research article approximate state transition matrix and...
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Research ArticleApproximate State Transition Matrix and Secular Orbit Model
M P Ramachandran
Flight Dynamics Group ISRO Satellite Centre Bangalore 560 017 India
Correspondence should be addressed to M P Ramachandran mpramagmailcom
Received 21 September 2014 Revised 24 February 2015 Accepted 24 February 2015
Academic Editor Christopher J Damaren
Copyright copy 2015 M P RamachandranThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The state transition matrix (STM) is a part of the onboard orbit determination system It is used to control the satellitersquos orbitalmotion to a predefined reference orbit Firstly in this paper a simple orbit model that captures the secular behavior of the orbitalmotion in the presence of all perturbation forces is derived Next an approximate STM to match the secular effects in the orbitdue to oblate earth effect and later in the presence of all perturbation forces is derived Numerical experiments are provided forillustration
1 Introduction
Autonomous orbit control in satellites is possible with thepresent onboard technological advancements The GlobalPositioning System receiver solution gives the satellite posi-tion measurement in Cartesian frame State propagatingequations alongwith themeasurement equations in the linearfilter then estimate the orbit State transition matrix (STM) isused in the state update equations A reference orbit modelis available onboard Using the receiver orbit solution theabsolute orbit control system then ensures the satellitemotionto this reference orbit model in the earth centered fixedCartesian reference frameThis control enables the satellite toachieve the required orientation too In orbit determinationsystem STM of two-body dynamics as suggested in [1] isusually used Yet it will be always desirable to match thecomplete dynamics especially to improve the accuracy andscalability of the navigation system [2]
The orbital motion of the satellite is made up of secularor mean motion along with short and long periodic motions[3 page 571] When we include the complete dynamicsas reference orbit we have to use continuous control Thisrequires more fuel Continuous maneuver can also disturbthe payload functioning On the other hand mean motion(without periodic motions) as a reference orbit is more suitedfor orbit control by impulse thrusting This is adopted information flying [4 Chap 10] besides that the mean motion
is used to derive the initial conditions It is noted thatorbit control is usually executed as a function of time [5]instead of true anomaly In the control system the statemeasurements in Cartesian frame are usually updated in timespace Subsequently the STMderived here then updates thesestates
STM henceforth shall mean absolute STM unless men-tioned It may be noted that Vallado [[3] page 748] hasdiscussed the STM for two-body orbital motion In [6] a STMincluding the oblate earth effects using equinoctial meanelements and then applying interpolation is obtained Thepresent note brings out a STM that is in Cartesian frameas an alternative to [6] and considers only secular effectsWe note in the literature that the STM that is in Cartesianframe is derived in [7] and it includes secular and periodiceffects Here the periodic effects are neglected Further theSTM derived here is extendable to accommodate secularalong track effects in the presence of all perturbations Thisis simpler than the expansion based method of deriving theSTM as in [8]
It is important to note that secular forces due to oblateeffect are considered in relative motion as in formation flyingwhich is based on geometric approach [9] The absolutetransition matrix derived here can further be used to deriverelative transition matrix as in [10] This work is beyond thescope of this paper
Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2015 Article ID 475742 6 pageshttpdxdoiorg1011552015475742
2 International Journal of Aerospace Engineering
2 Secular Acceleration
Consider the equation of motion
r = nabla119880 + ap (1)
where r denotes the second derivative with respect to timeof r = (119909 119910 119911) the position vector in the inertial frame Thedisturbing potential [11] is
119880(119903 120595 119911) = (
120583
119903
) minus (
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) 3sin2120575 minus 1
(2)
where 119869
2= 00010826 120575 is the instantaneous declination 119877
is the radius of the earth 120583 is the gravitational constant and119903 is the magnitude of the position vector r The vector ap =
(119886
119901119909 119886
119901119910 119886
119901119911) represents other perturbation forces due to the
