research article low-thrust transfer design of low
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Research ArticleLow-Thrust Transfer Design of Low-ObservableGeostationary Earth Orbit Satellite
Bing Hua12 and Zhujun Shao2
1State Key Laboratory of Virtual Reality Technology and Systems Beihang University Beijing 100091 China2College of Astronautics Nanjing University of Aeronautics amp Astronautics Nanjing 210016 China
Correspondence should be addressed to Bing Hua binghuanuaaeducn
Received 4 August 2015 Revised 10 November 2015 Accepted 24 November 2015
Academic Editor Paul Williams
Copyright copy 2015 B Hua and Z Shao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
With radar and surface-to-airmissiles posing an increasing threat to on-orbit spacecraft low-observable satellites play an importantrole in low-thrust transfersThis paper presents the design for a low-thrust geostationary earth orbit (GEO) transfer control strategywhich takes into consideration the low-observable constraint and discusses Earth shadow and perturbation A control parameteroptimization addresses the orbit transfer problem and five thrustmodes are used Simulation results show that themethod outlinedin this paper is simple and feasible and results in reduced transfer time with a small amount of calculation The method thereforeoffers a useful reference for low-thrust GEO transfer design
1 Introduction
Creation of a low-observable satellite is accomplished by low-observable technology which makes it difficult or impossibleto avoid detection of satellites in orbit by hostile enemy forces[1ndash6] Adding a low-observability module to the spacecraftrsquosoverall design is key to improving operational effectivenessand satellite survivability
A geostationary satellite is usually launched from groundto Low Earth Orbit (LEO) or Middle Earth Orbit (MEO)rather than to GEO Since the main threat to LEO and MEOsatellites comes from radar a great deal of emphasis in low-observability design is placed on radar analysis The radarcross section (RCS) is the designersrsquo only controllable factorin radar detection Contemporary work on low observabilityhas its roots in efforts at reducing the RCS of a spacecraftfalling into two categories low-observable shape design andflight attitude planning [1ndash8] The former is a simple way ofreducing the range at which radar can detect the spacecrafthowever contouring the surface of a spacecraft to reduce theRCS equally in all directions is not possible As a result thelatter is typically supplemented in designs in order to producebetter results
The application of electric propulsion in geostationaryorbit platforms is inevitable for the development of theaerospace Used for spacecraft applications in Earth orbitsuch as station-keeping orbit-raising and orbit transferBoeing-702sp is an all-electric propulsion satellite with aspecific impulse of more than 3800 seconds [9ndash12] A greatdeal of research has been directed toward solving the low-thrust transfer problem Most of that research has beenbased on optimal control theory The numerical solutionincorporates both direct and indirect methods due to theirhigh accuracy [13ndash16] As is typical with these methods thesolution is often difficult to derive requiring a complex initialguess and tedious iteration for convergence
Electric propulsion low thrust and highly specificimpulses have led to greatly improved fuel efficiency at theexpense of relatively long transfer times For these reasonssatellites can be easily caught by detection systems duringlow-thrust GEO transfer Existing research on low-thrusttransfer emphasizes optimum techniques giving little con-sideration to the satellite attitude constraint of control Thispaper models an optimization scheme of the low-thrusttransfer based on low-observable technology which can
Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 439815 12 pageshttpdxdoiorg1011552015439815
2 International Journal of Aerospace Engineering
Dynamic model Satellite low observability
Low-thrust GEO transfer design
Perturbation
Eclipse effects
Radar detection area
Thrust control angles and satellite
attitude
Transfer scheme involved space environment and radar
detection
Low-thrust GEO transfer design based on
parameter analysis
Low-thrust GEO transfer design
Modified equinoctial orbit model
Section 2 Section 3 Section 4
Figure 1 Paper structure
reduce the visibility of satellites Furthermore the opti-mization includes space environment (Earth shadow andperturbation) Offering the advantages of short time a smallamount of calculation and no complex initial guess themethod designed in this paper can provide a valuable refer-ence for low-thrust GEO transfer design
The paper is organized in five distinct sections as repre-sented in Figure 1The introduction above has described low-observable satellites and low-thrust transfer optimizationtechniques In Section 2 Earth shadow and perturbationin orbit transfer are analyzed In Section 3 a mathematicalmodel of low-observable constraint is demonstrated includ-ing the radar detection area and the relationship of thrustcontrol angle and attitude angle Section 4 presents a methodbased on control parameter analysis to solve minimum-timetransfer Finally we summarize the paper
2 Dynamic Model
21 Modified Equinoctial Orbit Model The dynamical equa-tions of motion for a thrusting spacecraft can be establishedby the classical orbital elements the semimajor axis 119886 theeccentricity 119890 the inclination 119894 the right ascension Ω theargument of perigee 120596 and the mean anomaly119872
119886 =
21198862119890 sin Vℎ
119891119903+
21198862119901
ℎ119903
119891119905
119890 =
1
ℎ
119901 sin V119891119903+
1
ℎ
[(119901 + 119903) cos V + 119903119890] 119891119905
119894 =
119903 cos (120596 + V)ℎ
119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(1)
where 119901 = 119886(1minus1198902) ℎ = radic120583119901 119899 = radic120583119886
2 119903 = 119901(1+119890 cos V)with Earthrsquos gravitational coefficient 120583 and the true anomalyV Performing analyses of transfer orbits using classicalorbital elements is a straightforward task but singularities areexhibited for zero eccentricity and inclinations of 0∘ and 90∘
To eliminate these deficiencies amodified set of equinoc-tial orbit elements (119901 119890
119909 119890119910 ℎ119910 ℎ119909 and 119871) is frequently used
119875 is the semilatus rectum (119890119909 119890119910) is the eccentricity vector
(ℎ119909 ℎ119910) is the inclination vector and 119871 is the true longitude
The relationship between the modified equinoctial elementsand the classical orbital elements is given by
119901 = 119886 (1 minus 1198902)
119890119909= 119890 cos (120596 + Ω)
119890119910= 119890 sin (120596 + Ω)
ℎ119909= tan(
119894
2
) cosΩ
ℎ119910= tan(
119894
2
) sinΩ
119871 = 120596 + Ω + V
(2)
The equations ofmotion written in terms of themodifiedequinoctial elements are
= radic
119901
120583
2119901
120596
119891119905
119890119909= radic
119901
120583
119891119903sin 119871 + [(1 + 120596) cos 119871 + 119890
119909]
119891119905
120596
minus (ℎ119909sin 119871 minus ℎ
119910cos 119871)
119890119910119891119899
120596
119890119910= radic
119901
120583
minus119891119903cos 119871 + [(1 + 120596) sin 119871 + 119890
119910]
119891119905
120596
+ (ℎ119909sin 119871 minus ℎ
119910cos 119871)
119890119909119891119899
120596
International Journal of Aerospace Engineering 3
rarr
T
rarr
Nf
rarr
R
O
120572 120573
Figure 2 Coordinate frame
Transfer orbit EarthSun
Os
Rs rsOe
r120579
Re
Figure 3 Cylindrical projection model
ℎ119909= radic
119901
120583
1199042119891119899
2120596
cos 119871
ℎ119910= radic
119901
120583
1199042119891119899
2120596
sin 119871
= radic120583119901(
120596
119901
)
2
+
1
120596
radic
119901
120583
(ℎ119909sin 119871 minus ℎ
119910cos 119871)119891
119899
(3)
with120596 = 1+119890119909cos 119871+119890
119910sin 119871 1199042 = 1+ℎ
2
119909+ℎ2
119910The acceleration
components 119891119903 119891119905 and 119891
119899are denoted in the RTN frame
(with 119891119903being the normal direction 119891
119905being the tangential
direction and 119891119899being the direction orthogonal to the orbit
plane) and 119891 is the thrust accelerationThe constraint on thecontrol is 119891 le 119891max = 119879max119898
To take into account the true acceleration of the thrustwe must consider the mass flow Its evolution is given by
119889119898
119889119905
= minus
119879
119868sp1198920 (4)
where 119879 is the thrust modulus 119868sp the specific impulse ofthruster and 119892
0the gravitational acceleration at sea-level and
1198920= 980665NsAs shown in Figure 2 thrust acceleration components
can also be defined by acceleration 119891 and two control angles(pitch-steering angle 120572 and yaw-steering angle 120573) whichdescribe the direction of the thrust vector in relation to
the velocity vector and the orbital plane respectively Theacceleration components are expressed by
119891119903= 119891 cos120573 sin120572
119891119905= 119891 cos120573 cos120572
119891119899= 119891 sin120573
(5)
with minus120587 le 120572 le 120587 and minus1205872 le 120573 le 1205872
22 Eclipse Effects To model the electric propulsion systemaccurately it is necessary tomodel the satellitersquos trajectory as itpasses through the shadow of Earth In particular the satelliteis at discharge which leads to high power consumption whenin the shadow In order to ensure the safety of spacecrafttherefore the thrust is 0 when the vehicle is in the shadowof Earth A simplified Earth-shadow model namely thecylindrical projection model shown in Figure 3 is used toestimate the location of the shadow [17ndash20]
Referring to the geometry illustrated in Figure 3 r119904is
defined as the vector from Earth to the Sun with norm r119904
and r as the vector from Earth to the satellite with norm r119877119890is the radius of Earth and 119877
119904is the radius of the Sun
The cone angle 120579 is defined by
cos 120579 = angr119904r =
r sdot r119904
(r 1003817100381710038171003817r119904
1003817100381710038171003817)
(6)
4 International Journal of Aerospace Engineering
Earth
Transfer orbit
Figure 4 Low-thrust transfer with no radar detection
The projected spacecraft position r is given by
r = r[
[
[
[
