mission to makethis approach into implementation Cielaszyk and Wie [12]employed a new LQR control method combined with a dis-turbance accommodating controller for Lissajous and Haloorbits maintenance based on LTI model Di Giamberardinoand Monaco [13] designed a nonlinear controller based onLTI model to solve the problem of tracking Halo orbit aboutthe 119871

2point However the LTI model includes only 1st

2nd or 3rd order term of the gravitational force and is onlyeffective in the neighborhood of the libration point As amatter of fact the higher the amplitude of the libration pointorbit the greater the influence of nonlinear factors

To improve the modeling accuracy LTV method isemployed instead of the LTI approach Gurfil and Kasdin [14]expended the LTI model to LTVmodel and designed a time-varying controller for formation flying in the Sun-Earth sys-tem Barcolona Group proposed the so-called Floquet ModeControl [15ndash17] which employs nonlinear techniques tocompute the invariant manifolds and determine the discreteimpulsive maneuvers to cancel deviations along the unstablecomponent based on LTV model Howell and Pernicka [18]developed a target-point approach to maintain the spacecraftwithin some torus about the nominal Halo orbit Further-more several other control strategies are developed to over-come the disadvantage of searching for an optimal controllerfor both Floquet Mode approach and target-point approachKulkarni et al [3] extended the traditional119867

infinframework to

periodic discrete LTV systems for stabilization of spacecraftflight in Halo orbits Wang et al [6] presented a nonlinearcontroller based on polynomial eigenstructure assignment ofLTV model for the control of Sun-Earth 119871

2point station-

keeping and formation flying without considering the systemuncertainties

Besides the LTI and LTV model-based method existingworks also involve methodologies directly developed fromoriginal nonlinear dynamical equation of motion Rahmaniet al [19] solved the problem of Halo orbit control usingthe optimal control theory and the variation of the extremetechnique Xin et al [4] used a suboptimal control technique(the 120579 minus 119863 technique) to complete the mission of multiplespacecraft formation flying in deep space about the 119871

2point

Marchand and Howell [20] employed the feedback lineariza-tion approach for formation flight in the vicinity of librationpoints Gurfil et al [21] presented a novel nonlinear adaptiveneural control methodology for deep-space formation flyingBai and Junkins [22] proposed a modified Chebyshev-Picardinteration method for station-keeping of 119871

2Halo orbits of

Earth-Moon system These nonlinear control methods haveimproved the control performance However the robustnessunder nonlinear system uncertainties and the request forlow computation burden are unreasonably neglected whichshould be considered before implementation

Consequently this paper derives an active disturbancerejection station-keeping control method consideringmainlythe following issues

(1) Proposing an error driven rather than model-basedcontrol law which takes into account system uncer-tainties unmodeled disturbance and orbit injectionerrors to achieve better robustness

(2) Considering output feedback from practical view-point rather than full information since only positionsignals of spacecraft can be measured

(3) Better station-keeping performance as well as simplercomputation burden

A new nonlinear station-keeping control law based onactive disturbance rejection control (ADRC) method whichrefers to the so-called active disturbance rejection station-keeping control (ADRSC) method in this paper is pro-posed and analyzed ADRC was firstly proposed by Han[23 24] and Sun [25] and has been successfully appliedin various industrial processes such as vehicle flight con-trol ship tracking control robot control and power plantcontrol

The remainder of this paper is organized as followsSection 2 presents the equations of motion and periodic ref-erence orbits about the Sun-Earth libration point Section 3presents an introduction to the theory of ADRC synthesisSection 4 presents an ADRSC method for nonlinear station-keeping of periodic orbits around collinear libration pointSection 5 carries out simulation results to validate the effec-tiveness of the new method Finally Section 6 concludes thepaper

2 Equation of Motion

In this section the dynamic models of spacecraft basedon the circular restricted three-body problem (CR3BP) areestablished Additionally the relative Halo reference orbitsderived from LP map method are presented

21 Dynamics of CR3BP The CR3BP is one of the mostcommon nonlinear models which investigate the relativemotion around the libration points As is shown in Figure 1the infinitesimal mass 119898 of spacecraft compared to the mass1198981of the Sun and the mass 119898

2of the Earth is assumed

It is further assumed that the two primary bodies rotateabout their barycenter in a circular orbit under a constantangular velocity 119908 A rotating coordinate system (119874119883 119884 119885)

is defined with the origin set at the barycenter of the Sun-Earth system The 119883-axis is directed from the Sun towardsthe EarthThe119885-axis is perpendicular to the plane of rotationand is positive when pointing upward The 119884-axis completesthe set to yield a right-hand reference frame Normalizationis performed by defining the distance 119877 between the Sunand the Earth as the unit AU of length the time of 1119908 asthe unit of time TU and the total mass of the Sun and theEarth as the unit ofmassNormalization is obtained by setting120583 = 119898

2(1198981+ 1198982) 1198982= 120583 and 119898

1= 1 minus 120583 thus 119898

1is

Mathematical Problems in Engineering 3

EarthX

Z

Y

OR

Spacecraft

j

i

km2

m1

r2r1

L1 120574

120596

Sun

Figure 1 Restricted three-body problem

located at (minus120583 0 0) and1198982is located at (1minus120583 0 0) The well-

known equations of motion for CR3BP can be written in thedimensionless form [1]

= 2 + 119883 minus(1 minus 120583) (119883 + 120583)

1199033

1

minus120583 (119883 minus (1 minus 120583))

1199033

2

= minus2 + 119884 minus(1 minus 120583)119884

1199033

1

minus120583119884

1199033

2

= minus(1 minus 120583)119885

1199033

1

minus120583119885

1199033

2

(1)

where the dot represents time derivative in the rotatingframe and the distance 119903

1= radic(119883 + 120583)

2+ 1198842 + 1198852 and 119903

2=

radic(119883 minus (1 minus 120583))2+ 1198842 + 1198852

22 Periodic Reference Orbits We can find libration pointsof the Sun-Earth system denoted by 119871

1ndash1198715 by setting all

derivatives in (1) to zero For the Sun-Earth system 1198711is

located at119883 = 0989986055 It has been proved that the threecollinear points 119871

1 1198712 and 119871

3 are unstable but controllable

[1] And a family of periodic orbits and quasi-periodic orbitsexist around the collinear points [2]

Periodic solutions of the nonlinear equations of motioncan be constructed using the method of successive approx-imations in conjunction with a technique similar to theLindstedt-Poincaremethod [26]These periodic solutions arefirstly presented by Richardson [27 28] and have been widelyquoted and used To obtain the periodic solutions around 119871

1

a new rotating coordinate system (1198711 119894 119895 119896) is defined with

the origin set at 1198711and the distance 120574 between the Earth and

1198711as the new unit of length which is also shown in Figure 1

The relationship between the (119874119883 119884 119885) reference frame andthe (119871

1 119894 119895 119896) reference frame is as follows

119894 =119883 minus (1 minus 120583 minus 120574)

120574 119895 =

119884

120574 119896 =

119885

120574 (2)

where 120574 = 0010010905 [27]

The equations of theHalo orbit to the third order are givenby

119909119903(119905) = 119886

211198602

119909+ 119886221198602

119911minus 119860119909cos (120582119905)

+ (119886231198602

119909minus 119886241198602

119911) cos (2120582119905)

+ (119886311198603minus 119886321198601199091198602

119911) cos (3120582119905)

119910119903(119905) = (119860

119910+ 119887331198603

119909+ (11988734

minus 11988735) 1198601199091198602

119911) sin (120582119905)

+ (119887211198602

119909minus 119887221198602

119911) sin (2120582119905)

+ (119887311198603

119909minus 119887321198601199091198602

119911) sin (3120582119905)

119911119903(119905) = minus 3119889

21119860119909119860119911+ 119860119911cos (120582119905)

+ 11988921119860119909119860119911cos (2120582119905)

+ (119889321198601199111198602

119909minus 119889311198603

119911) cos (3120582119905)

(3)

where 119909119903(119905) 119910119903(119905) and 119911

119903(119905) are the coordinates along 119894 119895 and

119896 axes The Halo orbit designed for ISEE-3 mission [27] isselected as one of the target station-keeping orbits The otherreference orbit is selected with higher vertical amplitudeThevalues of the various constants in (2) of both Halo orbits aregiven in Table 1

3 Introduction of ADRC

Due to complexity of modern systems more attention hasbeen paid on data-driven control scheme recently [29 30]The ADRC is a typical data-driven control It inherits fromPID using the error driven rather than model-based controllaw to eliminate errors

Hanrsquos ADRC consists of three parts a nonlinear trackingdifferentiator (TD) [31] which is used to arrange the idealtransient process of the system an extended state observer(ESO) [32] which could estimate the disturbance fromthe system output and then the ADRC compensates thedisturbances according to estimated values and a nonlinearstate error feedback (NLSEF) [24] which is used to get thecontrol input of the system

Consider the following system

1199091= 119910

1= 1199092

2= 119891 (119909

1 1199092 119908 (119905) 119905) + 119887119906

(4)

where 119910 is the output variable 119906 is the control input 119887 ismagnification factor and 120596(119905) is the external disturbance119891(1199091 1199092 119908(119905) 119905) includes three parts modeling dynamics

uncertain dynamics and disturbance The ADRC approachmakes an effort to compensate for the unknown dynamicsand the external disturbances in real time without an explicitmathematical expression

4 Mathematical Problems in Engineering

Table 1The values of the various constants in (2) for ISEE-3 (119885 = 110000 km) orbit and Halo orbit (119885 = 800000 km) near 1198711libration point

of the Sun-Earth system For ISEE-3 (119885 = 110000 km) orbit and for Halo orbit (119885 = 800000 km)

Parameters Value Parameters Value Parameters Value120582 208645 119896 322927 119886

21209270

11988622

248298 times 10minus1

11988623

minus905965 times 10minus1

11988624

minus104464 times 10minus1

11988631

793820 times 10minus1

11988632

826854 times 10minus2

11988721

minus492446 times 10minus1

11988722

607465 times 10minus2

11988731

885701 times 10minus1

11988732

230198 times 10minus2

11988733

minus284508 11988734

minus230206 11988735

minus18703711988921

minus346865 times 10minus1

11988931

190439 times 10minus2

11988932

398095 times 10minus1

31 TD The TD has the ability to track the given inputreference signalwith quick response andnoovershoot by pro-viding transition process for expected input V and differentialtrajectory of set value that is V

1 and its differential V

2

One feasible second-order TD can be designed as [31]

V1= V2

V2= fhan (V

1minus V (119905) V

2 119903 ℎ0)

(5)

where V(119905) denotes the control objective 119903 is speed factor anddecides tracking speed ℎ