inhomogeneous mass distribution of the earth third bodyforces due to sun and moon besides solar radiation pressureand atmosphere drag forces The potential is axisymmetricabout the 119911-axis and is independent of azimuth angle 120595 TheLagrangersquos planetary equation of motion is invoked and thefollowing relations are deduced The Keplerian elements areaveraged over an orbit The first-order secular motion thatneglects periodic effects is described by
119886 = 119886
0
119890 = 119890
0
119894 = 119894
0
(3a)
where 119886 119890 119894 are respectively the semimajor axis eccentricityand inclination are invariant over the duration of interest
119908 = 119908
0+ (
3
2
) 119869
2(
119877
119901
)
2
119899
2 minus (
5
2
) sin2 (119894) Δ119905
Ω = Ω
0minus (
3
2
) 119869
2(
119877
119901
)
2
119899
cos (119894) Δ119905
119899
= 119899
0[1 + (
3
2
) 119869
2(
119877
119901
)
2
[1 minus (
3
2
) sin2 (119894)) (1 minus 119890
2)
12
]
119899
0=
radic(
120583
119886
3)
119872 = 119872
0+ 119899
119905
(3b)
And 119908Ω119872 are argument of perigee the longitude ofascending node and mean anomaly respectively The equa-tion of the centre enables getting the true anomaly (119891)
119891 = 119872 + (2119890 minus (
1
4
) 119890
3) sin (119872) + (
5
4
) 119890
2 sin (2119872)
+ (
13
12
) 119890
3 sin (3119872) + 119900 (119890
4)
(4)
In (3a) and (3b) we note that
119901 = 119886 (1 minus 119890
2) (5)
Equation (4) is used when the eccentricity is not largeThe longitude of the ascending node Ω
0varies linearly
with incremental time Δ119905 Orbit models in the satellite forcontrol purposes need to have the cross-track motion that ispredominant due to 119869
2 The argument of perigee (119908) along
with the true anomaly (119891) gives the argument of latitude (120579)
which is
120579
2= 119908
2+ 119891
2 (6)
Here the subscript 2 has been added to denote the 119869
2model
When all forces are included and solution of (1) is obtainedthe instantaneous argument of latitude is denoted by 120579
119886 Here
in this paper a proposal ismade to add a polynomial functionto the argument of latitude 120579
2 in (6) over every orbit This
is a mean variation of the differential argument of latitudeand could be say a quadratic or cubic power of time and isdenoted as 120579
119891[[3] page 570 652] Here in this note 120579
119891is a
least squares fit over one orbital period and it accommodatesthe secular difference This is defined as
120579
119901
def= 120579
2+ 120579
119891
(7)
The residue between 120579
119886and 120579
119901is periodic which is inciden-
tally not required for control This correction enables (3a)and (3b) along with (7) to match the secular effects whenall perturbation is present to a reasonable accuracy speciallyalong track Next a STM that matches the orbit model in (3a)and (3b) and later the secular effect in the presence of allforces is derived
The following equations are used to transform from theorbital frame [119903 119905 119899] to the Cartesian frame [119909 119910 119911]
(
119909
119910
119911
) = (119860)(
119903
119905
119899
) (8)
119886
11= cos (120579) cos (Ω) minus cos (119894) sin (Ω) sin (120579)
119886
12= minus sin (120579) cos (Ω) minus cos (119894) sin (Ω) cos (120579)
119886
13= sin (119894) sin (120579)
119886
21= cos (120579) sin (Ω) + cos (119894) cos (Ω) sin (120579)
119886
22= minus sin (120579) sin (Ω) + cos (119894) cos (Ω) cos (120579)
119886
23= minus sin (119894) cos (Ω)
119886
31= sin (120579) sin (119894)
119886
32= sin (119894) cos (120579)
119886
33= cos (119894)
(9)
where unit vectors 119903
119905
119899
respectively are in the radialtangential that is along the direction of motion (along-track)and normal to the orbital plane (see Figure 1)
By using sin(120575) = 119903 sin(120579) sin(119894) in (2) the potential dueto oblate earth effect becomes
119880
2= minus(
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) 3sin2 (119894) sin2 (120579) minus 1
(10)
International Journal of Aerospace Engineering 3
z
x
y
Flight path
r
n
t
Figure 1 Orbit frame fromposition tangential and normal vectors
The acceleration is then obtained using the relations in (8)and is
nabla119880 = (minus
120583
119903
2) (119903
) minus (120583119877
2119869
2(
1
2
))(minus
15119911
2
119903
6) + (
3
119903
4) (119903
)
minus (120583119877
2119869
2(
1
2
))
6119911
119903
5 (119911
)
(11)
Equation (11) when substituted into (8) finally gives theaccelerations in the (119909 119910 119911) frame
119886
119909= minus
120583119909
(119903
3) [1 + (32) 119869
2(119877119903)
2(1 minus (5119911
2119903
2))]
119886
119910= (
119910
119909
) 119886
119909
119886
119911= minus
120583119911
(119903
3) [1 + (32) 119869
2(119877119903)
2(3 minus (5119911
2119903
2))]
(12)
Partial differentiation with respect to the state variables(119909 119910 119911) in (12) yields the STM that includes both secular andperiodic components (see [7])