cos V cosΩ cos120596 minus cos V sinΩ sin120596 cos 119894 minus sin V cosΩ cos120596 minus sin V sinΩ sin120596 cos 119894
cos V sinΩ cos120596 + cos V cosΩ sin120596 cos 119894 minus sin V sinΩ cos120596 + sin V cosΩ sin120596 cos 119894
cos V sin120596 sin 119894 + sin V cos120596 sin 119894
]
]
]
]
(7)
The shadow entry and exit locations are judged bylocating the cone terminators at the projected spacecraftlocation The shadow can only be found when the anglesatisfies
cos 120579 le 0
sin 120579 lt
119877119890
119903
(8)
23 Perturbation Earthrsquos oblateness atmospheric drag lightpressure secondary body and other factors in space can alsoperturb a satellitersquos motion Among these variables Earthrsquosoblateness is of vital importance for predicting the trajectoryof the satellite accurately The oblate Earth perturbation iscaused by the reality of Earthrsquos shape not being perfectlyspherical The impact of Earthrsquos oblateness due to 119869
2is always
taken into account in the engineering calculations [20 21]In terms of orbital elements the dynamical system is
119889119886
119889119905
= 0
119889119890
119889119905
= 0
119889119894
119889119905
= 0
119889Ω
119889119905
= minus
311989911986921198772
119890
2 (1 minus 1198902) 1198862cos 119894
119889120596
119889119905
=
311989911986921198772
119890
4 (1 minus 1198902) 1198862(5 cos2119894 minus 1)
119889119872
119889119905
= 119899 minus
311989911986921198772
119890
4 (1 minus 1198902)32
1198862
(1 minus 3 cos2119894)
(9)
The effect of Earthrsquos oblateness (due to 1198692) on the orbital
transfer can be included by appending the perturbation to therespective right-hand sides of (5)
119891119903=
3
2
1198692
119906
11990341198772
119864(3sin2119894 sin2119906 minus 1)
119891119905= minus
3
2
1198692
119906
11990341198772
119864sin2119894 sin 2119906
119891119899= minus1198692
119906
11990341198772
119864sin 2119894 sin 119906
(10)
3 Satellite Low Observability
The research activity presented here is focused on the opti-mum algorithm and does not take into account space envi-ronment analysis especially the low-observable constraintThe low-observable satellite is designed to minimize itsfrontal RCS requiring low-observable shape design and flightattitude adjustment [6ndash8] The satellite keeps its front towardEarth when it is flying over ground-based radar detectionareas
31 Radar Detection Area In order to avoid reflecting radarsignals directly the scanning range of the ground-based radarshould be modeled first Figure 4 shows low-thrust GEOsatellite transfer with no radar detection Figure 5 is anillustration of that GEO satellite being detected in low-thrusttransfer by a ground-based radar and the shadow is the radardetection area
International Journal of Aerospace Engineering 5
Radar detection area
Transfer orbit
Earth
Figure 5 Low-thrust transfer with radar detection
Due to the limited energy and probability of interceptradar repeats its search for the target in a narrow area whichismodeled as a specific coverage of yaw angle pitch angle andoperating distance rather than the entire airspace [22ndash24]
The latitude and longitude of the ground station aredefined respectively as 120582
119901and 120575
119901 119878 is the track of the sub-
satellite point whose right ascension and declination aredenoted as 120582 and 120575119882
119890is the rotation speed of Earth
Consider
120582 = 119886119903 tan (cos 119894 tan 119906) + Ω minus 119882119890119905
120575 = 119886119903 sin (sin 119894 sin 119906)
(11)
The radar pitch angle 120595ℎand radar yaw angle 120601
ℎare
obtained by the spherical triangle Respectively
tan120595ℎ=
119903 cos 120574 minus 119877119890
119903 cos 120574
sin120601ℎ=
sin (120582 minus 120582119901) sin (90
∘minus 120575)
sin 120574
(12)
where 119903 = (119886(1 minus 1198902)(1 + 119890cosV)) sdot 120574 is the geocentric angle
between the subsatellite point and observation points It sat-isfies
cos 120574 = sin 120575119901sin 120575 + cos 120575
119901cos 120575 cos (120582 minus 120582
119901) (13)
However their distance is
119877OP = radic(119877119890+ ℎ)2
+ 1198772
119890minus 2119877119890(119877119890+ ℎ) cos 120574 (14)
Suppose the radar yaw angle [1205901 1205902] pitch angle [120591
1 1205912]
and the operating distance 119903op
120595ℎisin [1205901 1205902]
120601ℎisin [1205911 1205912]
119877OP le 119903op
(15)
Only a satellite meeting these conditions has access to theradar detection area
32 Thrust Control Angels and Satellite Attitude Four thrust-ers are installed on the floor of the satellite as shown in
1 2 3 4
Frontallow RCS shape
design
Direction of solar irradiation
Thrusters
Figure 6 Thruster installation
Figure 6 Two work during orbit transfer and the other twoact as a backup systemThrust is perpendicular to the bottomof the satellite in low-thrust transferThe satellite has a frontalRCS shape design Assuming the maximum thrust is 119865 thenthrust components in the satellite coordinates are
[119865119909119865119910
119865119911]
119879
= [0 119865 0]
119879
(16)
The RTN frame is where the attitude of three axis-stabi-lized satellites is defined in which thrust components aredescribed as
[119865119909119900
119865yo 119865119911119900
]
119879
= [119865 cos120573 cos120572 119865 cos120573 sin120572 119865 sin120573]
119879
(17)
Satellite attitude is related to the order of three rotations[20] The yaw angle roll angle and pitch angle of the satelliteare recorded as 120601 120595 and 120579 and are derived in the orderof 3-1-2 Define attitude matrix 119860
312= 1198772(120579)1198771(120595)1198773(120601)
and its transpose matrix 119860minus1
312= 119877minus1
3(120601)119877minus1
1(120595)119877minus1
2(120579) The
relationship between satellite attitude and thrust controlangles is
6 International Journal of Aerospace Engineering
[
[
[
cos120573 cos120572sin120573
cos120573 sin120572
]
]
]
= 119860minus1
312
[
[
[
0
1
0
]
]
]
= 119877minus1
3(120601) 119877minus1
1(120595) 119877minus1
2(120579)
[
[
[
0
1
0
]
]
]
=[
[
[
cos120601 minus sin120601 0
sin120601 cos120601 0
0 0 1
]
]
]
[
[
[
1 0 0
0 cos120595 minus sin120595
0 sin120595 cos120595
]
]
]
[
[
[
cos 120579 0 sin 120579
0 1 0
minus sin 120579 0 cos 120579
]
]
]
[
[
[
0
1
0
]
]
]
=
[
[
[
[
cos120601 cos 120579 minus sin120601 sin120595 sin 120579 minus sin120601 cos120595 cos120601 sin 120579 + sin120601 sin120595 cos 120579
sin120601 cos 120579 + cos120601 sin120595 sin 120579 cos120601 cos120595 sin120601 sin 120579 minus cos120601 sin120595 cos 120579
minus sin120601 sin 120579 sin120595 cos120595 cos 120579
]
]
]
]
[
[
[
[
0
1
0
]
]
]
]
=
[
[
[
[
minus sin120601 cos120595
cos120601 cos120595
sin120595
]
]
]
]
(18)
It is ultimately expressed as
cos120573 cos120572 = minus sin120601 cos120595
sin120573 = cos120601 cos120595
cos120573 sin120572 = sin120595
(19)
The attitude adjustment range in the radar irradiationarea is assumed to be
120601min lt 120601 lt 120601max
120595min lt 120595 lt 120595max
120579min lt 120579 lt 120579max
(20)
The corresponding acceleration thrust components can beobtained by
119891119903= 119891 cos120573 sin120572 = minus119891 sin120601 cos120595
119891119905= 119891 cos120573 cos120572 = 119891 cos120601 cos120595
119891119899= 119891 sin120573 = 119891 sin120595
(21)
To realize low RCS toward Earth our paper sets theoptimal satellite attitude at a range of plusmn5
∘ which can beadjusted according to the real simulation and test Consider
120601 isin [minus5∘ 5∘]
120595 isin [minus5∘ 5∘]
120579 isin [175∘ 185∘]
(22)
4 Low-Thrust GEO Transfer Design
The goal of this paper is to design a minimum-time transferfor geostationary spacecraft equipped with electric propul-sion systems The transfer problem is thus to find an essen-tially bound control to reduce eccentricity and inclinationand raise the semimajor axis
After simplifying (1) they fall into
119886 =
21198862
ℎ
[119890 sin V119891119903+ (1 + 119890 cos V) 119891
119905]
119890 =
119901
ℎ
[sin V119891119903+ (cos V +
119890 + cos V1 + 119890 cos V
)119891119905]
119894 =
119903
ℎ
cos (120596 + V) 119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(23)
The acceleration component 119891119899only contributes to a
decrease of inclinationThe change of the semimajor axis andeccentricity are both related to 119891
119903and 119891
119905 119891119903and 119891
119905have a
greater effect on the semimajor axis than eccentricityFirst a parameter 119896 is introduced to isolate 119891
119899from 119891
119891119903119905
= radic1198912
119903+ 1198912
119905= 119891 sdot 119896
119891119899= 119891radic1 minus 119896
2
(24)
with 0 le 119896 le 1Then a control law that reduces eccentricity quickly is
derived by observing the time-rate equation for the semi-major axis and eccentricity Define 119890
1199011= sin V and 119890
1199012=
cos V + (119890 + cos V)(1 + 119890 cos V) The law is given by119891119903= 119891 sdot 119896
119891119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119903= 0
119891119905= 119891 sdot 119896
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
(25)
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
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DistributedSensor Networks
International Journal of
2 International Journal of Aerospace Engineering
Dynamic model Satellite low observability
Low-thrust GEO transfer design
Perturbation
Eclipse effects
Radar detection area
Thrust control angles and satellite
attitude
Transfer scheme involved space environment and radar
detection
Low-thrust GEO transfer design based on
parameter analysis
Low-thrust GEO transfer design
Modified equinoctial orbit model
Section 2 Section 3 Section 4
Figure 1 Paper structure
reduce the visibility of satellites Furthermore the opti-mization includes space environment (Earth shadow andperturbation) Offering the advantages of short time a smallamount of calculation and no complex initial guess themethod designed in this paper can provide a valuable refer-ence for low-thrust GEO transfer design
The paper is organized in five distinct sections as repre-sented in Figure 1The introduction above has described low-observable satellites and low-thrust transfer optimizationtechniques In Section 2 Earth shadow and perturbationin orbit transfer are analyzed In Section 3 a mathematicalmodel of low-observable constraint is demonstrated includ-ing the radar detection area and the relationship of thrustcontrol angle and attitude angle Section 4 presents a methodbased on control parameter analysis to solve minimum-timetransfer Finally we summarize the paper