0is filtering factor and fhan(V

1minus

V(119905) V2 119903 ℎ0) is as follows

119889 = 119903ℎ2

0

1198860= ℎ0V2

119910 = (V1minus V (119905)) + 119886

0

1198861= radic119889 (119889 + 8 |119889|)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119904119910=sign (119910 + 119889) minus sign (119910 minus 119889)

2

119886 = (1198860+ 119910 minus 119886

2) 119904119910+ 1198862

119904119886=sign (119886 + 119889) minus sign (119886 minus 119889)

2

fhan (V1minus V (119905) V

2 119903 ℎ0)

= minus119903 (119886

119889minus sign (119886)) 119904

119886minus 119903 sign (119886)

(6)

32 ESO ESO is used to estimate 119891(1199091 1199092 119908(119905) 119905) in real

time and tomake adjustments at each sampling point in a dig-ital controller An augmented variable 119909

3= 119891(119909

1 1199092 119908(119905) 119905)

is introduced in (4) Using 1199111 1199112 and 119911

3to estimate 119909

1 1199092

and 1199093 respectively a nonlinear observer is designed as [32]

119890 = 1199111minus 119910

1= 1199112minus 1205731119890

2= 1199113minus 1205732fal (119890 120572

1 120575) + 119887119906

3= minus1205733fal (119890 120572

2 120575)

(7)

where 1205731 1205732 and 120573

3are observer gains 119890 is the error and

fal(119909 120572 120575) is as follows

fal (119909 120572 120575) =

119909

1205751minus120572 |119909| le 120575

sign (119909) |119909|120572 |119909| gt 120575

(8)

33 NLSEF NLSEF generates control voltage 119906 for system byusing the errors between the output of ESO and TD

1198901= V1minus 1199111

1198902= V2minus 1199112

(9)

A nonlinear combination of errors signal can be con-structed as [24]

1199060= minusfhan (119890

1 1198881198902 120572 ℎ1) (10)

where the nonlinear coefficient 120572 is selected as 0 lt 120572 lt 1 and119888 is the proportional coefficients

The controller is designed as

119906 = 1199060minus1199113

119887 (11)

4 ADRSC for Station-Keeping

As is mentioned in Section 2 the reference Halo orbitaround collinear libration orbit is inherently unstable andstation-keeping control approach is designed under modeluncertainties and disturbances In this paper we extend theADRC for the ADRSCThe structure of ADRSC algorithm ispresented in Figure 2 ADRSC will be proposed and analyzedin this section from the following three aspects discrete TDdiscrete ESO and discrete NLSEF

41 Discrete TD for the Tracking of the Reference Orbit andthe SpacecraftTrajectory Assuming that only position signalsof spacecraft can be detected and used for feedback thereare two different kinds of discrete TDs designed for station-keeping control

411 TD I It is designed for reference Halo orbit tracking

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 119903119894 ℎ119894)

(12)

Mathematical Problems in Engineering 5

ESO

u

w

minus

minusminus

+Xr(t)r1

r

12

2 e2

e1

z1z2z3

u0 X(t)TD I

TD II

NLSEFSpacecraftnonlineardynamic

Disturbingforces

xr(t) =[xr(t) yr(t) zr(t)]T

Figure 2 The structure of ADRSC algorithm

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively V

119894(119896) denotes the reference orbit TD I provides

transition process for nominal orbit V119894and differential trajec-

tory of set position value that is V1198941and its differential V

1198942

412 TD II It is designed for flight position tracking andestimating of the velocity of the spacecraft

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 1199030119894 ℎ0119894)

(13)

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively For instance state vector [V11 V

12]119879 denotes the

estimation of [119883 ]119879 which is the position and velocity

vector of spacecraft trajectory along119883 axes V119894(119896) denotes the

flight trajectory of spacecraft And 1199030119894is the tracking speed

parameter and ℎ0119894is the filter factor which makes an effort

of filtering While the integration step is fixed increasingthe filtering factor will make the effort of filter better Theoutputs of TD II which are the spacecraft flight position andvelocity provide the inputs of ESO for feedback as describedin Figure 2

42 Discrete ESO for Estimating the Nonlinear DynamicsModel Uncertainties and Unmodeled Disturbance The equa-tions of station-keeping of Halo orbits using ADRSC can begenerally defined as

1198941= 1199091198942

1198942= 119892119894(X X) + 119908

119894+ 119906119894

119910 = 1199091198941

(14)

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectivelyX = [11990911 119909

2111990931]119879 denotes the position vector

of the spacecraft w = [1199081 1199082

1199083]119879 denotes the disturbance

vector and u = [1199061 1199062

1199063]119879 denotes the vector of station-

keeping control input g(XX) = [1198921 1198922

1198923]119879 represents

the uncertain dynamics vector respectively Note that (1)will be simulated as the real flight dynamic mode Hence

robust performance will be presented under such formationof dynamic model uncertainties

It is necessary to point out that a distinct improvementcan be obtained using ADRSC for station-keeping since thedynamic model does not need to be expressively knownwhich is different with the aforementioned studies DefineF(X X) = g(X X) + w In fact in the context of feedbackcontrol F(X X) is something to be overcome by the controlsignal and it is therefore denoted as the ldquototal disturbancerdquo Atthis point F(X X) is extended as an additional state variablethat is

1199091198943= 119865119894(X X) (15)

where 1198651 1198652 and 119865

3denote the corresponding axis com-

ponents of F(X X) respectively and let F(X X) = a(X X)where a(X X) is unknown

One can rewrite (14) as follows

1198941= 1199091198942

1198942= 1199091198943+ 119906119894

1198943= 119886119894

119910 = 1199091198941

(16)

where 1199091198941 1199091198942 and 119909

1198943represent the position velocity and

119865119894(X X) respectively 119894 = 1 2 3 Then one can use the

following discrete nonlinear observer 1199111198941 1199111198942 1199111198943 to estimate

state vector 1199091198941 1199091198942 1199091198943

119890119894= 119911119894minus 119910119894

1199111198941(119896 + 1) = 119911

1198941(119896) + ℎ times (119911

1198942(119896) minus 120573

1198941119890)

1199111198942(119896 + 1)

= 1199111198942(119896) + ℎ times (119891

0119894+ 1199111198943(119896) minus 120573

1198942times fal (119890 05 ℎ))

+ 119906 (119894)

1199111198943(119896 + 1) = 119911

1198943(119896) minus ℎ times 120573

1198943times fal (119890 025 ℎ)

(17)

43 Discrete NLSEF for Station-Keeping Since the statesunmodeled dynamics and disturbances have been estimatedby ESO presented in Section 42 state errors between the

6 Mathematical Problems in Engineering

output of ESO and TD I are combined for feedback controllaw large errors corresponding to low gains and small errorscorresponding to high gains

119906119894(119896 + 1) = minusfhan (119890

1198941 1198881198941198901198942 120572119894 ℎ119894) minus 1199111198943(119896) (18)

where 119888119894 120572119894 ℎ119894 are the NLSEF parameters described in (10)

1198901198941and 119890

1198942are the state errors described in (9) 119911

1198943denotes

unmodeled dynamics and disturbances 119865119894(X X) which is

estimated from (17)The parameter ℎ

119894of NLSEF which is also named filter

factor of fhan makes great contribution in NLSEF controlperformance With the integration step fixed ℎ

119894must be

selected properly It is noted that larger errors correspondingto larger ℎ

119894and smaller errors corresponding to smaller ℎ

119894will

make a better performanceIn this paper a parameter self-turning approach is firstly

proposed for ℎ119894selection as follows

ℎ119894=

119896log10radic

3

sum

119894=1

1198902

1198941+ ℎ0 radic

3

sum

119894=1

1198902

1198941gt 120585

119896log10120585 + ℎ0 radic

3

sum

119894=1

1198902

1198941le 120585

(19)

where the parameters 119896 ℎ0 and 120585 of self-turning ℎ

119894can

be easily chosen in practice as will be presented in thesimulation

44 Disturbing Forces In any actual mission the pertur-bation factors and the injection errors coupled with theinherent unstable nature of the Halo orbits around thecollinear libration points will cause a spacecraft to drift fromthe periodic reference orbit [33] A robust performance inthe presence of disturbance can be obtained by ADRSC andthe model of disturbance is not necessary for ADRSC as longas the disturbance is bounded Physically speaking deep-space disturbances are always bounded [21] To illustrate thecapability of ADRSC to reject unknowndisturbances hereweintroduce the perturbative forces due to the eccentric natureof the Earthrsquos orbit and the solar radiation pressure (SRP)disturbance

441 The Perturbative Forces due to the Eccentric Nature ofthe Earthrsquos Orbit The most important perturbative effects inthe CR3BP are the eccentric nature of the Earthrsquos orbit andthe gravitational force of the moon [3 28] In this paper thelargest perturbative force per unit mass dENE of spacecraftdue to the eccentric nature of the Earthrsquos orbit is taken intoaccount which has been given approximately in [3]

1003816100381610038161003816dENE1003816100381610038161003816 = 119866119872Earth

100381610038161003816100381610038161003816100381610038161003816

1

1199032119888

minus1

1199032119890

100381610038161003816100381610038161003816100381610038161003816

(20)

where 119903119890is the position vector of the Earth at the pericenter

of the Earthrsquos elliptical orbit from the libration point 119903119888is the

position vector similar to a circular orbit

442 SRP Disturbance In the deep-space mission SRP isanother disturbance to account for Here we adopt a widelyused model [34] According to this model the disturbanceacceleration to SRP is

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

2(_r1sdot n)2

n (21)

where 120573 is a parameter that depends on the coefficient ofreflectivity the area andmass of the spacecraft the solar fluxand the speed of light

_r1= r1r1 and n is the attitude

vector of the spacecraft in the rotating frame To simplify (21)we assume that n =

_r1 which yields

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

3r1 (22)

5 Simulation Results and Analysis

51 Simulation Scenario and Numerical Values To comparethe performance of ADRSC with linearization method thesimulation of an LQR controller [1] based on LTI systemfrom local linearization around 119871

1point of nonlinear system

equation (1) is also carried out In order to compare theperformance of ADRSC between low and large amplitudeorbits we select two orbits for simulation one orbit for theISEE-3 mission with vertical displacement 110000 km andthe other of 800000 km

The control forces for station-keeping are provided byan ionic engine with the maximum thrust 60mN whichis currently commercially available [35] The mass of thespacecraft is designed as119898 = 500 kg

The disturbing forces described in 44 are considered inthe simulation The eccentricity of Earth orbit 119890 = 0016675And the SRP parameter is calculated assuming that the solarflux is 119865

119904= 1358Wm2 the speed of light is 119888 = 3 times 10

8msthe cross-sectional area is 119878 = 35m2 and the coefficient ofreflectivity is 119902 = 06 which results in 120573 = 119878119865