Here the term cos(2120579) in (10) is periodic and neglectedthe remaining term that is secular is retained In (2) we have
119880 (119903 120575) = (
120583
119903
) minus (
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) (
3
2
) sin2 (119894) minus 1
(13)
Note that the potential in (13) is independent of (119911) Thenet acceleration then is obtained as in (11) by denoting 120572 =
minus120583119877
2119869
2(32)(1minus (32)sin2(119894)) in Cartesian frame and is given
by
119886
119909= (
120572119909
119903
5) minus
120583119909
119903
3
119886
119910= (
120572119910
119903
5) minus
120583119910
119903
3
119886
119911= (
120572119911
119903
5) minus
120583119911
119903
3
(14)
Then
120597119886
119909
120597119909
= (minus
3120572
119903
5) + (
15120572119909
2
119903
7) +
3120583119909
2
119903
5minus
120583
119903
3
120597119886
119909
120597119910
= (
15120572119909119910
119903
7) +
3120583119909119910
119903
5
120597119886
119909
120597119911
= (
15120572119909119911
119903
7) +
3120583119909119911
119903
5
120597119886
119910
120597119909
=
120597119886
119909
120597119910
120597119886
119910
120597119910
= (minus
3120572
119903
5) + (
15120572119909
2
119903
7) +
3120583119910
2
119903
5minus
120583
119903
3
120597119886
119910
120597119911
= (
15120572119910119911
119903
7) +
3120583119910119911
119903
5
120597119886
119911
120597119909
=
120597119886
119909
120597119911
120597119886
119911
120597119910
=
120597119886
119910
120597119911
120597119886
119911
120597119909
= (minus
3120572
119903
5) + (
15120572119911
2
119903
7) +
3120583119911
2
119903
5minus
120583
119903
3
(15)
It may be noted that when 120579 is substituted into (8) by 120579
2in
(3a) and (3b) or 120579
119901in (7) the accelerations derived in (15)
are still valid This implies that the STM (to be derived in thesection) with the accelerations derivatives in (15) are valid for(1) without considering periodic effects
3 Approximate STM
Next use the total acceleration in (14) and derive the approxi-mate STM following Markley [12] The STM is then obtainedapproximately based on Taylor expansion
Φ(119905 119905
0) = (
Φ
119903119903120593
119903V
ΦV119903 120593VV) (16)
With a knowledge of initial states as in (3a) and (3b) at 1199050 the
matrixΦ can be used to obtain state at the subsequent instantldquo119905rdquo using
119883 (119905) = Φ (119905 119905
0)119883 (119905
0) (17)
4 International Journal of Aerospace EngineeringD
evia
tion
(km
)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 2 Deviation in position for Molniya orbit
Dev
iatio
n (k
m)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 3 Deviation in position for sun-synchronous orbit
where 119883(119905) is differential of the states (119909 119910 119911 119909
1015840 119910
1015840 119911
1015840) at 1199050
Discarding higher order terms in (16) we have
Φ
119903119903= 119868 +
(2G0+ G) (Δ119905)
2
6
Φ
119903V = 119868Δ119905 +
(2G0+ G) (Δ119905)
3
12
ΦV119903 =(G0+ G) (Δ119905)
2
Φ
119903V = 119868 +
(G0+ 2G) (Δ119905)
2
6
(18)
The gradient matrix is
G =
(
(
120597119886
119909
120597119909
120597119886
119909
120597119910
120597119886
119909
120597119911
120597119886
119910
120597119909
120597119886
119910
120597119910
120597119886
119910
120597119911
120597119886
119911
120597119909
120597119886
119911
120597119910
120597119886
119909
120597119911
)
)
(19)
The matrices G0and G denote G(119905
0) and G(119905) respectively
and Δ119905 is (119905 minus 119905
0)
4 Illustration
STM is not to be used as a propagator and is used between theupdates of the state over the duration of orbit determination
However an experiment of propagation is carried out hereover certain duration to ensure that the present modelefficiently captures the secular effects The selected orbit isMolniya orbit with 119886 = 26554 kms 119890 = 072 and 119894 = 634
degrees This orbit is more eccentric and the short periodiceffects are predominant The top dotted line in Figure 2 isthe result that depicts the absolute deviation in position(distance) from the STM that includes short and periodiceffects given in [7] with respect to (3a) and (3b) The bottomcurve is the absolute deviation between the proposed STM(15)ndash(19) with respect to (3a) and (3b) again as function oftime Both have the same initial conditions This illustratesthat the proposed model is closer to the mean or secularorbital motion given in (3a) and (3b) in the presence of 119869
2
effect neglecting the numerical error due to propagationThenumerical propagation error due to step size is common inboth This validates that the STM derived in Section 3 usingthe partial derivatives of accelerations in (15) contains seculareffects alone