2 Dynamic Model
21 Modified Equinoctial Orbit Model The dynamical equa-tions of motion for a thrusting spacecraft can be establishedby the classical orbital elements the semimajor axis 119886 theeccentricity 119890 the inclination 119894 the right ascension Ω theargument of perigee 120596 and the mean anomaly119872
119886 =
21198862119890 sin Vℎ
119891119903+
21198862119901
ℎ119903
119891119905
119890 =
1
ℎ
119901 sin V119891119903+
1
ℎ
[(119901 + 119903) cos V + 119903119890] 119891119905
119894 =
119903 cos (120596 + V)ℎ
119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(1)
where 119901 = 119886(1minus1198902) ℎ = radic120583119901 119899 = radic120583119886
2 119903 = 119901(1+119890 cos V)with Earthrsquos gravitational coefficient 120583 and the true anomalyV Performing analyses of transfer orbits using classicalorbital elements is a straightforward task but singularities areexhibited for zero eccentricity and inclinations of 0∘ and 90∘
To eliminate these deficiencies amodified set of equinoc-tial orbit elements (119901 119890
119909 119890119910 ℎ119910 ℎ119909 and 119871) is frequently used
119875 is the semilatus rectum (119890119909 119890119910) is the eccentricity vector
(ℎ119909 ℎ119910) is the inclination vector and 119871 is the true longitude
The relationship between the modified equinoctial elementsand the classical orbital elements is given by
119901 = 119886 (1 minus 1198902)
119890119909= 119890 cos (120596 + Ω)
119890119910= 119890 sin (120596 + Ω)
ℎ119909= tan(
119894
2
) cosΩ
ℎ119910= tan(
119894
2
) sinΩ
119871 = 120596 + Ω + V
(2)
The equations ofmotion written in terms of themodifiedequinoctial elements are
= radic
119901
120583
2119901
120596
119891119905
119890119909= radic
119901
120583
119891119903sin 119871 + [(1 + 120596) cos 119871 + 119890
119909]
119891119905
120596
minus (ℎ119909sin 119871 minus ℎ
119910cos 119871)
119890119910119891119899
120596
119890119910= radic
119901
120583
minus119891119903cos 119871 + [(1 + 120596) sin 119871 + 119890
119910]
119891119905
120596
+ (ℎ119909sin 119871 minus ℎ
119910cos 119871)
119890119909119891119899
120596
International Journal of Aerospace Engineering 3
rarr
T
rarr
Nf
rarr
R
O
120572 120573
Figure 2 Coordinate frame
Transfer orbit EarthSun
Os
Rs rsOe
r120579
Re
Figure 3 Cylindrical projection model
ℎ119909= radic
119901
120583
1199042119891119899
2120596
cos 119871
ℎ119910= radic
119901
120583
1199042119891119899
2120596
sin 119871
= radic120583119901(
120596
119901
)
2
+
1
120596
radic
119901
120583
(ℎ119909sin 119871 minus ℎ
119910cos 119871)119891
119899
(3)
with120596 = 1+119890119909cos 119871+119890
119910sin 119871 1199042 = 1+ℎ
2
119909+ℎ2
119910The acceleration
components 119891119903 119891119905 and 119891
119899are denoted in the RTN frame
(with 119891119903being the normal direction 119891
119905being the tangential
direction and 119891119899being the direction orthogonal to the orbit
plane) and 119891 is the thrust accelerationThe constraint on thecontrol is 119891 le 119891max = 119879max119898
To take into account the true acceleration of the thrustwe must consider the mass flow Its evolution is given by
119889119898
119889119905
= minus
119879
119868sp1198920 (4)
where 119879 is the thrust modulus 119868sp the specific impulse ofthruster and 119892
0the gravitational acceleration at sea-level and
1198920= 980665NsAs shown in Figure 2 thrust acceleration components
can also be defined by acceleration 119891 and two control angles(pitch-steering angle 120572 and yaw-steering angle 120573) whichdescribe the direction of the thrust vector in relation to
the velocity vector and the orbital plane respectively Theacceleration components are expressed by
119891119903= 119891 cos120573 sin120572
119891119905= 119891 cos120573 cos120572
119891119899= 119891 sin120573
(5)
with minus120587 le 120572 le 120587 and minus1205872 le 120573 le 1205872
22 Eclipse Effects To model the electric propulsion systemaccurately it is necessary tomodel the satellitersquos trajectory as itpasses through the shadow of Earth In particular the satelliteis at discharge which leads to high power consumption whenin the shadow In order to ensure the safety of spacecrafttherefore the thrust is 0 when the vehicle is in the shadowof Earth A simplified Earth-shadow model namely thecylindrical projection model shown in Figure 3 is used toestimate the location of the shadow [17ndash20]
Referring to the geometry illustrated in Figure 3 r119904is
defined as the vector from Earth to the Sun with norm r119904
and r as the vector from Earth to the satellite with norm r119877119890is the radius of Earth and 119877
119904is the radius of the Sun
The cone angle 120579 is defined by
cos 120579 = angr119904r =
r sdot r119904
(r 1003817100381710038171003817r119904
1003817100381710038171003817)
(6)
4 International Journal of Aerospace Engineering
Earth
Transfer orbit
Figure 4 Low-thrust transfer with no radar detection
The projected spacecraft position r is given by
r = r[
[
[
[
cos V cosΩ cos120596 minus cos V sinΩ sin120596 cos 119894 minus sin V cosΩ cos120596 minus sin V sinΩ sin120596 cos 119894
cos V sinΩ cos120596 + cos V cosΩ sin120596 cos 119894 minus sin V sinΩ cos120596 + sin V cosΩ sin120596 cos 119894
cos V sin120596 sin 119894 + sin V cos120596 sin 119894
]
]
]
]
(7)
The shadow entry and exit locations are judged bylocating the cone terminators at the projected spacecraftlocation The shadow can only be found when the anglesatisfies
cos 120579 le 0
sin 120579 lt
119877119890
119903
(8)
23 Perturbation Earthrsquos oblateness atmospheric drag lightpressure secondary body and other factors in space can alsoperturb a satellitersquos motion Among these variables Earthrsquosoblateness is of vital importance for predicting the trajectoryof the satellite accurately The oblate Earth perturbation iscaused by the reality of Earthrsquos shape not being perfectlyspherical The impact of Earthrsquos oblateness due to 119869
2is always
taken into account in the engineering calculations [20 21]In terms of orbital elements the dynamical system is
119889119886
119889119905
= 0
119889119890
119889119905
= 0
119889119894
119889119905
= 0
119889Ω
119889119905
= minus
311989911986921198772
119890
2 (1 minus 1198902) 1198862cos 119894
119889120596
119889119905
=
311989911986921198772
119890
4 (1 minus 1198902) 1198862(5 cos2119894 minus 1)
119889119872
119889119905
= 119899 minus
311989911986921198772
119890
4 (1 minus 1198902)32
1198862
(1 minus 3 cos2119894)
(9)
The effect of Earthrsquos oblateness (due to 1198692) on the orbital
transfer can be included by appending the perturbation to therespective right-hand sides of (5)
119891119903=
3
2
1198692
119906
11990341198772
119864(3sin2119894 sin2119906 minus 1)
119891119905= minus
3
2
1198692
119906
11990341198772
119864sin2119894 sin 2119906
119891119899= minus1198692
119906
11990341198772
119864sin 2119894 sin 119906
(10)
3 Satellite Low Observability
The research activity presented here is focused on the opti-mum algorithm and does not take into account space envi-ronment analysis especially the low-observable constraintThe low-observable satellite is designed to minimize itsfrontal RCS requiring low-observable shape design and flightattitude adjustment [6ndash8] The satellite keeps its front towardEarth when it is flying over ground-based radar detectionareas
31 Radar Detection Area In order to avoid reflecting radarsignals directly the scanning range of the ground-based radarshould be modeled first Figure 4 shows low-thrust GEOsatellite transfer with no radar detection Figure 5 is anillustration of that GEO satellite being detected in low-thrusttransfer by a ground-based radar and the shadow is the radardetection area
International Journal of Aerospace Engineering 5
Radar detection area
Transfer orbit
Earth
Figure 5 Low-thrust transfer with radar detection
Due to the limited energy and probability of interceptradar repeats its search for the target in a narrow area whichismodeled as a specific coverage of yaw angle pitch angle andoperating distance rather than the entire airspace [22ndash24]
The latitude and longitude of the ground station aredefined respectively as 120582
119901and 120575
119901 119878 is the track of the sub-
satellite point whose right ascension and declination aredenoted as 120582 and 120575119882
119890is the rotation speed of Earth
Consider
120582 = 119886119903 tan (cos 119894 tan 119906) + Ω minus 119882119890119905
120575 = 119886119903 sin (sin 119894 sin 119906)
(11)
The radar pitch angle 120595ℎand radar yaw angle 120601
ℎare
obtained by the spherical triangle Respectively
tan120595ℎ=
119903 cos 120574 minus 119877119890
119903 cos 120574
sin120601ℎ=
sin (120582 minus 120582119901) sin (90
∘minus 120575)
sin 120574
(12)
where 119903 = (119886(1 minus 1198902)(1 + 119890cosV)) sdot 120574 is the geocentric angle
between the subsatellite point and observation points It sat-isfies
cos 120574 = sin 120575119901sin 120575 + cos 120575
119901cos 120575 cos (120582 minus 120582
119901) (13)
However their distance is
119877OP = radic(119877119890+ ℎ)2
+ 1198772
119890minus 2119877119890(119877119890+ ℎ) cos 120574 (14)
Suppose the radar yaw angle [1205901 1205902] pitch angle [120591
1 1205912]
and the operating distance 119903op
120595ℎisin [1205901 1205902]
120601ℎisin [1205911 1205912]
119877OP le 119903op
(15)
Only a satellite meeting these conditions has access to theradar detection area
32 Thrust Control Angels and Satellite Attitude Four thrust-ers are installed on the floor of the satellite as shown in
1 2 3 4
Frontallow RCS shape
design
Direction of solar irradiation
Thrusters
Figure 6 Thruster installation
Figure 6 Two work during orbit transfer and the other twoact as a backup systemThrust is perpendicular to the bottomof the satellite in low-thrust transferThe satellite has a frontalRCS shape design Assuming the maximum thrust is 119865 thenthrust components in the satellite coordinates are
[119865119909119865119910
119865119911]
119879
= [0 119865 0]
119879
(16)
The RTN frame is where the attitude of three axis-stabi-lized satellites is defined in which thrust components aredescribed as