119904(1 + 119902)(119898119888) =

5 times 10minus8ms2 [21] It is also assumed that the initial position

injection errors along the 119909 119910 and 119911 directions are 1000 kmand the velocity injection errors are 1ms The MATLABSimulink software is used for simulation The time step ℎ isset as 00001 TU for better accuracy of the spacecraft Thesimulation time is chosen as 1800 days almost 10 periods ofHalo motion about 119871

1libration point

In the simulation considering the system is not full statefeedback the TD is designed to track the spacecraft flightposition and obtain the velocity as well Figure 3 illustratesthe TD II tracking result of the spacecraft flight In Figure 3FP and FV denote the spacecraft flight position and velocityTP and TV denote the TD tracking position and velocityrespectively Figure 3 presents the accurately tracking ofposition signal and velocity signal along the 119909 119910 and 119911 axeswhich will be used for feedback control

52 ISEE-3 Halo Orbit (119885 = 110 000 km)

521 Related Parameters In dimensionless coordinates theinitial orbit parameters without injection errors of ISEE-3 are

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

ISEE-3 (110000 km) 0988872932669558 0 8109335378305802119890 minus 04 0 0008853797264729 0Halo (800000 km) 0989390221855232 0 0006248517078177 0 0012547011257083 0

given in Table 2 (in unit of AU for position and in unit ofAUTU for velocity)The parameters of ADRSC of ISEE-3 aregiven as follows 119903

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

522 Simulation Results Figures 4 and 5 illustrate thestation-keeping control results for ISEE-3 with orbit injectionerrors as well as disturbing forces which show that preferablestation-keeping accuracy can be obtained using ADRSC

through 60mNengine FromFigures 5(a) and 5(b) the errorsof relative position and relative velocity drop down veryquickly and steadily to zero within 5 days compared withLQR controller 14 days in Figures 6(a) and 6(b) The positionerrors along 119909 119910 and 119911 directions can be kept within 1 kmwhich is very small compared with the orbit altitude of theISEE-3 mission (110000 km) Also as shown in Figure 5(c)the control forces begin at 60mN to drive the relative positionerrors to zero but quickly reduce to the steady progressin 5 days Hence these results illustrate the effectiveness ofthe ADRSC of unstable Halo orbits near collinear librationpoints

In order to describe the results of ADRSCmore accuratelywhen the spacecraft moves steady on the Halo orbit here

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

we introduce the popular evaluation parameters for station-keeping [3 12 33] the velocity incrementsΔ119881

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

10038161003816100381610038161199061198941003816100381610038161003816 d119905 (119894 = 119909 119910 119911)

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

The velocity increment and themean absolute value of theposition errors of ISEE-3 station-keeping with orbit injectionerrors and disturbance are given in Table 3 It can be seenthat themean absolute position error is 0389 kmwhich illus-trates the higher precision characteristic for the spacecraftstation-keeping mission with ADRSC comparing with thatof LQR controller 2965 km shown in Table 3 Meanwhilethe velocity increment of ADRSC under disturbing forceand injection error is 36567msT which is approximatelyequal to that of LQR controller 36705msT Note thatthere is a trade-off between accuracy of the spacecraft tofollow the Halo orbit and the thrust usage [3] The earlierapproaches which use impulsive maneuvers are designed

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

for station-keeping missions and do not have stringent orbit-tracking requirements While high accuracy orbit station-keeping is definitely required for current missions which usecontinuous low-thrust propulsion

53 Halo Orbit (119885 = 800 000 km)

531 Related Parameters The parameters of ADRSC of Haloorbit (119885 = 800 000 km) are the same as ADRSC of ISEE-3mission except the parameters of self-turning ℎ

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

532 Simulation Results Figures 7 and 8 illustrate theADRSC results for Halo orbit (119885 = 800 000 km) with orbitinjection errors as well as disturbing force From Figures

8(a) and 8(b) one can find that ADRSC controller can drivethe relative position errors and velocity errors along 119909 119910and 119911 steadily to zero within 75 days compared with LQRcontroller 25 days in Figures 9(a) and 9(b) During the steadyperiodic the position errors are kept within 2 km whichillustrate high accurately maintained ability of ADRSC ofhigh Halo orbit

The velocity increment and the mean absolute value ofthe position errors of both ADRSC and LQR are presentedin Table 3 The position error of ADRSC is 1146 km which isgreatly less than LQR controller 17084 km Meanwhile thecontrol consumption of ADRSC is 195248msT which isalmost equal to that of LQR controller 195394msT

54 Summary From the above analyses and control resultsbetween ADRSC and LQR we can find the following

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Table 3 The velocity increments and the mean absolute value of the position errors of ADRSC and LQR control

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

ADRSC ISEE-3 (110000 km) 24872 25948 6721 36567 0135 0343 0119 0389LQR ISEE-3 (110000 km) 25030 25991 6727 36705 1991 2126 0555 2965ADRSC Halo (800000 km) 160770 81501 75049 195248 0380 0607 0895 1146LQR Halo (800000 km) 160815 81703 75115 195394 14325 6516 6647 17084

(1) TheADRSCwhich is an error driven controlmethodis adequate for the station-keeping control of unstableorbits without any knowledge about the spacecraftdynamic model

(2) The ADRSC presents praiseworthy station-keepingperformance with orbit injection errors as well as

unmodeled disturbances such as the SRP and theperturbative forces due to the eccentric nature of theEarthrsquos orbit using an ionic engine with maximumthrust 60mN

(3) The ADRSC approach has better station-keepingcontrol ability and higher orbit maintenance accuracy

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

(4) TheADRSC has better robust performance comparedwith the general station-keeping method dependingon spacecraft dynamic model

There has been a great deal of commendable research onunstable libration point orbits station-keeping as mentionedbefore A major point of departure between ADRSC methodand earlier approaches to station-keeping is that ADRSC is anerror driven rather than model-based control law which caninherently get across the dependency on model accuracy as

well as the drawbacks of linearization With the combinationof TD ESO and NLSEF the unmodeled disturbances as wellas the unmodeled system dynamic can be compensated inreal time Thus fast response-time requirement and highaccuracy of orbit maintenance requirement can be satisfiedby ADRSC

It is important to keep in mind not only the trackingaccuracy but also the robustness of the station-keepingcontroller as the space environment cannot be accuratelymodeled as well as the internal and external disturbanceADRSC extends the unmodeled spacecraft dynamic andthe disturbance as a state which can be estimated from

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

Figure 8 The relative position errors velocity errors and control input of ADRSC control of Halo orbit

ESO and then ldquorejectedrdquo from nonlinear feedback controlThus no matter the spacecraft system is model known orunknown linear or nonlinear time invariant or time variantwith disturbance or without ADRSC is able to show desiredperformance

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

output of nonlinear spacecraft dynamic system rather thanprecise nonlinear dynamic model or linearization The non-linear simulation results indicate that ADRSC is adequate forthe station-keeping of unstable orbits that take into account

the system uncertainties initial injection errors SRP andperturbations of the eccentric nature of the Earthrsquos orbitWith this ADRSC method the system control performanceis improved significantly by comparing with a general LQRmethod in the simulation The thrust required by the controllaw is also reasonable and can be implemented using acontinuous low-thrust propulsion device such as an ionengine ADRSC can be qualified for future missions whichrequire better performance more robustness and higherstation-keeping accuracy It is anticipated that this methodwill work for other unstable orbits such as Lissajous andHalo orbits near other collinear points as well as the for-mation flying on these orbits which are the focus for futurestudy

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Nature Science Founda-tion of China under Grant no 61004017 and by the Polish-Norwegian Research Programme in the frame of ProjectContract no Pol-Nor200957472013 The authors wouldlike to thank the editors and the anonymous reviewers fortheir keen and insightful comments which greatly improvethe paper

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

[2] K C Howell ldquoThree-dimensional periodic halo orbitsrdquo Celes-tial Mechanics vol 32 no 1 pp 53ndash71 1984

[3] J E Kulkarni M E Campbell and G E Dullerud ldquoStabiliza-tion of spacecraft flight in halo orbits an 119867

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

[4] M Xin S N Balakrishnan and H J Pernicka ldquoMultiplespacecraft formation control with 120579-D methodrdquo IET ControlTheory and Applications vol 1 no 2 pp 485ndash493 2007

[5] D Sheng X Yang and H R Karimi ldquoRobust control forautonomous spacecraft evacuation with model uncertainty and

14 Mathematical Problems in Engineering

upper bound of performance with constraintsrdquo MathematicalProblems in Engineering vol 2013 Article ID 589381 2013

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

[7] Z Li M Liu H R Karimi and X Cao ldquoObserver-based sta-bilization of spacecraft rendezvous with variable sampling andsensor nonlinearityrdquoMathematical Problems in Engineering vol2013 Article ID 902452 11 pages 2013

[8] Z LiM LiuH RKarimi andXCao ldquoSampled-data control ofspacecraft rendezvous with discontinuous lyapunov approachrdquoMathematical Problems in Engineering vol 2013 Article ID814271 10 pages 2013

[9] J V Breakwell ldquoInvestigation of halo satellite orbit controlrdquoTech Rep CR-132 858 NASA 1973

[10] J V Breakwell A A Kamel and M J Ratner ldquoStation-keepingfor a translunar communication stationrdquo Celestial Mechanicsvol 10 no 3 pp 357ndash373 1974

[11] J Erickson and A Glass ldquoImplementation of ISEE-3 trajectorycontrolrdquo in American Astronautical Society and American Insti-tute of Aeronautics and Astronautics Astrodynamics SpecialistConference Provincetown UK 1979

[12] D Cielaszyk and B Wie ldquoNew approach to halo orbit determi-nation and controlrdquo Journal of Guidance Control and Dynam-ics vol 19 no 2 pp 266ndash273 1996

[13] P Di Giamberardino and S Monaco ldquoOn halo orbits spacecraftstabilizationrdquo Acta Astronautica vol 38 no 12 pp 903ndash9251996

[14] P Gurfil and N J Kasdin ldquoStability and control of spacecraftformation flying in trajectories of the restricted three-bodyproblemrdquo Acta Astronautica vol 54 no 6 pp 433ndash453 2004

[15] C Simo G Gomez J Llibre R Martınez and J RodrıguezldquoOn the optimal station keeping control of halo orbitsrdquo ActaAstronautica vol 15 no 6 pp 391ndash397 1987

[16] G G Gomez J Llibre R Martınez and C Simo Dynamicsand Mission Design Near Libration Point Orbits vol 1 ofFundamentals The Case of Collinear Libration Points WorldScientific Singapore 2001

[17] G G Gomez Dynamics and Mission Design Near LibrationPoints vol 3 of Advanced Methods for Collinear Points WorldScientific 2001

[18] K C Howell and H J Pernicka ldquoStationkeeping method forlibration point trajectoriesrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 151ndash159 1993