Next a similar exercise as carried out in Figure 2 isconsidering a sun-synchronous polar orbit with 119886 = 71684119890 = 00011 and 119894 = 9851 The top dashed line in Figure 3is the result that depicts the deviation in position from theSTM that includes short and periodic effects The bottomcurve is the deviation in position from the proposed STMThe illustration confirms that the proposedmodel is closer to(3a) and (3b)
Next example is about illustrating the modeling of (7)The satellite orbit has 119886 = 72443 119890 = 00003 and 119894 = 20
degrees The satellites placed in such orbits are mostly usedfor metrological purposesThe payload sensor usually coversa wide area as in microwave remote sensing applicationsTheargument of latitude 120579
2corresponding to (3a) and (3b) is first
derived The satellite motion is also obtained as solution of(3a) and (3b) while considering complete perturbations asmentioned in (1) The instantaneous argument of latitude is120579
119886 From (3a) and (3b) the argument of latitude denoted as
120579
2 is obtainedThe residue (120579
119886minus120579
2) is plotted against time in
Figure 4 It can be seen that this deviation is secular and haslarge effect along the track
Thedifference between the argument of latitude of the fullforce model 120579
119886and 120579
2over one orbit that is 102 minutes in
duration is then fit for the secular effect by a polynomial in aleast square sense For in this case it is
120579
119891= 120579
119887minus 003624 + 00012119905 minus 0000013119905
2+ 0000000046119905
3
(20)
The same fit is proposed here to extend over successive orbitsHowever at the end of each orbital period the ordinate valueof difference 120579
119887is used for the next orbital period Here for
example the value of 120579
119887is minus0113 at the start This approach
of orbit model representation can be considered as analternative to the existing methods for onboard applications[12]
The difference between the argument latitude of the fullforce model 120579
119886and 120579
119891in (7) is computed This residue is
illustrated in Figure 5 over four orbits and is observed tobe periodic over each orbital period The initial states for
International Journal of Aerospace Engineering 5
Time (min)
(deg
)
09
07
05
03
01
minus01
minus03
0 102 204 306 408 510 612 714 816 918 1020
Figure 4 Difference in argument of latitude
Time (min)0 102 204 306 408
01
0
minus01
minus02
minus03Resid
ue (d
eg)
Residue in the argument of latitude
Figure 5 Residue with respect to the polynomial fit
the STM are obtained from the proposed orbit model whileusing (3a) and (3b) along with the correction in (20) Theabsolute deviation is with respect to the position of full forcesecular propagation model The plot in Figure 6 is similarto that in Figure 1 or Figure 2 The deviation in positionin the plot uses the STM in Section 3 and using (15) canbe seen to accommodate secular effects in the presence allperturbations
5 Application
Equations (3a) and (3b) describe the dynamics of the orbitingsatellite when oblate earth effect is considered The relativemotion based on geometric approach is given in [9] Theyindependently make use of the dynamics in (3a) and (3b)for both satellites The two satellites have identical (119886 119890 119894) in(3a) and distinct (119908Ω119872) in (3b) The relative motion asreproduced from [9] is
Δ119909 (Δ119905) = minus1 + 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (120575120579 minus 120575Ω)
+ 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (2120579
1+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119888 (120575120579) minus 119888 (2120579
119898+ 120575120579)]
Dev
iatio
n (k
m)
Time (s)
45
30
15
0
0 100 200 300 400 500 600 700 800 900 1000 1100
STM using (14)ndash(18)
Figure 6 Deviation in position
Δ119910 (Δ119905) = 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (120575120579 minus 120575Ω)
minus 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119904 (120575120579) + 119904 (2120579
119898+ 120575120579)]
Δ119911 (Δ119905) = minus119904 (119894
119898) 119904 (120575Ω) 119888 (120579
119904)
minus [119904 (
119894
119898
2
) 119888 (119894
119904) 119888 (120575Ω) minus 119888 (119894
119898) 119904 (119894
119904)] 119904 (120579
119904)
(21)
Here 120579
119898is the instantaneous argument of latitude that is 120579
2
of the main or chief satellite Similarly 120579
119904is the instantaneous
argument of latitude of the second or follower satellite andtheir difference (120579
119898minus 120579
119904) which is denoted as 120575120579 The
differences in the instantaneous longitude of the ascendingnode between them is 120575Ω The 119888 and 119904 functions denotecos and sin functions and 119894
119898and 119894
119904denote the inclinations
of the main and follower respectively Also Δ119905 denotes theincremental time as used in (3a) and (3b) In (21) Δ119909 Δ119910 Δ119911