[119865119909119900
119865yo 119865119911119900
]
119879
= [119865 cos120573 cos120572 119865 cos120573 sin120572 119865 sin120573]
119879
(17)
Satellite attitude is related to the order of three rotations[20] The yaw angle roll angle and pitch angle of the satelliteare recorded as 120601 120595 and 120579 and are derived in the orderof 3-1-2 Define attitude matrix 119860
312= 1198772(120579)1198771(120595)1198773(120601)
and its transpose matrix 119860minus1
312= 119877minus1
3(120601)119877minus1
1(120595)119877minus1
2(120579) The
relationship between satellite attitude and thrust controlangles is
6 International Journal of Aerospace Engineering
[
[
[
cos120573 cos120572sin120573
cos120573 sin120572
]
]
]
= 119860minus1
312
[
[
[
0
1
0
]
]
]
= 119877minus1
3(120601) 119877minus1
1(120595) 119877minus1
2(120579)
[
[
[
0
1
0
]
]
]
=[
[
[
cos120601 minus sin120601 0
sin120601 cos120601 0
0 0 1
]
]
]
[
[
[
1 0 0
0 cos120595 minus sin120595
0 sin120595 cos120595
]
]
]
[
[
[
cos 120579 0 sin 120579
0 1 0
minus sin 120579 0 cos 120579
]
]
]
[
[
[
0
1
0
]
]
]
=
[
[
[
[
cos120601 cos 120579 minus sin120601 sin120595 sin 120579 minus sin120601 cos120595 cos120601 sin 120579 + sin120601 sin120595 cos 120579
sin120601 cos 120579 + cos120601 sin120595 sin 120579 cos120601 cos120595 sin120601 sin 120579 minus cos120601 sin120595 cos 120579
minus sin120601 sin 120579 sin120595 cos120595 cos 120579
]
]
]
]
[
[
[
[
0
1
0
]
]
]
]
=
[
[
[
[
minus sin120601 cos120595
cos120601 cos120595
sin120595
]
]
]
]
(18)
It is ultimately expressed as
cos120573 cos120572 = minus sin120601 cos120595
sin120573 = cos120601 cos120595
cos120573 sin120572 = sin120595
(19)
The attitude adjustment range in the radar irradiationarea is assumed to be
120601min lt 120601 lt 120601max
120595min lt 120595 lt 120595max
120579min lt 120579 lt 120579max
(20)
The corresponding acceleration thrust components can beobtained by
119891119903= 119891 cos120573 sin120572 = minus119891 sin120601 cos120595
119891119905= 119891 cos120573 cos120572 = 119891 cos120601 cos120595
119891119899= 119891 sin120573 = 119891 sin120595
(21)
To realize low RCS toward Earth our paper sets theoptimal satellite attitude at a range of plusmn5
∘ which can beadjusted according to the real simulation and test Consider
120601 isin [minus5∘ 5∘]
120595 isin [minus5∘ 5∘]
120579 isin [175∘ 185∘]
(22)
4 Low-Thrust GEO Transfer Design
The goal of this paper is to design a minimum-time transferfor geostationary spacecraft equipped with electric propul-sion systems The transfer problem is thus to find an essen-tially bound control to reduce eccentricity and inclinationand raise the semimajor axis
After simplifying (1) they fall into
119886 =
21198862
ℎ
[119890 sin V119891119903+ (1 + 119890 cos V) 119891
119905]
119890 =
119901
ℎ
[sin V119891119903+ (cos V +
119890 + cos V1 + 119890 cos V
)119891119905]
119894 =
119903
ℎ
cos (120596 + V) 119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(23)
The acceleration component 119891119899only contributes to a
decrease of inclinationThe change of the semimajor axis andeccentricity are both related to 119891
119903and 119891
119905 119891119903and 119891
119905have a
greater effect on the semimajor axis than eccentricityFirst a parameter 119896 is introduced to isolate 119891
119899from 119891
119891119903119905
= radic1198912
119903+ 1198912
119905= 119891 sdot 119896
119891119899= 119891radic1 minus 119896
2
(24)
with 0 le 119896 le 1Then a control law that reduces eccentricity quickly is
derived by observing the time-rate equation for the semi-major axis and eccentricity Define 119890
1199011= sin V and 119890
1199012=
cos V + (119890 + cos V)(1 + 119890 cos V) The law is given by119891119903= 119891 sdot 119896
119891119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119903= 0
119891119905= 119891 sdot 119896
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
(25)
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
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DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 3
rarr
T
rarr
Nf
rarr
R
O
120572 120573
Figure 2 Coordinate frame
Transfer orbit EarthSun
Os
Rs rsOe
r120579
Re
Figure 3 Cylindrical projection model
ℎ119909= radic
119901
120583
1199042119891119899
2120596
cos 119871
ℎ119910= radic
119901
120583
1199042119891119899
2120596
sin 119871
= radic120583119901(
120596
119901
)
2
+
1
120596
radic
119901
120583
(ℎ119909sin 119871 minus ℎ
119910cos 119871)119891
119899
(3)
with120596 = 1+119890119909cos 119871+119890
119910sin 119871 1199042 = 1+ℎ
2
119909+ℎ2
119910The acceleration
components 119891119903 119891119905 and 119891
119899are denoted in the RTN frame
(with 119891119903being the normal direction 119891
119905being the tangential
direction and 119891119899being the direction orthogonal to the orbit
plane) and 119891 is the thrust accelerationThe constraint on thecontrol is 119891 le 119891max = 119879max119898
To take into account the true acceleration of the thrustwe must consider the mass flow Its evolution is given by
119889119898
119889119905
= minus
119879
119868sp1198920 (4)
where 119879 is the thrust modulus 119868sp the specific impulse ofthruster and 119892
0the gravitational acceleration at sea-level and
1198920= 980665NsAs shown in Figure 2 thrust acceleration components
can also be defined by acceleration 119891 and two control angles(pitch-steering angle 120572 and yaw-steering angle 120573) whichdescribe the direction of the thrust vector in relation to
the velocity vector and the orbital plane respectively Theacceleration components are expressed by
119891119903= 119891 cos120573 sin120572
119891119905= 119891 cos120573 cos120572
119891119899= 119891 sin120573
(5)
with minus120587 le 120572 le 120587 and minus1205872 le 120573 le 1205872
22 Eclipse Effects To model the electric propulsion systemaccurately it is necessary tomodel the satellitersquos trajectory as itpasses through the shadow of Earth In particular the satelliteis at discharge which leads to high power consumption whenin the shadow In order to ensure the safety of spacecrafttherefore the thrust is 0 when the vehicle is in the shadowof Earth A simplified Earth-shadow model namely thecylindrical projection model shown in Figure 3 is used toestimate the location of the shadow [17ndash20]
Referring to the geometry illustrated in Figure 3 r119904is
defined as the vector from Earth to the Sun with norm r119904
and r as the vector from Earth to the satellite with norm r119877119890is the radius of Earth and 119877
119904is the radius of the Sun
The cone angle 120579 is defined by
cos 120579 = angr119904r =
r sdot r119904
(r 1003817100381710038171003817r119904
1003817100381710038171003817)
(6)
4 International Journal of Aerospace Engineering
Earth
Transfer orbit
Figure 4 Low-thrust transfer with no radar detection
The projected spacecraft position r is given by
r = r[
[
[
[
cos V cosΩ cos120596 minus cos V sinΩ sin120596 cos 119894 minus sin V cosΩ cos120596 minus sin V sinΩ sin120596 cos 119894
cos V sinΩ cos120596 + cos V cosΩ sin120596 cos 119894 minus sin V sinΩ cos120596 + sin V cosΩ sin120596 cos 119894
cos V sin120596 sin 119894 + sin V cos120596 sin 119894
]
]
]
]
(7)
The shadow entry and exit locations are judged bylocating the cone terminators at the projected spacecraftlocation The shadow can only be found when the anglesatisfies
cos 120579 le 0
sin 120579 lt
119877119890
119903
(8)
23 Perturbation Earthrsquos oblateness atmospheric drag lightpressure secondary body and other factors in space can alsoperturb a satellitersquos motion Among these variables Earthrsquosoblateness is of vital importance for predicting the trajectoryof the satellite accurately The oblate Earth perturbation iscaused by the reality of Earthrsquos shape not being perfectlyspherical The impact of Earthrsquos oblateness due to 119869
2is always
taken into account in the engineering calculations [20 21]In terms of orbital elements the dynamical system is
119889119886
119889119905
= 0
119889119890
119889119905
= 0
119889119894
119889119905
= 0
119889Ω
119889119905
= minus
311989911986921198772
119890
2 (1 minus 1198902) 1198862cos 119894
119889120596
119889119905
=
311989911986921198772
119890
4 (1 minus 1198902) 1198862(5 cos2119894 minus 1)
119889119872
119889119905
= 119899 minus
311989911986921198772
119890
4 (1 minus 1198902)32
1198862
(1 minus 3 cos2119894)
(9)
The effect of Earthrsquos oblateness (due to 1198692) on the orbital
transfer can be included by appending the perturbation to therespective right-hand sides of (5)
119891119903=
3
2
1198692
119906
11990341198772
119864(3sin2119894 sin2119906 minus 1)
119891119905= minus
3
2
1198692
119906
11990341198772
119864sin2119894 sin 2119906
119891119899= minus1198692
119906
11990341198772
119864sin 2119894 sin 119906
(10)
3 Satellite Low Observability
The research activity presented here is focused on the opti-mum algorithm and does not take into account space envi-ronment analysis especially the low-observable constraintThe low-observable satellite is designed to minimize itsfrontal RCS requiring low-observable shape design and flightattitude adjustment [6ndash8] The satellite keeps its front towardEarth when it is flying over ground-based radar detectionareas
31 Radar Detection Area In order to avoid reflecting radarsignals directly the scanning range of the ground-based radarshould be modeled first Figure 4 shows low-thrust GEOsatellite transfer with no radar detection Figure 5 is anillustration of that GEO satellite being detected in low-thrusttransfer by a ground-based radar and the shadow is the radardetection area
International Journal of Aerospace Engineering 5
Radar detection area
Transfer orbit
Earth
Figure 5 Low-thrust transfer with radar detection
Due to the limited energy and probability of interceptradar repeats its search for the target in a narrow area whichismodeled as a specific coverage of yaw angle pitch angle andoperating distance rather than the entire airspace [22ndash24]
The latitude and longitude of the ground station aredefined respectively as 120582
119901and 120575
119901 119878 is the track of the sub-
satellite point whose right ascension and declination aredenoted as 120582 and 120575119882
119890is the rotation speed of Earth
Consider