[19] A Rahmani M-A Jalali and S Pourtakdoust ldquoOptimalapproach to halo orbit controlrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and Exhibit pp11ndash14 Austin Tex USA 2003

[20] B G Marchand and K C Howell ldquoControl strategies forformation flight in the vicinity of the libration pointsrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1210ndash12192005

[21] P Gurfil M Idan and N J Kasdin ldquoAdaptive neural control ofdeep-space formation flyingrdquo Journal of Guidance Control andDynamics vol 26 no 3 pp 491ndash501 2003

[22] X Bai and J L Junkins ldquoModified Chebyshev-Picard iterationmethods for station-keeping of translunar halo orbitsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 926158 18pages 2012

[23] J Han ldquoControl theory is it a model analysis or direct controlapproachrdquo Journal of Systems Science andMathematical Sciencevol 9 no 4 pp 328ndash335 1989

[24] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[25] D Sun ldquoComments on active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 54 no 6 pp3428ndash3429 2007

[26] G E Dullerud and F Paganini A Course in Robust ControlTheory vol 6 Springer New York NY USA 2000

[27] D L Richardson ldquoHalo orbit formulation for the isee-3 mis-sionrdquo Journal of Guidance Control and Dynamics vol 3 no 6pp 543ndash548 1980

[28] D L Richardson ldquoAnalytic construction of periodic orbitsabout the collinear pointsrdquo Celestial Mechanics vol 22 no 3pp 241ndash253 1980

[29] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark tennesseeeastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

[30] S Yin S XDingAHAbandan Sari andHHao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[31] J Han and W Wang ldquoNonlinear tracking-differentiatorrdquo Jour-nal of Systems Science and Mathematical Science vol 14 no 2pp 177ndash183 1994

[32] J Han ldquoThe ldquoextended state observerrdquo of a class of uncertainsystemsrdquo Control and Decision vol 10 no 1 pp 85ndash88 1995

[33] H Peng J Zhao Z Wu and W Zhong ldquoOptimal periodiccontroller for formation flying on libration point orbitsrdquo ActaAstronautica vol 69 no 7-8 pp 537ndash550 2011

[34] P Gurfil ldquoControl-theoretic analysis of low-thrust orbital trans-fer using orbital elementsrdquo Journal of Guidance Control andDynamics vol 26 no 6 pp 979ndash983 2003

[35] M Martinez-Sanchez and J E Pollard ldquoSpacecraft electricpropulsion-an overviewrdquo Journal of Propulsion and Power vol14 no 5 pp 688ndash699 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

EarthX

Z

Y

OR

Spacecraft

j

i

km2

m1

r2r1

L1 120574

120596

Sun

Figure 1 Restricted three-body problem

located at (minus120583 0 0) and1198982is located at (1minus120583 0 0) The well-

known equations of motion for CR3BP can be written in thedimensionless form [1]

= 2 + 119883 minus(1 minus 120583) (119883 + 120583)

1199033

1

minus120583 (119883 minus (1 minus 120583))

1199033

2

= minus2 + 119884 minus(1 minus 120583)119884

1199033

1

minus120583119884

1199033

2

= minus(1 minus 120583)119885

1199033

1

minus120583119885

1199033

2

(1)

where the dot represents time derivative in the rotatingframe and the distance 119903

1= radic(119883 + 120583)

2+ 1198842 + 1198852 and 119903

2=

radic(119883 minus (1 minus 120583))2+ 1198842 + 1198852

22 Periodic Reference Orbits We can find libration pointsof the Sun-Earth system denoted by 119871

1ndash1198715 by setting all

derivatives in (1) to zero For the Sun-Earth system 1198711is

located at119883 = 0989986055 It has been proved that the threecollinear points 119871

1 1198712 and 119871

3 are unstable but controllable

[1] And a family of periodic orbits and quasi-periodic orbitsexist around the collinear points [2]

Periodic solutions of the nonlinear equations of motioncan be constructed using the method of successive approx-imations in conjunction with a technique similar to theLindstedt-Poincaremethod [26]These periodic solutions arefirstly presented by Richardson [27 28] and have been widelyquoted and used To obtain the periodic solutions around 119871

1

a new rotating coordinate system (1198711 119894 119895 119896) is defined with

the origin set at 1198711and the distance 120574 between the Earth and

1198711as the new unit of length which is also shown in Figure 1

The relationship between the (119874119883 119884 119885) reference frame andthe (119871

1 119894 119895 119896) reference frame is as follows

119894 =119883 minus (1 minus 120583 minus 120574)

120574 119895 =

119884

120574 119896 =

119885

120574 (2)

where 120574 = 0010010905 [27]

The equations of theHalo orbit to the third order are givenby

119909119903(119905) = 119886

211198602

119909+ 119886221198602

119911minus 119860119909cos (120582119905)

+ (119886231198602

119909minus 119886241198602

119911) cos (2120582119905)

+ (119886311198603minus 119886321198601199091198602

119911) cos (3120582119905)

119910119903(119905) = (119860

119910+ 119887331198603

119909+ (11988734

minus 11988735) 1198601199091198602

119911) sin (120582119905)

+ (119887211198602

119909minus 119887221198602

119911) sin (2120582119905)

+ (119887311198603

119909minus 119887321198601199091198602

119911) sin (3120582119905)

119911119903(119905) = minus 3119889

21119860119909119860119911+ 119860119911cos (120582119905)

+ 11988921119860119909119860119911cos (2120582119905)

+ (119889321198601199111198602

119909minus 119889311198603

119911) cos (3120582119905)

(3)

where 119909119903(119905) 119910119903(119905) and 119911

119903(119905) are the coordinates along 119894 119895 and

119896 axes The Halo orbit designed for ISEE-3 mission [27] isselected as one of the target station-keeping orbits The otherreference orbit is selected with higher vertical amplitudeThevalues of the various constants in (2) of both Halo orbits aregiven in Table 1

3 Introduction of ADRC

Due to complexity of modern systems more attention hasbeen paid on data-driven control scheme recently [29 30]The ADRC is a typical data-driven control It inherits fromPID using the error driven rather than model-based controllaw to eliminate errors

Hanrsquos ADRC consists of three parts a nonlinear trackingdifferentiator (TD) [31] which is used to arrange the idealtransient process of the system an extended state observer(ESO) [32] which could estimate the disturbance fromthe system output and then the ADRC compensates thedisturbances according to estimated values and a nonlinearstate error feedback (NLSEF) [24] which is used to get thecontrol input of the system

Consider the following system

1199091= 119910

1= 1199092

2= 119891 (119909

1 1199092 119908 (119905) 119905) + 119887119906

(4)

where 119910 is the output variable 119906 is the control input 119887 ismagnification factor and 120596(119905) is the external disturbance119891(1199091 1199092 119908(119905) 119905) includes three parts modeling dynamics

uncertain dynamics and disturbance The ADRC approachmakes an effort to compensate for the unknown dynamicsand the external disturbances in real time without an explicitmathematical expression

4 Mathematical Problems in Engineering

Table 1The values of the various constants in (2) for ISEE-3 (119885 = 110000 km) orbit and Halo orbit (119885 = 800000 km) near 1198711libration point

of the Sun-Earth system For ISEE-3 (119885 = 110000 km) orbit and for Halo orbit (119885 = 800000 km)

Parameters Value Parameters Value Parameters Value120582 208645 119896 322927 119886

21209270

11988622

248298 times 10minus1

11988623

minus905965 times 10minus1

11988624

minus104464 times 10minus1

11988631

793820 times 10minus1

11988632

826854 times 10minus2

11988721

minus492446 times 10minus1

11988722

607465 times 10minus2

11988731

885701 times 10minus1

11988732

230198 times 10minus2

11988733

minus284508 11988734

minus230206 11988735

minus18703711988921

minus346865 times 10minus1

11988931

190439 times 10minus2

11988932

398095 times 10minus1

31 TD The TD has the ability to track the given inputreference signalwith quick response andnoovershoot by pro-viding transition process for expected input V and differentialtrajectory of set value that is V

1 and its differential V

2

One feasible second-order TD can be designed as [31]

V1= V2

V2= fhan (V

1minus V (119905) V

2 119903 ℎ0)

(5)

where V(119905) denotes the control objective 119903 is speed factor anddecides tracking speed ℎ

0is filtering factor and fhan(V

1minus

V(119905) V2 119903 ℎ0) is as follows

119889 = 119903ℎ2

0

1198860= ℎ0V2

119910 = (V1minus V (119905)) + 119886

0

1198861= radic119889 (119889 + 8 |119889|)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119904119910=sign (119910 + 119889) minus sign (119910 minus 119889)

2

119886 = (1198860+ 119910 minus 119886

2) 119904119910+ 1198862

119904119886=sign (119886 + 119889) minus sign (119886 minus 119889)

2

fhan (V1minus V (119905) V

2 119903 ℎ0)

= minus119903 (119886

119889minus sign (119886)) 119904

119886minus 119903 sign (119886)

(6)

32 ESO ESO is used to estimate 119891(1199091 1199092 119908(119905) 119905) in real

time and tomake adjustments at each sampling point in a dig-ital controller An augmented variable 119909

3= 119891(119909

1 1199092 119908(119905) 119905)

is introduced in (4) Using 1199111 1199112 and 119911

3to estimate 119909

1 1199092

and 1199093 respectively a nonlinear observer is designed as [32]

119890 = 1199111minus 119910

1= 1199112minus 1205731119890

2= 1199113minus 1205732fal (119890 120572

1 120575) + 119887119906

3= minus1205733fal (119890 120572

2 120575)

(7)

where 1205731 1205732 and 120573

3are observer gains 119890 is the error and

fal(119909 120572 120575) is as follows

fal (119909 120572 120575) =

119909

1205751minus120572 |119909| le 120575

sign (119909) |119909|120572 |119909| gt 120575

(8)

33 NLSEF NLSEF generates control voltage 119906 for system byusing the errors between the output of ESO and TD

1198901= V1minus 1199111

1198902= V2minus 1199112

(9)

A nonlinear combination of errors signal can be con-structed as [24]

1199060= minusfhan (119890

1 1198881198902 120572 ℎ1) (10)

where the nonlinear coefficient 120572 is selected as 0 lt 120572 lt 1 and119888 is the proportional coefficients

The controller is designed as

119906 = 1199060minus1199113

119887 (11)

4 ADRSC for Station-Keeping

As is mentioned in Section 2 the reference Halo orbitaround collinear libration orbit is inherently unstable andstation-keeping control approach is designed under modeluncertainties and disturbances In this paper we extend theADRC for the ADRSCThe structure of ADRSC algorithm ispresented in Figure 2 ADRSC will be proposed and analyzedin this section from the following three aspects discrete TDdiscrete ESO and discrete NLSEF