are relative positions of the second satellite in the orbit frameof the first
Here we shall outline the application of the secularapproximation 120579
119901from (7) when substituted into (21) This
is to enhance (21) to match the secular effects particularlyalong the track while considering all perturbations To do sowe replace the argument of latitude angle 120579
2for both satellites
in (21) by an appropriate 120579
119901 computed individually using (20)
and (7) as in Figure 6 for the main and secondary satellitesThese computed 120579
119901are then substituted into (21) as 120579
119898
and 120579
119904
respectively for the main and secondary satellite Now (21) isenhanced to match the secular effects particularly along thetrack of the full forcemodel Itmay be noted that in formationflying (21) is used for guidance and can also be used to derivethe initial conditions
6 International Journal of Aerospace Engineering
6 Conclusion
Approximate orbit model that captures the secular motion inthe argument of latitude while considering all perturbationforces is obtained Approximate state transition matrix isfirst derived for the orbital motion that matches the secularmotion with the effects of oblate earth This STM can alsoaccommodate the approximate orbit model with all forcesNumerical behavior of the state transition matrix and theorbit models has been provided Finally a direct applicationof the orbit model in relativemotion of two satellites has beenindicated The suggested approach is simpler for onboardimplementation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author acknowledges all the three anonymous refereesfor their comments that have enhanced the presentationTheauthor thanks Mr N S Gopinath Group Director and Dr SK Shivakumar Director from ISRO Satellite Centre for theirencouragements
References
[1] E Gill OMontenbruck andK Brieszlig ldquoGPS-based autonomousnavigation for the BIRD satelliterdquo in Proceedings of the 15thInternational Symposium on Spaceflight Dynamics BiarritzFrance June 2000
[2] S D Amico J-S Ardaens andOMontenbruck ldquoNavigation offormation flying spacecraft using GPS the PRISMA technologydemonstrationrdquo in Proceedings of the 22nd International Techni-cal Meeting of the Satellite Division of the Institute of Navigation(ION GNSS rsquo09) pp 1427ndash1441 Savannah Ga USA September2009
[3] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers Dordrecht The Netherlands2001
[4] K T Alfriend S R Vadali P Gurfil J P How and L S BregerSpacecraft Formation Flying Elsevier Oxford UK 2010
[5] H Schaub S R Vadali J L Junkins and K T Alfriend ldquoSpace-craft formation flying control using mean orbit elementsrdquo TheJournal of the Astronautical Sciences vol 48 no 1 pp 69ndash872001
[6] J S Shaver Formulation and evaluation of parallel algorithms fororbit determination problem [PhD thesis] Department of Aero-nautics Massachusetts Institute of Technology CambridgeMass USA 1980
[7] A P M Chiaradia H K K Kuga and A F B A PradoldquoComparison between two methods to calculate the transitionmatrix of orbit motionrdquoMathematical Problems in Engineeringvol 2012 Article ID 768973 12 pages 2012
[8] Y Tsuda ldquoState transitionmatrix approximation with geometrypreservation for general perturbed orbitsrdquo Acta Astronauticavol 68 no 7-8 pp 1051ndash1061 2011
[9] S R Vadali ldquoAn analytical solution of relative motion ofsatellitesrdquo in Proceedings of the Dynamics and Control of Systemsand Structures in Space Conference Cranfield UK July 2002
[10] C J Damaren ldquoAlmost periodic relative orbits under J2pertur-
bationsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 221 no 5 pp 767ndash774 2007
[11] A E Roy Orbital Motion Adam Hilger Bristol UK 1982[12] F L Markley ldquoApproximate Cartesian state transition matrixrdquo
The Journal of Astronautical Sciences vol 34 no 2 pp 161ndash1691986
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Shock and Vibration
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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
2 International Journal of Aerospace Engineering
2 Secular Acceleration
Consider the equation of motion
r = nabla119880 + ap (1)
where r denotes the second derivative with respect to timeof r = (119909 119910 119911) the position vector in the inertial frame Thedisturbing potential [11] is
119880(119903 120595 119911) = (
120583
119903
) minus (
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) 3sin2120575 minus 1
(2)
where 119869
2= 00010826 120575 is the instantaneous declination 119877
is the radius of the earth 120583 is the gravitational constant and119903 is the magnitude of the position vector r The vector ap =
(119886
119901119909 119886
119901119910 119886
119901119911) represents other perturbation forces due to the