120582 = 119886119903 tan (cos 119894 tan 119906) + Ω minus 119882119890119905
120575 = 119886119903 sin (sin 119894 sin 119906)
(11)
The radar pitch angle 120595ℎand radar yaw angle 120601
ℎare
obtained by the spherical triangle Respectively
tan120595ℎ=
119903 cos 120574 minus 119877119890
119903 cos 120574
sin120601ℎ=
sin (120582 minus 120582119901) sin (90
∘minus 120575)
sin 120574
(12)
where 119903 = (119886(1 minus 1198902)(1 + 119890cosV)) sdot 120574 is the geocentric angle
between the subsatellite point and observation points It sat-isfies
cos 120574 = sin 120575119901sin 120575 + cos 120575
119901cos 120575 cos (120582 minus 120582
119901) (13)
However their distance is
119877OP = radic(119877119890+ ℎ)2
+ 1198772
119890minus 2119877119890(119877119890+ ℎ) cos 120574 (14)
Suppose the radar yaw angle [1205901 1205902] pitch angle [120591
1 1205912]
and the operating distance 119903op
120595ℎisin [1205901 1205902]
120601ℎisin [1205911 1205912]
119877OP le 119903op
(15)
Only a satellite meeting these conditions has access to theradar detection area
32 Thrust Control Angels and Satellite Attitude Four thrust-ers are installed on the floor of the satellite as shown in
1 2 3 4
Frontallow RCS shape
design
Direction of solar irradiation
Thrusters
Figure 6 Thruster installation
Figure 6 Two work during orbit transfer and the other twoact as a backup systemThrust is perpendicular to the bottomof the satellite in low-thrust transferThe satellite has a frontalRCS shape design Assuming the maximum thrust is 119865 thenthrust components in the satellite coordinates are
[119865119909119865119910
119865119911]
119879
= [0 119865 0]
119879
(16)
The RTN frame is where the attitude of three axis-stabi-lized satellites is defined in which thrust components aredescribed as
[119865119909119900
119865yo 119865119911119900
]
119879
= [119865 cos120573 cos120572 119865 cos120573 sin120572 119865 sin120573]
119879
(17)
Satellite attitude is related to the order of three rotations[20] The yaw angle roll angle and pitch angle of the satelliteare recorded as 120601 120595 and 120579 and are derived in the orderof 3-1-2 Define attitude matrix 119860
312= 1198772(120579)1198771(120595)1198773(120601)
and its transpose matrix 119860minus1
312= 119877minus1
3(120601)119877minus1
1(120595)119877minus1
2(120579) The
relationship between satellite attitude and thrust controlangles is
6 International Journal of Aerospace Engineering
[
[
[
cos120573 cos120572sin120573
cos120573 sin120572
]
]
]
= 119860minus1
312
[
[
[
0
1
0
]
]
]
= 119877minus1
3(120601) 119877minus1
1(120595) 119877minus1
2(120579)
[
[
[
0
1
0
]
]
]
=[
[
[
cos120601 minus sin120601 0
sin120601 cos120601 0
0 0 1
]
]
]
[
[
[
1 0 0
0 cos120595 minus sin120595
0 sin120595 cos120595
]
]
]
[
[
[
cos 120579 0 sin 120579
0 1 0
minus sin 120579 0 cos 120579
]
]
]
[
[
[
0
1
0
]
]
]
=
[
[
[
[
cos120601 cos 120579 minus sin120601 sin120595 sin 120579 minus sin120601 cos120595 cos120601 sin 120579 + sin120601 sin120595 cos 120579
sin120601 cos 120579 + cos120601 sin120595 sin 120579 cos120601 cos120595 sin120601 sin 120579 minus cos120601 sin120595 cos 120579
minus sin120601 sin 120579 sin120595 cos120595 cos 120579
]
]
]
]
[
[
[
[
0
1
0
]
]
]
]
=
[
[
[
[
minus sin120601 cos120595
cos120601 cos120595
sin120595
]
]
]
]
(18)
It is ultimately expressed as
cos120573 cos120572 = minus sin120601 cos120595
sin120573 = cos120601 cos120595
cos120573 sin120572 = sin120595
(19)
The attitude adjustment range in the radar irradiationarea is assumed to be
120601min lt 120601 lt 120601max
120595min lt 120595 lt 120595max
120579min lt 120579 lt 120579max
(20)
The corresponding acceleration thrust components can beobtained by
119891119903= 119891 cos120573 sin120572 = minus119891 sin120601 cos120595
119891119905= 119891 cos120573 cos120572 = 119891 cos120601 cos120595
119891119899= 119891 sin120573 = 119891 sin120595
(21)
To realize low RCS toward Earth our paper sets theoptimal satellite attitude at a range of plusmn5
∘ which can beadjusted according to the real simulation and test Consider
120601 isin [minus5∘ 5∘]
120595 isin [minus5∘ 5∘]
120579 isin [175∘ 185∘]
(22)
4 Low-Thrust GEO Transfer Design
The goal of this paper is to design a minimum-time transferfor geostationary spacecraft equipped with electric propul-sion systems The transfer problem is thus to find an essen-tially bound control to reduce eccentricity and inclinationand raise the semimajor axis
After simplifying (1) they fall into
119886 =
21198862
ℎ
[119890 sin V119891119903+ (1 + 119890 cos V) 119891
119905]
119890 =
119901
ℎ
[sin V119891119903+ (cos V +
119890 + cos V1 + 119890 cos V
)119891119905]
119894 =
119903
ℎ
cos (120596 + V) 119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(23)
The acceleration component 119891119899only contributes to a
decrease of inclinationThe change of the semimajor axis andeccentricity are both related to 119891
119903and 119891
119905 119891119903and 119891
119905have a
greater effect on the semimajor axis than eccentricityFirst a parameter 119896 is introduced to isolate 119891
119899from 119891
119891119903119905
= radic1198912
119903+ 1198912
119905= 119891 sdot 119896
119891119899= 119891radic1 minus 119896
2
(24)
with 0 le 119896 le 1Then a control law that reduces eccentricity quickly is
derived by observing the time-rate equation for the semi-major axis and eccentricity Define 119890
1199011= sin V and 119890
1199012=
cos V + (119890 + cos V)(1 + 119890 cos V) The law is given by119891119903= 119891 sdot 119896
119891119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119903= 0
119891119905= 119891 sdot 119896
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
(25)
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
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Shock and Vibration
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
4 International Journal of Aerospace Engineering
Earth
Transfer orbit
Figure 4 Low-thrust transfer with no radar detection
The projected spacecraft position r is given by
r = r[
[
[
[
cos V cosΩ cos120596 minus cos V sinΩ sin120596 cos 119894 minus sin V cosΩ cos120596 minus sin V sinΩ sin120596 cos 119894
cos V sinΩ cos120596 + cos V cosΩ sin120596 cos 119894 minus sin V sinΩ cos120596 + sin V cosΩ sin120596 cos 119894
cos V sin120596 sin 119894 + sin V cos120596 sin 119894
]
]
]
]
(7)
The shadow entry and exit locations are judged bylocating the cone terminators at the projected spacecraftlocation The shadow can only be found when the anglesatisfies
cos 120579 le 0
sin 120579 lt
119877119890
119903
(8)
23 Perturbation Earthrsquos oblateness atmospheric drag lightpressure secondary body and other factors in space can alsoperturb a satellitersquos motion Among these variables Earthrsquosoblateness is of vital importance for predicting the trajectoryof the satellite accurately The oblate Earth perturbation iscaused by the reality of Earthrsquos shape not being perfectlyspherical The impact of Earthrsquos oblateness due to 119869
2is always
taken into account in the engineering calculations [20 21]In terms of orbital elements the dynamical system is
119889119886
119889119905
= 0
119889119890
119889119905
= 0
119889119894
119889119905
= 0
119889Ω
119889119905
= minus
311989911986921198772
119890
2 (1 minus 1198902) 1198862cos 119894
119889120596
119889119905
=
311989911986921198772
119890
4 (1 minus 1198902) 1198862(5 cos2119894 minus 1)
119889119872
119889119905
= 119899 minus
311989911986921198772
119890
4 (1 minus 1198902)32
1198862
(1 minus 3 cos2119894)
(9)
The effect of Earthrsquos oblateness (due to 1198692) on the orbital
transfer can be included by appending the perturbation to therespective right-hand sides of (5)
119891119903=
3
2
1198692
119906
11990341198772
119864(3sin2119894 sin2119906 minus 1)
119891119905= minus
3
2
1198692
119906
11990341198772
119864sin2119894 sin 2119906
119891119899= minus1198692
119906
11990341198772
119864sin 2119894 sin 119906
(10)
3 Satellite Low Observability
The research activity presented here is focused on the opti-mum algorithm and does not take into account space envi-ronment analysis especially the low-observable constraintThe low-observable satellite is designed to minimize itsfrontal RCS requiring low-observable shape design and flightattitude adjustment [6ndash8] The satellite keeps its front towardEarth when it is flying over ground-based radar detectionareas
31 Radar Detection Area In order to avoid reflecting radarsignals directly the scanning range of the ground-based radarshould be modeled first Figure 4 shows low-thrust GEOsatellite transfer with no radar detection Figure 5 is anillustration of that GEO satellite being detected in low-thrusttransfer by a ground-based radar and the shadow is the radardetection area
International Journal of Aerospace Engineering 5
Radar detection area
Transfer orbit
Earth
Figure 5 Low-thrust transfer with radar detection
Due to the limited energy and probability of interceptradar repeats its search for the target in a narrow area whichismodeled as a specific coverage of yaw angle pitch angle andoperating distance rather than the entire airspace [22ndash24]
The latitude and longitude of the ground station aredefined respectively as 120582
119901and 120575
119901 119878 is the track of the sub-
satellite point whose right ascension and declination aredenoted as 120582 and 120575119882
119890is the rotation speed of Earth
Consider
120582 = 119886119903 tan (cos 119894 tan 119906) + Ω minus 119882119890119905
120575 = 119886119903 sin (sin 119894 sin 119906)
(11)
The radar pitch angle 120595ℎand radar yaw angle 120601
ℎare
obtained by the spherical triangle Respectively
tan120595ℎ=
119903 cos 120574 minus 119877119890
119903 cos 120574
sin120601ℎ=
sin (120582 minus 120582119901) sin (90
∘minus 120575)
sin 120574
(12)
where 119903 = (119886(1 minus 1198902)(1 + 119890cosV)) sdot 120574 is the geocentric angle
between the subsatellite point and observation points It sat-isfies
cos 120574 = sin 120575119901sin 120575 + cos 120575
119901cos 120575 cos (120582 minus 120582
119901) (13)
However their distance is
119877OP = radic(119877119890+ ℎ)2
+ 1198772
119890minus 2119877119890(119877119890+ ℎ) cos 120574 (14)
Suppose the radar yaw angle [1205901 1205902] pitch angle [120591
1 1205912]
and the operating distance 119903op