41 Discrete TD for the Tracking of the Reference Orbit andthe SpacecraftTrajectory Assuming that only position signalsof spacecraft can be detected and used for feedback thereare two different kinds of discrete TDs designed for station-keeping control

411 TD I It is designed for reference Halo orbit tracking

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 119903119894 ℎ119894)

(12)

Mathematical Problems in Engineering 5

ESO

u

w

minus

minusminus

+Xr(t)r1

r

12

2 e2

e1

z1z2z3

u0 X(t)TD I

TD II

NLSEFSpacecraftnonlineardynamic

Disturbingforces

xr(t) =[xr(t) yr(t) zr(t)]T

Figure 2 The structure of ADRSC algorithm

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively V

119894(119896) denotes the reference orbit TD I provides

transition process for nominal orbit V119894and differential trajec-

tory of set position value that is V1198941and its differential V

1198942

412 TD II It is designed for flight position tracking andestimating of the velocity of the spacecraft

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 1199030119894 ℎ0119894)

(13)

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively For instance state vector [V11 V

12]119879 denotes the

estimation of [119883 ]119879 which is the position and velocity

vector of spacecraft trajectory along119883 axes V119894(119896) denotes the

flight trajectory of spacecraft And 1199030119894is the tracking speed

parameter and ℎ0119894is the filter factor which makes an effort

of filtering While the integration step is fixed increasingthe filtering factor will make the effort of filter better Theoutputs of TD II which are the spacecraft flight position andvelocity provide the inputs of ESO for feedback as describedin Figure 2

42 Discrete ESO for Estimating the Nonlinear DynamicsModel Uncertainties and Unmodeled Disturbance The equa-tions of station-keeping of Halo orbits using ADRSC can begenerally defined as

1198941= 1199091198942

1198942= 119892119894(X X) + 119908

119894+ 119906119894

119910 = 1199091198941

(14)

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectivelyX = [11990911 119909

2111990931]119879 denotes the position vector

of the spacecraft w = [1199081 1199082

1199083]119879 denotes the disturbance

vector and u = [1199061 1199062

1199063]119879 denotes the vector of station-

keeping control input g(XX) = [1198921 1198922

1198923]119879 represents

the uncertain dynamics vector respectively Note that (1)will be simulated as the real flight dynamic mode Hence

robust performance will be presented under such formationof dynamic model uncertainties

It is necessary to point out that a distinct improvementcan be obtained using ADRSC for station-keeping since thedynamic model does not need to be expressively knownwhich is different with the aforementioned studies DefineF(X X) = g(X X) + w In fact in the context of feedbackcontrol F(X X) is something to be overcome by the controlsignal and it is therefore denoted as the ldquototal disturbancerdquo Atthis point F(X X) is extended as an additional state variablethat is

1199091198943= 119865119894(X X) (15)

where 1198651 1198652 and 119865

3denote the corresponding axis com-

ponents of F(X X) respectively and let F(X X) = a(X X)where a(X X) is unknown

One can rewrite (14) as follows

1198941= 1199091198942

1198942= 1199091198943+ 119906119894

1198943= 119886119894

119910 = 1199091198941

(16)

where 1199091198941 1199091198942 and 119909

1198943represent the position velocity and

119865119894(X X) respectively 119894 = 1 2 3 Then one can use the

following discrete nonlinear observer 1199111198941 1199111198942 1199111198943 to estimate

state vector 1199091198941 1199091198942 1199091198943

119890119894= 119911119894minus 119910119894

1199111198941(119896 + 1) = 119911

1198941(119896) + ℎ times (119911

1198942(119896) minus 120573

1198941119890)

1199111198942(119896 + 1)

= 1199111198942(119896) + ℎ times (119891

0119894+ 1199111198943(119896) minus 120573

1198942times fal (119890 05 ℎ))

+ 119906 (119894)

1199111198943(119896 + 1) = 119911

1198943(119896) minus ℎ times 120573

1198943times fal (119890 025 ℎ)

(17)

43 Discrete NLSEF for Station-Keeping Since the statesunmodeled dynamics and disturbances have been estimatedby ESO presented in Section 42 state errors between the

6 Mathematical Problems in Engineering

output of ESO and TD I are combined for feedback controllaw large errors corresponding to low gains and small errorscorresponding to high gains

119906119894(119896 + 1) = minusfhan (119890

1198941 1198881198941198901198942 120572119894 ℎ119894) minus 1199111198943(119896) (18)

where 119888119894 120572119894 ℎ119894 are the NLSEF parameters described in (10)

1198901198941and 119890

1198942are the state errors described in (9) 119911

1198943denotes

unmodeled dynamics and disturbances 119865119894(X X) which is

estimated from (17)The parameter ℎ

119894of NLSEF which is also named filter

factor of fhan makes great contribution in NLSEF controlperformance With the integration step fixed ℎ

119894must be

selected properly It is noted that larger errors correspondingto larger ℎ

119894and smaller errors corresponding to smaller ℎ

119894will

make a better performanceIn this paper a parameter self-turning approach is firstly

proposed for ℎ119894selection as follows

ℎ119894=

119896log10radic

3

sum

119894=1

1198902

1198941+ ℎ0 radic

3

sum

119894=1

1198902

1198941gt 120585

119896log10120585 + ℎ0 radic

3

sum

119894=1

1198902

1198941le 120585

(19)

where the parameters 119896 ℎ0 and 120585 of self-turning ℎ

119894can

be easily chosen in practice as will be presented in thesimulation

44 Disturbing Forces In any actual mission the pertur-bation factors and the injection errors coupled with theinherent unstable nature of the Halo orbits around thecollinear libration points will cause a spacecraft to drift fromthe periodic reference orbit [33] A robust performance inthe presence of disturbance can be obtained by ADRSC andthe model of disturbance is not necessary for ADRSC as longas the disturbance is bounded Physically speaking deep-space disturbances are always bounded [21] To illustrate thecapability of ADRSC to reject unknowndisturbances hereweintroduce the perturbative forces due to the eccentric natureof the Earthrsquos orbit and the solar radiation pressure (SRP)disturbance

441 The Perturbative Forces due to the Eccentric Nature ofthe Earthrsquos Orbit The most important perturbative effects inthe CR3BP are the eccentric nature of the Earthrsquos orbit andthe gravitational force of the moon [3 28] In this paper thelargest perturbative force per unit mass dENE of spacecraftdue to the eccentric nature of the Earthrsquos orbit is taken intoaccount which has been given approximately in [3]

1003816100381610038161003816dENE1003816100381610038161003816 = 119866119872Earth

100381610038161003816100381610038161003816100381610038161003816

1

1199032119888

minus1

1199032119890

100381610038161003816100381610038161003816100381610038161003816

(20)

where 119903119890is the position vector of the Earth at the pericenter

of the Earthrsquos elliptical orbit from the libration point 119903119888is the

position vector similar to a circular orbit

442 SRP Disturbance In the deep-space mission SRP isanother disturbance to account for Here we adopt a widelyused model [34] According to this model the disturbanceacceleration to SRP is

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

2(_r1sdot n)2

n (21)

where 120573 is a parameter that depends on the coefficient ofreflectivity the area andmass of the spacecraft the solar fluxand the speed of light

_r1= r1r1 and n is the attitude

vector of the spacecraft in the rotating frame To simplify (21)we assume that n =

_r1 which yields

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

3r1 (22)

5 Simulation Results and Analysis

51 Simulation Scenario and Numerical Values To comparethe performance of ADRSC with linearization method thesimulation of an LQR controller [1] based on LTI systemfrom local linearization around 119871

1point of nonlinear system

equation (1) is also carried out In order to compare theperformance of ADRSC between low and large amplitudeorbits we select two orbits for simulation one orbit for theISEE-3 mission with vertical displacement 110000 km andthe other of 800000 km

The control forces for station-keeping are provided byan ionic engine with the maximum thrust 60mN whichis currently commercially available [35] The mass of thespacecraft is designed as119898 = 500 kg

The disturbing forces described in 44 are considered inthe simulation The eccentricity of Earth orbit 119890 = 0016675And the SRP parameter is calculated assuming that the solarflux is 119865

119904= 1358Wm2 the speed of light is 119888 = 3 times 10

8msthe cross-sectional area is 119878 = 35m2 and the coefficient ofreflectivity is 119902 = 06 which results in 120573 = 119878119865

119904(1 + 119902)(119898119888) =

5 times 10minus8ms2 [21] It is also assumed that the initial position

injection errors along the 119909 119910 and 119911 directions are 1000 kmand the velocity injection errors are 1ms The MATLABSimulink software is used for simulation The time step ℎ isset as 00001 TU for better accuracy of the spacecraft Thesimulation time is chosen as 1800 days almost 10 periods ofHalo motion about 119871

1libration point

In the simulation considering the system is not full statefeedback the TD is designed to track the spacecraft flightposition and obtain the velocity as well Figure 3 illustratesthe TD II tracking result of the spacecraft flight In Figure 3FP and FV denote the spacecraft flight position and velocityTP and TV denote the TD tracking position and velocityrespectively Figure 3 presents the accurately tracking ofposition signal and velocity signal along the 119909 119910 and 119911 axeswhich will be used for feedback control

52 ISEE-3 Halo Orbit (119885 = 110 000 km)

521 Related Parameters In dimensionless coordinates theinitial orbit parameters without injection errors of ISEE-3 are

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

ISEE-3 (110000 km) 0988872932669558 0 8109335378305802119890 minus 04 0 0008853797264729 0Halo (800000 km) 0989390221855232 0 0006248517078177 0 0012547011257083 0

given in Table 2 (in unit of AU for position and in unit ofAUTU for velocity)The parameters of ADRSC of ISEE-3 aregiven as follows 119903

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

522 Simulation Results Figures 4 and 5 illustrate thestation-keeping control results for ISEE-3 with orbit injectionerrors as well as disturbing forces which show that preferablestation-keeping accuracy can be obtained using ADRSC

through 60mNengine FromFigures 5(a) and 5(b) the errorsof relative position and relative velocity drop down veryquickly and steadily to zero within 5 days compared withLQR controller 14 days in Figures 6(a) and 6(b) The positionerrors along 119909 119910 and 119911 directions can be kept within 1 kmwhich is very small compared with the orbit altitude of theISEE-3 mission (110000 km) Also as shown in Figure 5(c)the control forces begin at 60mN to drive the relative positionerrors to zero but quickly reduce to the steady progressin 5 days Hence these results illustrate the effectiveness ofthe ADRSC of unstable Halo orbits near collinear librationpoints

In order to describe the results of ADRSCmore accuratelywhen the spacecraft moves steady on the Halo orbit here

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

we introduce the popular evaluation parameters for station-keeping [3 12 33] the velocity incrementsΔ119881