inhomogeneous mass distribution of the earth third bodyforces due to sun and moon besides solar radiation pressureand atmosphere drag forces The potential is axisymmetricabout the 119911-axis and is independent of azimuth angle 120595 TheLagrangersquos planetary equation of motion is invoked and thefollowing relations are deduced The Keplerian elements areaveraged over an orbit The first-order secular motion thatneglects periodic effects is described by
119886 = 119886
0
119890 = 119890
0
119894 = 119894
0
(3a)
where 119886 119890 119894 are respectively the semimajor axis eccentricityand inclination are invariant over the duration of interest
119908 = 119908
0+ (
3
2
) 119869
2(
119877
119901
)
2
119899
2 minus (
5
2
) sin2 (119894) Δ119905
Ω = Ω
0minus (
3
2
) 119869
2(
119877
119901
)
2
119899
cos (119894) Δ119905
119899
= 119899
0[1 + (
3
2
) 119869
2(
119877
119901
)
2
[1 minus (
3
2
) sin2 (119894)) (1 minus 119890
2)
12
]
119899
0=
radic(
120583
119886
3)
119872 = 119872
0+ 119899
119905
(3b)
And 119908Ω119872 are argument of perigee the longitude ofascending node and mean anomaly respectively The equa-tion of the centre enables getting the true anomaly (119891)
119891 = 119872 + (2119890 minus (
1
4
) 119890
3) sin (119872) + (
5
4
) 119890
2 sin (2119872)
+ (
13
12
) 119890
3 sin (3119872) + 119900 (119890
4)
(4)
In (3a) and (3b) we note that
119901 = 119886 (1 minus 119890
2) (5)
Equation (4) is used when the eccentricity is not largeThe longitude of the ascending node Ω
0varies linearly
with incremental time Δ119905 Orbit models in the satellite forcontrol purposes need to have the cross-track motion that ispredominant due to 119869
2 The argument of perigee (119908) along
with the true anomaly (119891) gives the argument of latitude (120579)
which is
120579
2= 119908
2+ 119891
2 (6)
Here the subscript 2 has been added to denote the 119869
2model
When all forces are included and solution of (1) is obtainedthe instantaneous argument of latitude is denoted by 120579
119886 Here
in this paper a proposal ismade to add a polynomial functionto the argument of latitude 120579
2 in (6) over every orbit This
is a mean variation of the differential argument of latitudeand could be say a quadratic or cubic power of time and isdenoted as 120579
119891[[3] page 570 652] Here in this note 120579
119891is a
least squares fit over one orbital period and it accommodatesthe secular difference This is defined as
120579
119901
def= 120579
2+ 120579
119891
(7)
The residue between 120579
119886and 120579
119901is periodic which is inciden-
tally not required for control This correction enables (3a)and (3b) along with (7) to match the secular effects whenall perturbation is present to a reasonable accuracy speciallyalong track Next a STM that matches the orbit model in (3a)and (3b) and later the secular effect in the presence of allforces is derived
The following equations are used to transform from theorbital frame [119903 119905 119899] to the Cartesian frame [119909 119910 119911]
(
119909
119910
119911
) = (119860)(
119903
119905
119899
) (8)
119886
11= cos (120579) cos (Ω) minus cos (119894) sin (Ω) sin (120579)
119886
12= minus sin (120579) cos (Ω) minus cos (119894) sin (Ω) cos (120579)
119886
13= sin (119894) sin (120579)
119886
21= cos (120579) sin (Ω) + cos (119894) cos (Ω) sin (120579)
119886
22= minus sin (120579) sin (Ω) + cos (119894) cos (Ω) cos (120579)
119886
23= minus sin (119894) cos (Ω)
119886
31= sin (120579) sin (119894)
119886
32= sin (119894) cos (120579)
119886
33= cos (119894)
(9)
where unit vectors 119903
119905
119899
respectively are in the radialtangential that is along the direction of motion (along-track)and normal to the orbital plane (see Figure 1)
By using sin(120575) = 119903 sin(120579) sin(119894) in (2) the potential dueto oblate earth effect becomes
119880
2= minus(
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) 3sin2 (119894) sin2 (120579) minus 1
(10)
International Journal of Aerospace Engineering 3
z
x
y
Flight path
r
n
t
Figure 1 Orbit frame fromposition tangential and normal vectors
The acceleration is then obtained using the relations in (8)and is
nabla119880 = (minus
120583
119903
2) (119903
) minus (120583119877
2119869
2(
1
2
))(minus
15119911
2
119903
6) + (
3
119903
4) (119903
)
minus (120583119877
2119869
2(
1
2
))
6119911
119903
5 (119911
)
(11)
Equation (11) when substituted into (8) finally gives theaccelerations in the (119909 119910 119911) frame
119886
119909= minus
120583119909
(119903
3) [1 + (32) 119869
2(119877119903)
2(1 minus (5119911
2119903
2))]
119886
119910= (
119910
119909
) 119886
119909
119886
119911= minus
120583119911
(119903
3) [1 + (32) 119869
2(119877119903)
2(3 minus (5119911
2119903
2))]
(12)
Partial differentiation with respect to the state variables(119909 119910 119911) in (12) yields the STM that includes both secular andperiodic components (see [7])