120595ℎisin [1205901 1205902]
120601ℎisin [1205911 1205912]
119877OP le 119903op
(15)
Only a satellite meeting these conditions has access to theradar detection area
32 Thrust Control Angels and Satellite Attitude Four thrust-ers are installed on the floor of the satellite as shown in
1 2 3 4
Frontallow RCS shape
design
Direction of solar irradiation
Thrusters
Figure 6 Thruster installation
Figure 6 Two work during orbit transfer and the other twoact as a backup systemThrust is perpendicular to the bottomof the satellite in low-thrust transferThe satellite has a frontalRCS shape design Assuming the maximum thrust is 119865 thenthrust components in the satellite coordinates are
[119865119909119865119910
119865119911]
119879
= [0 119865 0]
119879
(16)
The RTN frame is where the attitude of three axis-stabi-lized satellites is defined in which thrust components aredescribed as
[119865119909119900
119865yo 119865119911119900
]
119879
= [119865 cos120573 cos120572 119865 cos120573 sin120572 119865 sin120573]
119879
(17)
Satellite attitude is related to the order of three rotations[20] The yaw angle roll angle and pitch angle of the satelliteare recorded as 120601 120595 and 120579 and are derived in the orderof 3-1-2 Define attitude matrix 119860
312= 1198772(120579)1198771(120595)1198773(120601)
and its transpose matrix 119860minus1
312= 119877minus1
3(120601)119877minus1
1(120595)119877minus1
2(120579) The
relationship between satellite attitude and thrust controlangles is
6 International Journal of Aerospace Engineering
[
[
[
cos120573 cos120572sin120573
cos120573 sin120572
]
]
]
= 119860minus1
312
[
[
[
0
1
0
]
]
]
= 119877minus1
3(120601) 119877minus1
1(120595) 119877minus1
2(120579)
[
[
[
0
1
0
]
]
]
=[
[
[
cos120601 minus sin120601 0
sin120601 cos120601 0
0 0 1
]
]
]
[
[
[
1 0 0
0 cos120595 minus sin120595
0 sin120595 cos120595
]
]
]
[
[
[
cos 120579 0 sin 120579
0 1 0
minus sin 120579 0 cos 120579
]
]
]
[
[
[
0
1
0
]
]
]
=
[
[
[
[
cos120601 cos 120579 minus sin120601 sin120595 sin 120579 minus sin120601 cos120595 cos120601 sin 120579 + sin120601 sin120595 cos 120579
sin120601 cos 120579 + cos120601 sin120595 sin 120579 cos120601 cos120595 sin120601 sin 120579 minus cos120601 sin120595 cos 120579
minus sin120601 sin 120579 sin120595 cos120595 cos 120579
]
]
]
]
[
[
[
[
0
1
0
]
]
]
]
=
[
[
[
[
minus sin120601 cos120595
cos120601 cos120595
sin120595
]
]
]
]
(18)
It is ultimately expressed as
cos120573 cos120572 = minus sin120601 cos120595
sin120573 = cos120601 cos120595
cos120573 sin120572 = sin120595
(19)
The attitude adjustment range in the radar irradiationarea is assumed to be
120601min lt 120601 lt 120601max
120595min lt 120595 lt 120595max
120579min lt 120579 lt 120579max
(20)
The corresponding acceleration thrust components can beobtained by
119891119903= 119891 cos120573 sin120572 = minus119891 sin120601 cos120595
119891119905= 119891 cos120573 cos120572 = 119891 cos120601 cos120595
119891119899= 119891 sin120573 = 119891 sin120595
(21)
To realize low RCS toward Earth our paper sets theoptimal satellite attitude at a range of plusmn5
∘ which can beadjusted according to the real simulation and test Consider
120601 isin [minus5∘ 5∘]
120595 isin [minus5∘ 5∘]
120579 isin [175∘ 185∘]
(22)
4 Low-Thrust GEO Transfer Design
The goal of this paper is to design a minimum-time transferfor geostationary spacecraft equipped with electric propul-sion systems The transfer problem is thus to find an essen-tially bound control to reduce eccentricity and inclinationand raise the semimajor axis
After simplifying (1) they fall into
119886 =
21198862
ℎ
[119890 sin V119891119903+ (1 + 119890 cos V) 119891
119905]
119890 =
119901
ℎ
[sin V119891119903+ (cos V +
119890 + cos V1 + 119890 cos V
)119891119905]
119894 =
119903
ℎ
cos (120596 + V) 119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(23)
The acceleration component 119891119899only contributes to a
decrease of inclinationThe change of the semimajor axis andeccentricity are both related to 119891
119903and 119891
119905 119891119903and 119891
119905have a
greater effect on the semimajor axis than eccentricityFirst a parameter 119896 is introduced to isolate 119891
119899from 119891
119891119903119905
= radic1198912
119903+ 1198912
119905= 119891 sdot 119896
119891119899= 119891radic1 minus 119896
2
(24)
with 0 le 119896 le 1Then a control law that reduces eccentricity quickly is
derived by observing the time-rate equation for the semi-major axis and eccentricity Define 119890
1199011= sin V and 119890
1199012=
cos V + (119890 + cos V)(1 + 119890 cos V) The law is given by119891119903= 119891 sdot 119896
119891119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119903= 0
119891119905= 119891 sdot 119896
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
(25)
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
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DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 5
Radar detection area
Transfer orbit
Earth
Figure 5 Low-thrust transfer with radar detection
Due to the limited energy and probability of interceptradar repeats its search for the target in a narrow area whichismodeled as a specific coverage of yaw angle pitch angle andoperating distance rather than the entire airspace [22ndash24]
The latitude and longitude of the ground station aredefined respectively as 120582
119901and 120575
119901 119878 is the track of the sub-
satellite point whose right ascension and declination aredenoted as 120582 and 120575119882
119890is the rotation speed of Earth
Consider
120582 = 119886119903 tan (cos 119894 tan 119906) + Ω minus 119882119890119905
120575 = 119886119903 sin (sin 119894 sin 119906)
(11)
The radar pitch angle 120595ℎand radar yaw angle 120601
ℎare
obtained by the spherical triangle Respectively
tan120595ℎ=
119903 cos 120574 minus 119877119890
119903 cos 120574
sin120601ℎ=
sin (120582 minus 120582119901) sin (90
∘minus 120575)
sin 120574
(12)
where 119903 = (119886(1 minus 1198902)(1 + 119890cosV)) sdot 120574 is the geocentric angle
between the subsatellite point and observation points It sat-isfies
cos 120574 = sin 120575119901sin 120575 + cos 120575
119901cos 120575 cos (120582 minus 120582
119901) (13)
However their distance is
119877OP = radic(119877119890+ ℎ)2
+ 1198772
119890minus 2119877119890(119877119890+ ℎ) cos 120574 (14)
Suppose the radar yaw angle [1205901 1205902] pitch angle [120591
1 1205912]
and the operating distance 119903op
120595ℎisin [1205901 1205902]
120601ℎisin [1205911 1205912]
119877OP le 119903op
(15)
Only a satellite meeting these conditions has access to theradar detection area
32 Thrust Control Angels and Satellite Attitude Four thrust-ers are installed on the floor of the satellite as shown in
1 2 3 4
Frontallow RCS shape
design
Direction of solar irradiation
Thrusters
Figure 6 Thruster installation
Figure 6 Two work during orbit transfer and the other twoact as a backup systemThrust is perpendicular to the bottomof the satellite in low-thrust transferThe satellite has a frontalRCS shape design Assuming the maximum thrust is 119865 thenthrust components in the satellite coordinates are
[119865119909119865119910
119865119911]
119879
= [0 119865 0]
119879
(16)
The RTN frame is where the attitude of three axis-stabi-lized satellites is defined in which thrust components aredescribed as
[119865119909119900
119865yo 119865119911119900
]
119879
= [119865 cos120573 cos120572 119865 cos120573 sin120572 119865 sin120573]
119879
(17)
Satellite attitude is related to the order of three rotations[20] The yaw angle roll angle and pitch angle of the satelliteare recorded as 120601 120595 and 120579 and are derived in the orderof 3-1-2 Define attitude matrix 119860
312= 1198772(120579)1198771(120595)1198773(120601)
and its transpose matrix 119860minus1
312= 119877minus1
3(120601)119877minus1
1(120595)119877minus1
2(120579) The
relationship between satellite attitude and thrust controlangles is
6 International Journal of Aerospace Engineering
[
[
[
cos120573 cos120572sin120573
cos120573 sin120572
]
]
]
= 119860minus1
312
[
[
[
0
1
0
]
]
]
= 119877minus1
3(120601) 119877minus1
1(120595) 119877minus1
2(120579)
[
[
[
0
1
0
]
]
]
=[
[
[
cos120601 minus sin120601 0
sin120601 cos120601 0
0 0 1
]
]
]
[
[
[
1 0 0
0 cos120595 minus sin120595
0 sin120595 cos120595
]
]
]
[
[
[
cos 120579 0 sin 120579
0 1 0
minus sin 120579 0 cos 120579
]
]
]
[
[
[
0
1
0
]
]
]
=
[
[
[
[
cos120601 cos 120579 minus sin120601 sin120595 sin 120579 minus sin120601 cos120595 cos120601 sin 120579 + sin120601 sin120595 cos 120579
sin120601 cos 120579 + cos120601 sin120595 sin 120579 cos120601 cos120595 sin120601 sin 120579 minus cos120601 sin120595 cos 120579
minus sin120601 sin 120579 sin120595 cos120595 cos 120579
]
]
]
]
[
[
[
[
0
1
0
]
]
]
]
=
[
[
[
[
minus sin120601 cos120595
cos120601 cos120595
sin120595
]
]
]
]
(18)
It is ultimately expressed as
cos120573 cos120572 = minus sin120601 cos120595
sin120573 = cos120601 cos120595
cos120573 sin120572 = sin120595
(19)
The attitude adjustment range in the radar irradiationarea is assumed to be
120601min lt 120601 lt 120601max
120595min lt 120595 lt 120595max
120579min lt 120579 lt 120579max
(20)
The corresponding acceleration thrust components can beobtained by
119891119903= 119891 cos120573 sin120572 = minus119891 sin120601 cos120595
119891119905= 119891 cos120573 cos120572 = 119891 cos120601 cos120595
119891119899= 119891 sin120573 = 119891 sin120595
(21)
To realize low RCS toward Earth our paper sets theoptimal satellite attitude at a range of plusmn5
∘ which can beadjusted according to the real simulation and test Consider
120601 isin [minus5∘ 5∘]
120595 isin [minus5∘ 5∘]
120579 isin [175∘ 185∘]
(22)
4 Low-Thrust GEO Transfer Design
The goal of this paper is to design a minimum-time transferfor geostationary spacecraft equipped with electric propul-sion systems The transfer problem is thus to find an essen-tially bound control to reduce eccentricity and inclinationand raise the semimajor axis
After simplifying (1) they fall into
119886 =
21198862
ℎ
[119890 sin V119891119903+ (1 + 119890 cos V) 119891
119905]
119890 =
119901
ℎ
[sin V119891119903+ (cos V +
119890 + cos V1 + 119890 cos V
)119891119905]
119894 =
119903
ℎ
cos (120596 + V) 119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(23)
The acceleration component 119891119899only contributes to a