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

10038161003816100381610038161199061198941003816100381610038161003816 d119905 (119894 = 119909 119910 119911)

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

The velocity increment and themean absolute value of theposition errors of ISEE-3 station-keeping with orbit injectionerrors and disturbance are given in Table 3 It can be seenthat themean absolute position error is 0389 kmwhich illus-trates the higher precision characteristic for the spacecraftstation-keeping mission with ADRSC comparing with thatof LQR controller 2965 km shown in Table 3 Meanwhilethe velocity increment of ADRSC under disturbing forceand injection error is 36567msT which is approximatelyequal to that of LQR controller 36705msT Note thatthere is a trade-off between accuracy of the spacecraft tofollow the Halo orbit and the thrust usage [3] The earlierapproaches which use impulsive maneuvers are designed

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

for station-keeping missions and do not have stringent orbit-tracking requirements While high accuracy orbit station-keeping is definitely required for current missions which usecontinuous low-thrust propulsion

53 Halo Orbit (119885 = 800 000 km)

531 Related Parameters The parameters of ADRSC of Haloorbit (119885 = 800 000 km) are the same as ADRSC of ISEE-3mission except the parameters of self-turning ℎ

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

532 Simulation Results Figures 7 and 8 illustrate theADRSC results for Halo orbit (119885 = 800 000 km) with orbitinjection errors as well as disturbing force From Figures

8(a) and 8(b) one can find that ADRSC controller can drivethe relative position errors and velocity errors along 119909 119910and 119911 steadily to zero within 75 days compared with LQRcontroller 25 days in Figures 9(a) and 9(b) During the steadyperiodic the position errors are kept within 2 km whichillustrate high accurately maintained ability of ADRSC ofhigh Halo orbit

The velocity increment and the mean absolute value ofthe position errors of both ADRSC and LQR are presentedin Table 3 The position error of ADRSC is 1146 km which isgreatly less than LQR controller 17084 km Meanwhile thecontrol consumption of ADRSC is 195248msT which isalmost equal to that of LQR controller 195394msT

54 Summary From the above analyses and control resultsbetween ADRSC and LQR we can find the following

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Table 3 The velocity increments and the mean absolute value of the position errors of ADRSC and LQR control

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

ADRSC ISEE-3 (110000 km) 24872 25948 6721 36567 0135 0343 0119 0389LQR ISEE-3 (110000 km) 25030 25991 6727 36705 1991 2126 0555 2965ADRSC Halo (800000 km) 160770 81501 75049 195248 0380 0607 0895 1146LQR Halo (800000 km) 160815 81703 75115 195394 14325 6516 6647 17084

(1) TheADRSCwhich is an error driven controlmethodis adequate for the station-keeping control of unstableorbits without any knowledge about the spacecraftdynamic model

(2) The ADRSC presents praiseworthy station-keepingperformance with orbit injection errors as well as

unmodeled disturbances such as the SRP and theperturbative forces due to the eccentric nature of theEarthrsquos orbit using an ionic engine with maximumthrust 60mN

(3) The ADRSC approach has better station-keepingcontrol ability and higher orbit maintenance accuracy

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

(4) TheADRSC has better robust performance comparedwith the general station-keeping method dependingon spacecraft dynamic model

There has been a great deal of commendable research onunstable libration point orbits station-keeping as mentionedbefore A major point of departure between ADRSC methodand earlier approaches to station-keeping is that ADRSC is anerror driven rather than model-based control law which caninherently get across the dependency on model accuracy as

well as the drawbacks of linearization With the combinationof TD ESO and NLSEF the unmodeled disturbances as wellas the unmodeled system dynamic can be compensated inreal time Thus fast response-time requirement and highaccuracy of orbit maintenance requirement can be satisfiedby ADRSC

It is important to keep in mind not only the trackingaccuracy but also the robustness of the station-keepingcontroller as the space environment cannot be accuratelymodeled as well as the internal and external disturbanceADRSC extends the unmodeled spacecraft dynamic andthe disturbance as a state which can be estimated from

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

Figure 8 The relative position errors velocity errors and control input of ADRSC control of Halo orbit

ESO and then ldquorejectedrdquo from nonlinear feedback controlThus no matter the spacecraft system is model known orunknown linear or nonlinear time invariant or time variantwith disturbance or without ADRSC is able to show desiredperformance

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

output of nonlinear spacecraft dynamic system rather thanprecise nonlinear dynamic model or linearization The non-linear simulation results indicate that ADRSC is adequate forthe station-keeping of unstable orbits that take into account

the system uncertainties initial injection errors SRP andperturbations of the eccentric nature of the Earthrsquos orbitWith this ADRSC method the system control performanceis improved significantly by comparing with a general LQRmethod in the simulation The thrust required by the controllaw is also reasonable and can be implemented using acontinuous low-thrust propulsion device such as an ionengine ADRSC can be qualified for future missions whichrequire better performance more robustness and higherstation-keeping accuracy It is anticipated that this methodwill work for other unstable orbits such as Lissajous andHalo orbits near other collinear points as well as the for-mation flying on these orbits which are the focus for futurestudy

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Nature Science Founda-tion of China under Grant no 61004017 and by the Polish-Norwegian Research Programme in the frame of ProjectContract no Pol-Nor200957472013 The authors wouldlike to thank the editors and the anonymous reviewers fortheir keen and insightful comments which greatly improvethe paper

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

[2] K C Howell ldquoThree-dimensional periodic halo orbitsrdquo Celes-tial Mechanics vol 32 no 1 pp 53ndash71 1984

[3] J E Kulkarni M E Campbell and G E Dullerud ldquoStabiliza-tion of spacecraft flight in halo orbits an 119867

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

[4] M Xin S N Balakrishnan and H J Pernicka ldquoMultiplespacecraft formation control with 120579-D methodrdquo IET ControlTheory and Applications vol 1 no 2 pp 485ndash493 2007

[5] D Sheng X Yang and H R Karimi ldquoRobust control forautonomous spacecraft evacuation with model uncertainty and

14 Mathematical Problems in Engineering

upper bound of performance with constraintsrdquo MathematicalProblems in Engineering vol 2013 Article ID 589381 2013

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

[7] Z Li M Liu H R Karimi and X Cao ldquoObserver-based sta-bilization of spacecraft rendezvous with variable sampling andsensor nonlinearityrdquoMathematical Problems in Engineering vol2013 Article ID 902452 11 pages 2013

[8] Z LiM LiuH RKarimi andXCao ldquoSampled-data control ofspacecraft rendezvous with discontinuous lyapunov approachrdquoMathematical Problems in Engineering vol 2013 Article ID814271 10 pages 2013

[9] J V Breakwell ldquoInvestigation of halo satellite orbit controlrdquoTech Rep CR-132 858 NASA 1973

[10] J V Breakwell A A Kamel and M J Ratner ldquoStation-keepingfor a translunar communication stationrdquo Celestial Mechanicsvol 10 no 3 pp 357ndash373 1974

[11] J Erickson and A Glass ldquoImplementation of ISEE-3 trajectorycontrolrdquo in American Astronautical Society and American Insti-tute of Aeronautics and Astronautics Astrodynamics SpecialistConference Provincetown UK 1979

[12] D Cielaszyk and B Wie ldquoNew approach to halo orbit determi-nation and controlrdquo Journal of Guidance Control and Dynam-ics vol 19 no 2 pp 266ndash273 1996

[13] P Di Giamberardino and S Monaco ldquoOn halo orbits spacecraftstabilizationrdquo Acta Astronautica vol 38 no 12 pp 903ndash9251996

[14] P Gurfil and N J Kasdin ldquoStability and control of spacecraftformation flying in trajectories of the restricted three-bodyproblemrdquo Acta Astronautica vol 54 no 6 pp 433ndash453 2004

[15] C Simo G Gomez J Llibre R Martınez and J RodrıguezldquoOn the optimal station keeping control of halo orbitsrdquo ActaAstronautica vol 15 no 6 pp 391ndash397 1987

[16] G G Gomez J Llibre R Martınez and C Simo Dynamicsand Mission Design Near Libration Point Orbits vol 1 ofFundamentals The Case of Collinear Libration Points WorldScientific Singapore 2001

[17] G G Gomez Dynamics and Mission Design Near LibrationPoints vol 3 of Advanced Methods for Collinear Points WorldScientific 2001

[18] K C Howell and H J Pernicka ldquoStationkeeping method forlibration point trajectoriesrdquo Journal of Guidance Control andDynamics vol 16 no 1 pp 151ndash159 1993

[19] A Rahmani M-A Jalali and S Pourtakdoust ldquoOptimalapproach to halo orbit controlrdquo in Proceedings of the AIAAGuidance Navigation and Control Conference and Exhibit pp11ndash14 Austin Tex USA 2003

[20] B G Marchand and K C Howell ldquoControl strategies forformation flight in the vicinity of the libration pointsrdquo Journal ofGuidance Control and Dynamics vol 28 no 6 pp 1210ndash12192005

[21] P Gurfil M Idan and N J Kasdin ldquoAdaptive neural control ofdeep-space formation flyingrdquo Journal of Guidance Control andDynamics vol 26 no 3 pp 491ndash501 2003

[22] X Bai and J L Junkins ldquoModified Chebyshev-Picard iterationmethods for station-keeping of translunar halo orbitsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 926158 18pages 2012

[23] J Han ldquoControl theory is it a model analysis or direct controlapproachrdquo Journal of Systems Science andMathematical Sciencevol 9 no 4 pp 328ndash335 1989

[24] J Han ldquoFromPID to active disturbance rejection controlrdquo IEEETransactions on Industrial Electronics vol 56 no 3 pp 900ndash906 2009

[25] D Sun ldquoComments on active disturbance rejection controlrdquoIEEE Transactions on Industrial Electronics vol 54 no 6 pp3428ndash3429 2007

[26] G E Dullerud and F Paganini A Course in Robust ControlTheory vol 6 Springer New York NY USA 2000

[27] D L Richardson ldquoHalo orbit formulation for the isee-3 mis-sionrdquo Journal of Guidance Control and Dynamics vol 3 no 6pp 543ndash548 1980

[28] D L Richardson ldquoAnalytic construction of periodic orbitsabout the collinear pointsrdquo Celestial Mechanics vol 22 no 3pp 241ndash253 1980

[29] S Yin S X Ding A Haghani H Hao and P Zhang ldquoAcomparison study of basic data-driven fault diagnosis andprocess monitoring methods on the benchmark tennesseeeastman processrdquo Journal of Process Control vol 22 no 9 pp1567ndash1581 2012

[30] S Yin S XDingAHAbandan Sari andHHao ldquoData-drivenmonitoring for stochastic systems and its application on batchprocessrdquo International Journal of Systems Science vol 44 no 7pp 1366ndash1376 2013