Here the term cos(2120579) in (10) is periodic and neglectedthe remaining term that is secular is retained In (2) we have
119880 (119903 120575) = (
120583
119903
) minus (
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) (
3
2
) sin2 (119894) minus 1
(13)
Note that the potential in (13) is independent of (119911) Thenet acceleration then is obtained as in (11) by denoting 120572 =
minus120583119877
2119869
2(32)(1minus (32)sin2(119894)) in Cartesian frame and is given
by
119886
119909= (
120572119909
119903
5) minus
120583119909
119903
3
119886
119910= (
120572119910
119903
5) minus
120583119910
119903
3
119886
119911= (
120572119911
119903
5) minus
120583119911
119903
3
(14)
Then
120597119886
119909
120597119909
= (minus
3120572
119903
5) + (
15120572119909
2
119903
7) +
3120583119909
2
119903
5minus
120583
119903
3
120597119886
119909
120597119910
= (
15120572119909119910
119903
7) +
3120583119909119910
119903
5
120597119886
119909
120597119911
= (
15120572119909119911
119903
7) +
3120583119909119911
119903
5
120597119886
119910
120597119909
=
120597119886
119909
120597119910
120597119886
119910
120597119910
= (minus
3120572
119903
5) + (
15120572119909
2
119903
7) +
3120583119910
2
119903
5minus
120583
119903
3
120597119886
119910
120597119911
= (
15120572119910119911
119903
7) +
3120583119910119911
119903
5
120597119886
119911
120597119909
=
120597119886
119909
120597119911
120597119886
119911
120597119910
=
120597119886
119910
120597119911
120597119886
119911
120597119909
= (minus
3120572
119903
5) + (
15120572119911
2
119903
7) +
3120583119911
2
119903
5minus
120583
119903
3
(15)
It may be noted that when 120579 is substituted into (8) by 120579
2in
(3a) and (3b) or 120579
119901in (7) the accelerations derived in (15)
are still valid This implies that the STM (to be derived in thesection) with the accelerations derivatives in (15) are valid for(1) without considering periodic effects
3 Approximate STM
Next use the total acceleration in (14) and derive the approxi-mate STM following Markley [12] The STM is then obtainedapproximately based on Taylor expansion
Φ(119905 119905
0) = (
Φ
119903119903120593
119903V
ΦV119903 120593VV) (16)
With a knowledge of initial states as in (3a) and (3b) at 1199050 the
matrixΦ can be used to obtain state at the subsequent instantldquo119905rdquo using
119883 (119905) = Φ (119905 119905
0)119883 (119905
0) (17)
4 International Journal of Aerospace EngineeringD
evia
tion
(km
)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 2 Deviation in position for Molniya orbit
Dev
iatio
n (k
m)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 3 Deviation in position for sun-synchronous orbit
where 119883(119905) is differential of the states (119909 119910 119911 119909
1015840 119910
1015840 119911
1015840) at 1199050
Discarding higher order terms in (16) we have
Φ
119903119903= 119868 +
(2G0+ G) (Δ119905)
2
6
Φ
119903V = 119868Δ119905 +
(2G0+ G) (Δ119905)
3
12
ΦV119903 =(G0+ G) (Δ119905)
2
Φ
119903V = 119868 +
(G0+ 2G) (Δ119905)
2
6
(18)
The gradient matrix is
G =
(
(
120597119886
119909
120597119909
120597119886
119909
120597119910
120597119886
119909
120597119911
120597119886
119910
120597119909
120597119886
119910
120597119910
120597119886
119910
120597119911
120597119886
119911
120597119909
120597119886
119911
120597119910
120597119886
119909
120597119911
)
)
(19)
The matrices G0and G denote G(119905
0) and G(119905) respectively
and Δ119905 is (119905 minus 119905
0)
4 Illustration
STM is not to be used as a propagator and is used between theupdates of the state over the duration of orbit determination
However an experiment of propagation is carried out hereover certain duration to ensure that the present modelefficiently captures the secular effects The selected orbit isMolniya orbit with 119886 = 26554 kms 119890 = 072 and 119894 = 634
degrees This orbit is more eccentric and the short periodiceffects are predominant The top dotted line in Figure 2 isthe result that depicts the absolute deviation in position(distance) from the STM that includes short and periodiceffects given in [7] with respect to (3a) and (3b) The bottomcurve is the absolute deviation between the proposed STM(15)ndash(19) with respect to (3a) and (3b) again as function oftime Both have the same initial conditions This illustratesthat the proposed model is closer to the mean or secularorbital motion given in (3a) and (3b) in the presence of 119869
2
effect neglecting the numerical error due to propagationThenumerical propagation error due to step size is common inboth This validates that the STM derived in Section 3 usingthe partial derivatives of accelerations in (15) contains seculareffects alone