decrease of inclinationThe change of the semimajor axis andeccentricity are both related to 119891
119903and 119891
119905 119891119903and 119891
119905have a
greater effect on the semimajor axis than eccentricityFirst a parameter 119896 is introduced to isolate 119891
119899from 119891
119891119903119905
= radic1198912
119903+ 1198912
119905= 119891 sdot 119896
119891119899= 119891radic1 minus 119896
2
(24)
with 0 le 119896 le 1Then a control law that reduces eccentricity quickly is
derived by observing the time-rate equation for the semi-major axis and eccentricity Define 119890
1199011= sin V and 119890
1199012=
cos V + (119890 + cos V)(1 + 119890 cos V) The law is given by119891119903= 119891 sdot 119896
119891119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119903= 0
119891119905= 119891 sdot 119896
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
(25)
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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International Journal of
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Navigation and Observation
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DistributedSensor Networks
International Journal of
6 International Journal of Aerospace Engineering
[
[
[
cos120573 cos120572sin120573
cos120573 sin120572
]
]
]
= 119860minus1
312
[
[
[
0
1
0
]
]
]
= 119877minus1
3(120601) 119877minus1
1(120595) 119877minus1
2(120579)
[
[
[
0
1
0
]
]
]
=[
[
[
cos120601 minus sin120601 0
sin120601 cos120601 0
0 0 1
]
]
]
[
[
[
1 0 0
0 cos120595 minus sin120595
0 sin120595 cos120595
]
]
]
[
[
[
cos 120579 0 sin 120579
0 1 0
minus sin 120579 0 cos 120579
]
]
]
[
[
[
0
1
0
]
]
]
=
[
[
[
[
cos120601 cos 120579 minus sin120601 sin120595 sin 120579 minus sin120601 cos120595 cos120601 sin 120579 + sin120601 sin120595 cos 120579
sin120601 cos 120579 + cos120601 sin120595 sin 120579 cos120601 cos120595 sin120601 sin 120579 minus cos120601 sin120595 cos 120579
minus sin120601 sin 120579 sin120595 cos120595 cos 120579
]
]
]
]
[
[
[
[
0
1
0
]
]
]
]
=
[
[
[
[
minus sin120601 cos120595
cos120601 cos120595
sin120595
]
]
]
]
(18)
It is ultimately expressed as
cos120573 cos120572 = minus sin120601 cos120595
sin120573 = cos120601 cos120595
cos120573 sin120572 = sin120595
(19)
The attitude adjustment range in the radar irradiationarea is assumed to be
120601min lt 120601 lt 120601max
120595min lt 120595 lt 120595max
120579min lt 120579 lt 120579max
(20)
The corresponding acceleration thrust components can beobtained by
119891119903= 119891 cos120573 sin120572 = minus119891 sin120601 cos120595
119891119905= 119891 cos120573 cos120572 = 119891 cos120601 cos120595
119891119899= 119891 sin120573 = 119891 sin120595
(21)
To realize low RCS toward Earth our paper sets theoptimal satellite attitude at a range of plusmn5
∘ which can beadjusted according to the real simulation and test Consider
120601 isin [minus5∘ 5∘]
120595 isin [minus5∘ 5∘]
120579 isin [175∘ 185∘]
(22)
4 Low-Thrust GEO Transfer Design
The goal of this paper is to design a minimum-time transferfor geostationary spacecraft equipped with electric propul-sion systems The transfer problem is thus to find an essen-tially bound control to reduce eccentricity and inclinationand raise the semimajor axis
After simplifying (1) they fall into
119886 =
21198862
ℎ
[119890 sin V119891119903+ (1 + 119890 cos V) 119891
119905]
119890 =
119901
ℎ
[sin V119891119903+ (cos V +
119890 + cos V1 + 119890 cos V
)119891119905]
119894 =
119903
ℎ
cos (120596 + V) 119891119899
Ω =
119903 sin (120596 + V)ℎ sin 119894
119891119899
= minus
119901 cos Vℎ119890
119891119903+
(119901 + 119903) sin Vℎ119890
119891119905
+
119903 sin V (120596 + V) cos 119894ℎ sin 119894
119891119899
= 119899 +
1
119886119890ℎ
[(119901 cos V minus 2119890119903) 119891119903minus (119901 + 119903) sin V119891
119905]
(23)
The acceleration component 119891119899only contributes to a
decrease of inclinationThe change of the semimajor axis andeccentricity are both related to 119891
119903and 119891
119905 119891119903and 119891
119905have a
greater effect on the semimajor axis than eccentricityFirst a parameter 119896 is introduced to isolate 119891
119899from 119891
119891119903119905
= radic1198912
119903+ 1198912
119905= 119891 sdot 119896
119891119899= 119891radic1 minus 119896
2
(24)
with 0 le 119896 le 1Then a control law that reduces eccentricity quickly is
derived by observing the time-rate equation for the semi-major axis and eccentricity Define 119890
1199011= sin V and 119890
1199012=
cos V + (119890 + cos V)(1 + 119890 cos V) The law is given by119891119903= 119891 sdot 119896
119891119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119903= 0
119891119905= 119891 sdot 119896
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
(25)
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Active and Passive Electronic Components
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RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 7
Table 1 Thrust modes in low-thrust transfer
Direction Modulus
Condition Mode 1 Mode 2 Mode 3 Mode 4 Mode 5119896 = 1 119896 = 1 119896 = 1 Earth shadow Low observability
119894119901= cos(120596 + V)119894119901gt 0 119891
119899= minus119891radic1 minus 119896
2
119891119899= 0 119891
119899= 0 119891
119899= 0
119894119901le 0 119891
119899= 119891radic1 minus 119896
2
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816gt
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
119891119899= 0
1198901199011
ge 0 119891119903= minus119891 sdot 119896 119891
119905= 0 119891
119903= minus119891 119891
119905= 0
119891119903= 119891
119891119905= 0
119891119903= 0 119891
119903= 0 119891
119905= 0
1198901199011
lt 0 119891119903= 119891 sdot 119896 119891
119905= 0 119891
119903= 119891 119891
119905= 0 119891
119905= 0 119891
119903= 0 119891
119905= 0
100381610038161003816100381610038161198901199011
10038161003816100381610038161003816le
100381610038161003816100381610038161198901199012
10038161003816100381610038161003816
1198901199012
ge 0 119891119903= 0 119891
119905= minus119891 sdot 119896 119891
119903= 0 119891
119905= minus119891 119891
119903= 0 119891
119905= 0
1198901199012
lt 0 119891119903= 0 119891
119905= 119891 sdot 119896 119891
119903= 0 119891
119905= 119891 119891
119903= 0 119891
119905= 119891
frfn ft fn
fr
ft
ftfn
minusft
minusfr
minusfr
minusfn
Figure 7 Reduction of inclination
Five thrust modes and both their correspondingmodulusand direction are summed up in Table 1
As stated our transfer problem is parameterized andthe control is given according to 119894
119901 1198901199011 and 119890
1199012 which are
determined by the argument of perigee and the true anomalyBecause the parameter 119896 is between 0 and 1 it is convenientto optimize it to minimize the transfer time by a simpletraversal which is a selection of 119896 from 0 to 1 by varying it in aprefixed step As Table 1 details fully our method for achiev-ing terminal orbit is performed in three steps
Step 1 (reduction of inclination) The thrust mode relies onmode 1 to reduce inclination by rational 119896 along with thesimultaneous targeting of the semimajor axis and eccentricityThe thrust form is shown in Figure 7
Step 2 (raising the semimajor axis) Mode 2 represents a lawused to control the semimajor axis and the eccentricity whenthe inclination reaches the target value The thrust operatesas shown in Figure 8
Step 3 (reducing eccentricity) While the semimajor axisreaches its target 119891
119905remains at 0 to ensure the stability of the
semimajor axis 119891119903is used to reduce eccentricity in mode 3
as illustrated in Figure 9
Step 4 Mode 4 is the thrust model for Earth shadow accord-ing to formula (9) to realize real-time judgment of shadow inwhich thrust is 0
fr
ft
minusft
minusfr
Figure 8 Raising semimajor axis
frfr
minusfrminusfr
Figure 9 Reducing eccentricity
Step 5 Mode 5 is used for the radar detection area to realizesatellite low observability The satellite adjusts to its bestattitude corresponding to thrust components 119891
119903= 0 119891
119899= 0
and if 1198901199012
lt 0 119891119905= 119891 otherwise 119891
119905= 0 because 119891
119905= 119891 con-
tributes to both the decrease of eccentricity and the increaseof the semimajor axis when 119890
1199012lt 0 (Figure 10)
In short the method described in this paper determinesa control depending on a parameter and the state of currentorbit Our control law is simple with only one uncertaincontrol parameter whose optimization is so convenient thatour minimum-time transfer problem is greatly simplified
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 International Journal of Aerospace Engineering
Thruster operation mode
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Estimation of eclipse
Cylindrical projection eclipse model
Relationship of thrust control angels and satellite attitude
Radar detection area(pitch angle yaw angle and distance)
Orb
it el
emen
ts u
pdat
e in
the l
ow-th
rust
GEO
tran
sfer
Modified equinoctial orbit model
No eclipseand noradar
Eclipse
No eclipsebut radar
i decreases a increases e decreases
i remains a increases e decreases
i remains a remains e decreases
fr ft and fn
fr ft and fn
Perturbation Earthrsquos (J2) oblateness
Figure 10 Overall scheme
5 Simulation
51 Initial Parameter The ideal attitude of the satellite wasassumed to have the range of its yaw angle roll angle andpitch angle as [minus5
∘ 5∘] [minus5∘ 5∘] and [175∘ 185∘] The yaw
angle of each radar ranges from minus60∘ to 60∘ pitch angleranges from 0 to 90∘ and operating distance was 15000 kmThey were located in the following places (longitudelati-tude) (65∘E 35∘N) (30∘E 75∘N) (75∘E 75∘N) (120∘E 75∘N)(165∘E 75∘N) (15∘E 75∘N) (60∘W 75∘N) (105∘W 75∘N) and(150∘W 75∘N) (30∘E 75∘S) (75∘E 75∘S) (120∘E 75∘S) (165∘E75∘S) (15∘E 75∘S) (60∘W 75∘S) (105∘W 75∘S) and (150∘W75∘S) (30∘E 45∘N) (75∘E 45∘N) (120∘E 45∘N) (165∘E45∘N) (15∘E 45∘N) (60∘W45∘N) (105∘W45∘N) and (150∘W45∘N) (30∘E 45∘S) (75∘E 45∘N) (120∘E 45∘S) (165∘E 45∘S)(15∘E 45∘S) (60∘W 45∘S) (105∘W 45∘S) and (150∘W 45∘S)(30∘E 15∘N) (75∘E 15∘N) (120∘E 15∘N) (165∘E 15∘N) (15∘E15∘N) (60∘W 15∘N) (105∘W 15∘N) and (150∘W 15∘N) (30∘E15∘S) (75∘E 15∘S) (120∘E 15∘S) (165∘E 15∘S) (15∘E 15∘S)(60∘W 15∘S) (105∘W 15∘S) and (150∘W 15∘S)
The satellitersquos initial mass 1198980= 2600 kg was assumed to