[31] J Han and W Wang ldquoNonlinear tracking-differentiatorrdquo Jour-nal of Systems Science and Mathematical Science vol 14 no 2pp 177ndash183 1994

[32] J Han ldquoThe ldquoextended state observerrdquo of a class of uncertainsystemsrdquo Control and Decision vol 10 no 1 pp 85ndash88 1995

[33] H Peng J Zhao Z Wu and W Zhong ldquoOptimal periodiccontroller for formation flying on libration point orbitsrdquo ActaAstronautica vol 69 no 7-8 pp 537ndash550 2011

[34] P Gurfil ldquoControl-theoretic analysis of low-thrust orbital trans-fer using orbital elementsrdquo Journal of Guidance Control andDynamics vol 26 no 6 pp 979ndash983 2003

[35] M Martinez-Sanchez and J E Pollard ldquoSpacecraft electricpropulsion-an overviewrdquo Journal of Propulsion and Power vol14 no 5 pp 688ndash699 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Parameters Value Parameters Value Parameters Value120582 208645 119896 322927 119886

21209270

11988622

248298 times 10minus1

11988623

minus905965 times 10minus1

11988624

minus104464 times 10minus1

11988631

793820 times 10minus1

11988632

826854 times 10minus2

11988721

minus492446 times 10minus1

11988722

607465 times 10minus2

11988731

885701 times 10minus1

11988732

230198 times 10minus2

11988733

minus284508 11988734

minus230206 11988735

minus18703711988921

minus346865 times 10minus1

11988931

190439 times 10minus2

11988932

398095 times 10minus1

1 and its differential V

2

One feasible second-order TD can be designed as [31]

V1= V2

V2= fhan (V

1minus V (119905) V

2 119903 ℎ0)

(5)

where V(119905) denotes the control objective 119903 is speed factor anddecides tracking speed ℎ

0is filtering factor and fhan(V

1minus

V(119905) V2 119903 ℎ0) is as follows

119889 = 119903ℎ2

0

1198860= ℎ0V2

119910 = (V1minus V (119905)) + 119886

0

1198861= radic119889 (119889 + 8 |119889|)

1198862= 1198860+sign (119910) (119886

1minus 119889)

2

119904119910=sign (119910 + 119889) minus sign (119910 minus 119889)

2

119886 = (1198860+ 119910 minus 119886

2) 119904119910+ 1198862

119904119886=sign (119886 + 119889) minus sign (119886 minus 119889)

2

fhan (V1minus V (119905) V

2 119903 ℎ0)

= minus119903 (119886

119889minus sign (119886)) 119904

119886minus 119903 sign (119886)

(6)

32 ESO ESO is used to estimate 119891(1199091 1199092 119908(119905) 119905) in real

3= 119891(119909

1 1199092 119908(119905) 119905)

is introduced in (4) Using 1199111 1199112 and 119911

3to estimate 119909

1 1199092

and 1199093 respectively a nonlinear observer is designed as [32]

119890 = 1199111minus 119910

1= 1199112minus 1205731119890

2= 1199113minus 1205732fal (119890 120572

1 120575) + 119887119906

3= minus1205733fal (119890 120572

2 120575)

(7)

where 1205731 1205732 and 120573

3are observer gains 119890 is the error and

fal(119909 120572 120575) is as follows

fal (119909 120572 120575) =

119909

1205751minus120572 |119909| le 120575

sign (119909) |119909|120572 |119909| gt 120575

(8)

1198901= V1minus 1199111

1198902= V2minus 1199112

(9)

A nonlinear combination of errors signal can be con-structed as [24]

1199060= minusfhan (119890

1 1198881198902 120572 ℎ1) (10)

The controller is designed as

119906 = 1199060minus1199113

119887 (11)

4 ADRSC for Station-Keeping

411 TD I It is designed for reference Halo orbit tracking

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 119903119894 ℎ119894)

(12)

Mathematical Problems in Engineering 5

ESO

u

w

minus

minusminus

+Xr(t)r1

r

12

2 e2

e1

z1z2z3

u0 X(t)TD I

TD II

NLSEFSpacecraftnonlineardynamic

Disturbingforces

xr(t) =[xr(t) yr(t) zr(t)]T

Figure 2 The structure of ADRSC algorithm

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively V

119894(119896) denotes the reference orbit TD I provides

transition process for nominal orbit V119894and differential trajec-

tory of set position value that is V1198941and its differential V

1198942

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 1199030119894 ℎ0119894)

(13)

12]119879 denotes the

estimation of [119883 ]119879 which is the position and velocity

vector of spacecraft trajectory along119883 axes V119894(119896) denotes the

flight trajectory of spacecraft And 1199030119894is the tracking speed

parameter and ℎ0119894is the filter factor which makes an effort

1198941= 1199091198942

1198942= 119892119894(X X) + 119908

119894+ 119906119894

119910 = 1199091198941

(14)

2111990931]119879 denotes the position vector

of the spacecraft w = [1199081 1199082

1199083]119879 denotes the disturbance

vector and u = [1199061 1199062

1199063]119879 denotes the vector of station-

keeping control input g(XX) = [1198921 1198922

1198923]119879 represents

robust performance will be presented under such formationof dynamic model uncertainties

1199091198943= 119865119894(X X) (15)

where 1198651 1198652 and 119865

3denote the corresponding axis com-

ponents of F(X X) respectively and let F(X X) = a(X X)where a(X X) is unknown

One can rewrite (14) as follows

1198941= 1199091198942

1198942= 1199091198943+ 119906119894

1198943= 119886119894

119910 = 1199091198941

(16)

where 1199091198941 1199091198942 and 119909

1198943represent the position velocity and

119865119894(X X) respectively 119894 = 1 2 3 Then one can use the

following discrete nonlinear observer 1199111198941 1199111198942 1199111198943 to estimate

state vector 1199091198941 1199091198942 1199091198943

119890119894= 119911119894minus 119910119894

1199111198941(119896 + 1) = 119911

1198941(119896) + ℎ times (119911

1198942(119896) minus 120573

1198941119890)

1199111198942(119896 + 1)

= 1199111198942(119896) + ℎ times (119891

0119894+ 1199111198943(119896) minus 120573

1198942times fal (119890 05 ℎ))

+ 119906 (119894)

1199111198943(119896 + 1) = 119911

1198943(119896) minus ℎ times 120573

1198943times fal (119890 025 ℎ)

(17)

6 Mathematical Problems in Engineering

119906119894(119896 + 1) = minusfhan (119890

1198941 1198881198941198901198942 120572119894 ℎ119894) minus 1199111198943(119896) (18)

where 119888119894 120572119894 ℎ119894 are the NLSEF parameters described in (10)

1198901198941and 119890

1198942are the state errors described in (9) 119911

1198943denotes

unmodeled dynamics and disturbances 119865119894(X X) which is

estimated from (17)The parameter ℎ

119894of NLSEF which is also named filter

119894must be

selected properly It is noted that larger errors correspondingto larger ℎ

119894and smaller errors corresponding to smaller ℎ

119894will

make a better performanceIn this paper a parameter self-turning approach is firstly

proposed for ℎ119894selection as follows

ℎ119894=

119896log10radic

3

sum

119894=1

1198902

1198941+ ℎ0 radic

3

sum

119894=1

1198902

1198941gt 120585

119896log10120585 + ℎ0 radic

3

sum

119894=1

1198902

1198941le 120585

(19)

where the parameters 119896 ℎ0 and 120585 of self-turning ℎ

119894can

be easily chosen in practice as will be presented in thesimulation

1003816100381610038161003816dENE1003816100381610038161003816 = 119866119872Earth

100381610038161003816100381610038161003816100381610038161003816

1

1199032119888

minus1

1199032119890

100381610038161003816100381610038161003816100381610038161003816

(20)

where 119903119890is the position vector of the Earth at the pericenter

of the Earthrsquos elliptical orbit from the libration point 119903119888is the

position vector similar to a circular orbit

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

2(_r1sdot n)2

n (21)

_r1= r1r1 and n is the attitude

vector of the spacecraft in the rotating frame To simplify (21)we assume that n =

_r1 which yields

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

3r1 (22)

5 Simulation Results and Analysis

1point of nonlinear system

119904= 1358Wm2 the speed of light is 119888 = 3 times 10

119904(1 + 119902)(119898119888) =

5 times 10minus8ms2 [21] It is also assumed that the initial position

1libration point

52 ISEE-3 Halo Orbit (119885 = 110 000 km)

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

53 Halo Orbit (119885 = 800 000 km)

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

ESO

u

w

minus

minusminus

+Xr(t)r1

r

12

2 e2

e1

z1z2z3

u0 X(t)TD I

TD II

NLSEFSpacecraftnonlineardynamic

Disturbingforces

xr(t) =[xr(t) yr(t) zr(t)]T

Figure 2 The structure of ADRSC algorithm

where 119894 = 1 2 3 denote the 119883 119884 and 119885 axes componentrespectively V

119894(119896) denotes the reference orbit TD I provides

transition process for nominal orbit V119894and differential trajec-

tory of set position value that is V1198941and its differential V

1198942

119890 = V1198941(119896) minus V

119894(119896)

V1198941(119896 + 1) = V

1198941(119896) + ℎ times V

1198942(119896)

V1198942(119896 + 1) = V

1198942(119896) + ℎ timesfhan (119890 V

1198942(119896) 1199030119894 ℎ0119894)

(13)

12]119879 denotes the

estimation of [119883 ]119879 which is the position and velocity

vector of spacecraft trajectory along119883 axes V119894(119896) denotes the

flight trajectory of spacecraft And 1199030119894is the tracking speed

parameter and ℎ0119894is the filter factor which makes an effort

1198941= 1199091198942

1198942= 119892119894(X X) + 119908

119894+ 119906119894

119910 = 1199091198941

(14)

2111990931]119879 denotes the position vector

of the spacecraft w = [1199081 1199082

1199083]119879 denotes the disturbance

vector and u = [1199061 1199062

1199063]119879 denotes the vector of station-

keeping control input g(XX) = [1198921 1198922

1198923]119879 represents

robust performance will be presented under such formationof dynamic model uncertainties

1199091198943= 119865119894(X X) (15)

where 1198651 1198652 and 119865

3denote the corresponding axis com-

ponents of F(X X) respectively and let F(X X) = a(X X)where a(X X) is unknown

One can rewrite (14) as follows

1198941= 1199091198942

1198942= 1199091198943+ 119906119894

1198943= 119886119894

119910 = 1199091198941

(16)

where 1199091198941 1199091198942 and 119909

1198943represent the position velocity and

119865119894(X X) respectively 119894 = 1 2 3 Then one can use the

following discrete nonlinear observer 1199111198941 1199111198942 1199111198943 to estimate

state vector 1199091198941 1199091198942 1199091198943

119890119894= 119911119894minus 119910119894

1199111198941(119896 + 1) = 119911

1198941(119896) + ℎ times (119911

1198942(119896) minus 120573

1198941119890)

1199111198942(119896 + 1)

= 1199111198942(119896) + ℎ times (119891

0119894+ 1199111198943(119896) minus 120573

1198942times fal (119890 05 ℎ))

+ 119906 (119894)

1199111198943(119896 + 1) = 119911

1198943(119896) minus ℎ times 120573

1198943times fal (119890 025 ℎ)

(17)

6 Mathematical Problems in Engineering

119906119894(119896 + 1) = minusfhan (119890

1198941 1198881198941198901198942 120572119894 ℎ119894) minus 1199111198943(119896) (18)

where 119888119894 120572119894 ℎ119894 are the NLSEF parameters described in (10)

1198901198941and 119890

1198942are the state errors described in (9) 119911

1198943denotes

unmodeled dynamics and disturbances 119865119894(X X) which is

estimated from (17)The parameter ℎ

119894of NLSEF which is also named filter

119894must be

selected properly It is noted that larger errors correspondingto larger ℎ

119894and smaller errors corresponding to smaller ℎ

119894will

make a better performanceIn this paper a parameter self-turning approach is firstly

proposed for ℎ119894selection as follows

ℎ119894=

119896log10radic

3

sum

119894=1

1198902

1198941+ ℎ0 radic

3

sum

119894=1

1198902

1198941gt 120585

119896log10120585 + ℎ0 radic

3

sum

119894=1

1198902

1198941le 120585

(19)

where the parameters 119896 ℎ0 and 120585 of self-turning ℎ

119894can

be easily chosen in practice as will be presented in thesimulation

1003816100381610038161003816dENE1003816100381610038161003816 = 119866119872Earth

100381610038161003816100381610038161003816100381610038161003816

1

1199032119888

minus1

1199032119890

100381610038161003816100381610038161003816100381610038161003816

(20)

where 119903119890is the position vector of the Earth at the pericenter

of the Earthrsquos elliptical orbit from the libration point 119903119888is the

position vector similar to a circular orbit

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

2(_r1sdot n)2

n (21)

_r1= r1r1 and n is the attitude

vector of the spacecraft in the rotating frame To simplify (21)we assume that n =

_r1 which yields

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

3r1 (22)

5 Simulation Results and Analysis

1point of nonlinear system

119904= 1358Wm2 the speed of light is 119888 = 3 times 10

119904(1 + 119902)(119898119888) =

5 times 10minus8ms2 [21] It is also assumed that the initial position

1libration point

52 ISEE-3 Halo Orbit (119885 = 110 000 km)

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

53 Halo Orbit (119885 = 800 000 km)

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

119906119894(119896 + 1) = minusfhan (119890

1198941 1198881198941198901198942 120572119894 ℎ119894) minus 1199111198943(119896) (18)

where 119888119894 120572119894 ℎ119894 are the NLSEF parameters described in (10)

1198901198941and 119890

1198942are the state errors described in (9) 119911

1198943denotes

unmodeled dynamics and disturbances 119865119894(X X) which is

estimated from (17)The parameter ℎ

119894of NLSEF which is also named filter

119894must be

selected properly It is noted that larger errors correspondingto larger ℎ

119894and smaller errors corresponding to smaller ℎ

119894will

make a better performanceIn this paper a parameter self-turning approach is firstly

proposed for ℎ119894selection as follows

ℎ119894=

119896log10radic

3

sum

119894=1

1198902

1198941+ ℎ0 radic

3

sum

119894=1

1198902

1198941gt 120585

119896log10120585 + ℎ0 radic

3

sum

119894=1

1198902

1198941le 120585

(19)

where the parameters 119896 ℎ0 and 120585 of self-turning ℎ

119894can

be easily chosen in practice as will be presented in thesimulation

1003816100381610038161003816dENE1003816100381610038161003816 = 119866119872Earth

100381610038161003816100381610038161003816100381610038161003816

1

1199032119888

minus1

1199032119890

100381610038161003816100381610038161003816100381610038161003816

(20)

where 119903119890is the position vector of the Earth at the pericenter

of the Earthrsquos elliptical orbit from the libration point 119903119888is the

position vector similar to a circular orbit

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

2(_r1sdot n)2

n (21)

_r1= r1r1 and n is the attitude

vector of the spacecraft in the rotating frame To simplify (21)we assume that n =

_r1 which yields

dSRP = minus120573

1003817100381710038171003817r11003817100381710038171003817

3r1 (22)

5 Simulation Results and Analysis

1point of nonlinear system

119904= 1358Wm2 the speed of light is 119888 = 3 times 10

119904(1 + 119902)(119898119888) =

5 times 10minus8ms2 [21] It is also assumed that the initial position

1libration point

52 ISEE-3 Halo Orbit (119885 = 110 000 km)

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

53 Halo Orbit (119885 = 800 000 km)

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 005 01

0

01

200 400 600 800 1000 1200 1400 1600 1800minus5

0

5

01

200 400 600 800 1000 1200 1400 1600 180001

200 400 600 800 1000 1200 1400 1600 180001

FPTP

FVTV

FPTP

FVTV

minus001

0

001

0

005

01

minus2

0

2

Days

Days

Days

Days

Days

Days

minus01

0 005 01

0 005 01

0

01

minus01

times10minus3

times10minus3

TDx

TDy

TDz

Figure 3 The TD tracking result

Table 2 Initial orbits parameters of Halo missions without injection errors

Mission orbits 1198830

1198840

1198850

0

0

0

119894= 1000 119903

0119894= 2000 120573

12= 12057322

= 30000012057311

= 12057321

= 12057313

= 12057323

= 5000 12057332

= 100000 12057331

= 12057333

=

2000 119888119894= 05 120572

119894= 200 119894 = 1 2 3 119896 = 00005 ℎ

0= minus00004

and 120585 = 10 The weight matrices of the LQR controller areselected as 119876 = diag3 times 10

8 3 times 10

8 3 times 10

8 3 times 10

6 3 times

106 3 times 10

6 119877 = I

3

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

53 Halo Orbit (119885 = 800 000 km)

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0985

099

0995

minus5

0

5minus1

0

1

x (AU)

y (AU)

z (A

U)

times10minus3

times10minus3

L1

lowast

(a)y

(AU

)

x (AU)0988 099 0992 0994minus5

0

5times10minus3

(b)

z (A

U)

x (AU)0988 099 0992 0994

minus1

minus05

0

05

1times10minus3

(c)

z (A

U)

y (AU)minus5 0 5

minus1

minus05

0

05

1times10minus3

times10minus3

(d)

Figure 4 ADRSC of ISEE-3 (119885 = 110 000 km)

119909Δ119881119910 and Δ119881

119911

(in unit ofmsT) and themean absolute value of the positionerrors 119890

119909 119890119910 and 119890

119911(in unit of km) as follows [33]

Δ119881119894=

1

119879int

119879

0

Δ119881 = radicΔ1198812119909+ Δ1198812119910+ Δ1198812119911

119890119894= |Δ119894|mean (119894 = 119909 119910 119911) 119890 = radic1198902

119909+ 1198902119910+ 1198902119911

(23)

where 119879 is the orbitalperiod

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

53 Halo Orbit (119885 = 800 000 km)

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus1

0

1

20t (days)t (days)

(a)

0 5 10 15 20minus10

0

10

Velo

city

erro

rs (m

s)

minus5

0

5

500 1000 150020t (days)t (days)

times10minus4

(b)

0 5 10 15 20

xy

z

xy

z

500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

t (days)t (days)

(c)

Figure 5 The relative position errors velocity errors and control input of ADRSC control of ISEE-3

53 Halo Orbit (119885 = 800 000 km)

119894 119896 = 00120

ℎ0= minus0011 Thus the ADRSC can be easily taken into prac-

ticeTheweight matrices of the LQR controller are selected as119876 = diag4 times 10

8 4 times 10

8 4 times 10

8 106 106 106 119877 = I

3

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 5 10 15

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus10

0

10

15t (days)t (days)

(a)

0 5 10 15 500 1000 150015minus5

0

5

Velo

city

erro

r (m

s)

minus005

0

005

t (days) t (days)

(b)

0 5 10 15 500 1000 150015

minus50

0

50

Con

trol i

nput

(mN

)

minus5

0

5

xy

z

xy

z

t (days) t (days)

(c)

Figure 6 The relative position errors velocity errors and control input of LQR control of ISEE-3

Mission orbits ΔV119909

ΔV119910

ΔV119911

ΔV 119890119909

119890119910

119890119911

119890

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

098

099

1

minus001

0

001minus001

0

001

x (AU)

y (AU)

z (A

U)

(a)

0985 099 0995minus001

minus0005

0

0005

001

x (AU)

y (A

U)

(b)

0985 099 0995minus5

0

5

10

x (AU)

times10minus3

z (A

U)

(c)

minus001 0 001minus5

0

5

10times10minus3

z (A

U)

y (AU)

times10minus3

(d)

Figure 7 ADRSC control of Halo orbit (119885 = 800 000 km)

compared with LQR controller despite of the samelevel of both the control methods thrust usage

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

0 5 10 15 20

minus1000

0

1000Po

sitio

n er

rors

(km

)

500 1000 1500minus4

minus2

0

2

20t (days) t (days)

(a)

0 5 10 15 20minus5

0

5

Velo

city

erro

rs (m

s)

minus1

0

1

2

500 1000 150020

times10minus3

t (days) t (days)

(b)

0 5 10 15 20 500 1000 150020minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

xy

z

xy

z

t (days) t (days)

(c)

6 Conclusion

A successful ADRSC approach for station-keeping in theSun-Earth 119871

1point is accomplished using the input and

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

0 10 20 30

minus1000

0

1000Po

sitio

n er

ror (

km)

500 1000 1500minus50

0

50

30t (days)t (days)

(a)

0 10 20 30minus2

0

2

Velo

city

erro

r (m

s)

minus005

0

005

500 1000 150030t (days)t (days)

(b)

0 10 20 30

minus50

0

50

Con

trol i

nput

(mN

)

minus10

0

10

20

500 1000 150030t (days)t (days)

xy

z

xy

z

(c)

Figure 9 The relative position errors velocity errors and control input of LQR control of Halo orbit

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

References

[1] B Wie Space Vehicle Dynamics and Control AIAA 1998

infinapproachrdquo IEEE

Transactions on Control Systems Technology vol 14 no 3 pp572ndash578 2006

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

[6] F Wang X Chen A Tsourdos B A White and X Cao ldquoSun-earth 119871

2point formation control using polynomial eigenstruc-

ture assignmentrdquo Acta Astronautica vol 76 no 3 pp 26ndash362012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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