be equipped with four thrusters with a specific impulse of119868sp = 1600 s and a maximum thrust of 119865 = 80mNThe GTOdeparture date was fixed as 1 June 2008 for shadow calcula-tions Table 2 displays the initial GTO and GEO boundaryconditions
52 Thrust and Control Angles Figures 11 and 12 show thetime-evolution of thrust and thrust control angles in theorbit transfer in which Earth shadow and oblateness (119869
2)
effects were considered with no low-observable constraint
Table 2 Orbit parameters
Orbit parameters Initial orbit Final orbitGTO GEO
Semimajor axis 119886 (km) 24731 42164Inclination 119894 (deg) 285 0Eccentricity 119890 07194 0Right ascension Ω (deg) 17104 FreeArgument of perigee 120596 (deg) 1796 FreeMean anomaly 119872 (deg) 29044 Free
In contrast Figures 13 and 14 show results taking intoconsideration all of the aforementioned factors
As can be seen in Figures 11ndash14 the satellite flew over theradar detection area three times in these six orbit periodsrespectively 6263 times 10
6sim 6294 times 10
6 s 6363 times 106sim
6377 times 106 s and 644 times 10
6sim 6465 times 10
6 s The thrustcomponents were 119891
119903= 0 119891
119905= 119891 and 119891
119899= 0 The thruster
operated in mode 5 to maintain low observability of thesatellite (Table 3)
53 Numerical Results To demonstrate the validity of ourproposed method four scenarios are presented in Table 4Case 1 represents a transfer based on the optimal controlmethod l using Pontryaginrsquos maximum principle to find theoptimum control law [14] Cases 2 and 3 represent transfersbased on our method Case 3 involves Earth shadow andoblateness (119869
2) effects Cases 4 and 5 involve Earth shadow
oblateness (1198692) effects and low observability Case 4 considers
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 9fr
(N)
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
minus05
0
05
Figure 11 Time-evolution of thrust with no radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0
angl
e (de
g)
Pitc
h-ste
erin
g 100
Figure 12 Time-evolution of thrust control angles with no radardetection
only one radar detection area (65∘E 35∘N) while Case 5includes other 48 areas
The results of these four cases are summarized in detail inTable 4 The time-evolution of state for Case 3 is presented inFigures 15ndash17The time-evolution of state for Case 4 is shownin Figures 18ndash20The time-evolution of state for Case 5 is pre-sented in Figures 21ndash23 GTO-GEO transfer times were 230days and 245 days for the optimal method and our methodrespectively Case 4 led to a 288-day transfer requiring 507 kgof fuel Case 5 was a 410-day transfer consuming 723 kgof fuel Compared with the results of the optimal controlour method clearly offered performance near the results ofoptimal method with no ground estimation was particularlyflexible when orbit elements changed and presented no two-point boundary problem and a low computational burden
6 Conclusion
A new method based on control parameter analysis has beenused to design a low-thrust orbit transferThe optimization ofthe parameter introduced here is utilized to solve the optimal
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
times106
ft
(N) 05
0
minus05
fn
(N) 02
0
minus02
fr
(N)
minus05
0
05
Figure 13 Time-evolution of thrust with radar detection
Time (s)62 625 63 635 64 645 65
Time (s)62 625 63 635 64 645 65
times106
times106
Yaw
-ste
erin
gan
gle (
deg
)
50
0
minus50
minus100
0an
gle (
deg
)Pi
tch-
steer
ing 100
Figure 14 Time-evolution of thrust control angles with radardetection
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 15 Time-evolution of the semimajor axis of Case 3
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 International Journal of Aerospace Engineering
Table 3 Low-thrust transfer with low observability
Startstop time (s) Total time (s) Thrust componets N Control angels1 6263 times 10
6sim 6294 times 10
6 s 31 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
2 6363 times 106sim 6377 times 10
6 s 14 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
3 644 times 106sim 6465 times 10
6 s 25 times 104
119891119903= 0 119891
119905= 119891 and 119891
119899= 0 120572 = 0 120573 = 0
Table 4 Fuel consumption and transfer days
Mission GTO rarr GEOCase 1 Case 2 Case 3 Case 4 Case 5
Fuel consumption Δ119898 (kg) 404 431 443 507 723Transfer days Δ119905 (d) 230 245 251 288 410
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 16 Time-evolution of the eccentricity of Case 3
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
50 100 150 200 2500Time (d)
Figure 17 Time-evolution of the inclination of Case 3
control problem Satellite low observability was considered inGEO transfers and the effects of Earth shadow and oblatenesswere also included in our method
This approach possesses three important features (1)The approach is characterized by simplicity and feasibility
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
50 100 150 200 2500Time (d)
Figure 18 Time-evolution of the semimajor axis of Case 4
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
50 100 150 200 2500Time (d)
Figure 19 Time-evolution of the eccentricity of Case 4
on-board real-time solution less calculation and no initialguess for convergence (2) the influence of Earth shadow andperturbation is taken into account proving its strong faulttolerance and (3) a simple but effective approach is used to
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of Aerospace Engineering 11
0
5
10
15
20
25
30In
clina
tion
(deg
)
50 100 150 200 2500Time (d)
Figure 20 Time-evolution of the inclination of Case 4
times107
2
25
3
35
4
45
Sem
imaj
or ax
is (m
)
100 200 300 4000Time (d)
Figure 21 Time-evolution of the semimajor axis of Case 5
0
01
02
03
04
05
06
07
08
Ecce
ntric
ity
100 200 300 4000Time (d)
Figure 22 Time-evolution of the eccentricity of Case 5
0
5
10
15
20
25
30
Incli
natio
n (d
eg)
100 200 300 4000Time (d)
Figure 23 Time-evolution of the inclination of Case 5
achieve satellite low observability and reduce the threat ofradar In general this trajectorymethodwould serve as a use-ful preliminary design for GEO mission designers
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The paper is supported by the open funding project ofState Key Laboratory of Virtual Reality Technology and Sys-tems Beihang University (Grant no BUAA-VR-14KF-06)and National Natural Science Foundation of China (no61203188)
References
[1] C Wang and W Liao ldquoDevelopment status and application inspace attack-defense of stealthy satelliterdquo Aerospace ElectronicWarfare vol 27 no 4 pp 17ndash19 2011
[2] K Su and J Zhou ldquoDevelopment of satellite stealth technologyrdquoStealth Technology no 3 pp 9ndash14 2008
[3] J Zhang Y Gen X Qin and Y Zhao ldquoAnalysis of detection effi-ciency of space target surveillance radar to low-orbit satelliterdquoRadar Science and Technology vol 7 no 1 pp 10ndash14 2009
[4] L Meng X Zhang and Y Hou ldquoDevelopment analysis on lowdetectionable characteristics of satelliterdquo Aerospace ElectronicWarfare vol 26 no 1 pp 33ndash36 2010
[5] C Wang M Shane and T Nguyen ldquoRadio frequency interfer-ence from ground radars to a geostationary satelliterdquo in Pro-ceedings of the 20th AIAA International Communication SatelliteSystems Conference and Exhibit AIAA 2002-1948 MontrealCanada May 2002
[6] K SuKey technology research onmicro-satellite low observability[PhD thesis] Nanjing University of Aeronautics and Astronau-tics Nanjing China 2011
[7] D Micheli R Pastore G Giannini A Vricella and MMarchetti ldquoLow-cost low-observable satellites made of carbon
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 International Journal of Aerospace Engineering
nanostructured multilayersrdquo in Proceedings of the 53rd IsraelAnnual Conferenceon Aerospace Sciences Tel Aviv Israel March2013
[8] K Su and J J Zhou ldquoFlight attitude planning for low observablemicro-satellite shieldsrdquo Aeronautic et Astronautic Since vol 32no 4 pp 720ndash728 2011
[9] X Bian and Y Wei ldquoKey techniques for the application ofthe electric propulsion system in the geostationary satelliteplatformrdquo Aerospace Control and Application vol 34 no 1 pp20ndash24 2008
[10] P Cui D Qiao and H Cui Trajectory Design and Optimizationof Deep Space Exploration Science Press 2013
[11] G Tang H Zhang andGWangDynamics and Control of Low-Thrust Orbital Maneuver Science Press 2013
[12] B A Conway Spacecraft Trajectory Optimization CambridgeUniversity Press Cambridge UK 2010
[13] J T Betts ldquoSurvey of numerical methods for trajectory opti-mizationrdquo Journal of Guidance Control and Dynamics vol 21no 2 pp 193ndash207 1998
[14] J Fourcade S Geffroy and R Epenoy ldquoAn averaging optimalcontrol tool for low-thrust minimum-time transfersrdquo in LowThrust Trajectory Optimization CNES 2000
[15] G T Huntington Advancement and analysis of a Gauss pseu-dospectral transcription for optimal control problems [PhDthesis] University of Florida Gainesville Fla USA 2007
[16] K Yu and M Xu ldquoDesign for orbit-transfer based on ion pro-pulsionrdquo Aerospace Control and Application vol 39 no 5 pp39ndash42 2013
[17] C A Kluever ldquoUsing edelbaumrsquos method to compute Low-Thrust transfers with earth-shadow eclipsesrdquo Journal of Guid-ance Control and Dynamics vol 34 no 1 pp 300ndash303 2011
[18] J T Betts ldquoOptimal low-thrust orbit transfers with eclipsingrdquoOptimal Control Applications ampMethods vol 36 no 2 pp 218ndash240 2015
[19] J Ding F Qiu and Z Wei ldquoCalculating and analysis of satelliteumbra based on Cartesian coordinaterdquo Science of Surveying andMapping no 1 pp 39ndash40 2010
[20] X Xi WWang and Y Gao Fundamentals of Near-Earth Space-craft Obit National University of Defense Technology Press2003
[21] J Yang Q Fan W Yang and S Chen Dynamics and Control ofSpacecraft Aerospace Press 1999
[22] K Hu and J Luo ldquoPrinciple analysis of ground based surveil-lance radar to space targetrdquo Space Electronic Technology vol 6no 4 pp 66ndash70 2009
[23] J Zhang and G Yang ldquoAnalysis of detection efficiency of spacetarget surveillance radar to low-orbit satelliterdquo Radar Scienceand Technology vol 7 no 1 pp 10ndash13 2009
[24] R Zhang Attitude Dynamics and Control of Satellite Orbit Bei-hang University Press 1998
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of