Next a similar exercise as carried out in Figure 2 isconsidering a sun-synchronous polar orbit with 119886 = 71684119890 = 00011 and 119894 = 9851 The top dashed line in Figure 3is the result that depicts the deviation in position from theSTM that includes short and periodic effects The bottomcurve is the deviation in position from the proposed STMThe illustration confirms that the proposedmodel is closer to(3a) and (3b)
Next example is about illustrating the modeling of (7)The satellite orbit has 119886 = 72443 119890 = 00003 and 119894 = 20
degrees The satellites placed in such orbits are mostly usedfor metrological purposesThe payload sensor usually coversa wide area as in microwave remote sensing applicationsTheargument of latitude 120579
2corresponding to (3a) and (3b) is first
derived The satellite motion is also obtained as solution of(3a) and (3b) while considering complete perturbations asmentioned in (1) The instantaneous argument of latitude is120579
119886 From (3a) and (3b) the argument of latitude denoted as
120579
2 is obtainedThe residue (120579
119886minus120579
2) is plotted against time in
Figure 4 It can be seen that this deviation is secular and haslarge effect along the track
Thedifference between the argument of latitude of the fullforce model 120579
119886and 120579
2over one orbit that is 102 minutes in
duration is then fit for the secular effect by a polynomial in aleast square sense For in this case it is
120579
119891= 120579
119887minus 003624 + 00012119905 minus 0000013119905
2+ 0000000046119905
3
(20)
The same fit is proposed here to extend over successive orbitsHowever at the end of each orbital period the ordinate valueof difference 120579
119887is used for the next orbital period Here for
example the value of 120579
119887is minus0113 at the start This approach
of orbit model representation can be considered as analternative to the existing methods for onboard applications[12]
The difference between the argument latitude of the fullforce model 120579
119886and 120579
119891in (7) is computed This residue is
illustrated in Figure 5 over four orbits and is observed tobe periodic over each orbital period The initial states for
International Journal of Aerospace Engineering 5
Time (min)
(deg
)
09
07
05
03
01
minus01
minus03
0 102 204 306 408 510 612 714 816 918 1020
Figure 4 Difference in argument of latitude
Time (min)0 102 204 306 408
01
0
minus01
minus02
minus03Resid
ue (d
eg)
Residue in the argument of latitude
Figure 5 Residue with respect to the polynomial fit
the STM are obtained from the proposed orbit model whileusing (3a) and (3b) along with the correction in (20) Theabsolute deviation is with respect to the position of full forcesecular propagation model The plot in Figure 6 is similarto that in Figure 1 or Figure 2 The deviation in positionin the plot uses the STM in Section 3 and using (15) canbe seen to accommodate secular effects in the presence allperturbations
5 Application
Equations (3a) and (3b) describe the dynamics of the orbitingsatellite when oblate earth effect is considered The relativemotion based on geometric approach is given in [9] Theyindependently make use of the dynamics in (3a) and (3b)for both satellites The two satellites have identical (119886 119890 119894) in(3a) and distinct (119908Ω119872) in (3b) The relative motion asreproduced from [9] is
Δ119909 (Δ119905) = minus1 + 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (120575120579 minus 120575Ω)
+ 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (2120579
1+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119888 (120575120579) minus 119888 (2120579
119898+ 120575120579)]
Dev
iatio
n (k
m)
Time (s)
45
30
15
0
0 100 200 300 400 500 600 700 800 900 1000 1100
STM using (14)ndash(18)
Figure 6 Deviation in position
Δ119910 (Δ119905) = 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (120575120579 minus 120575Ω)
minus 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119904 (120575120579) + 119904 (2120579
119898+ 120575120579)]
Δ119911 (Δ119905) = minus119904 (119894
119898) 119904 (120575Ω) 119888 (120579
119904)
minus [119904 (
119894
119898
2
) 119888 (119894
119904) 119888 (120575Ω) minus 119888 (119894
119898) 119904 (119894
119904)] 119904 (120579
119904)
(21)
Here 120579
119898is the instantaneous argument of latitude that is 120579
2
of the main or chief satellite Similarly 120579
119904is the instantaneous
argument of latitude of the second or follower satellite andtheir difference (120579
119898minus 120579
119904) which is denoted as 120575120579 The
differences in the instantaneous longitude of the ascendingnode between them is 120575Ω The 119888 and 119904 functions denotecos and sin functions and 119894
119898and 119894
119904denote the inclinations
of the main and follower respectively Also Δ119905 denotes theincremental time as used in (3a) and (3b) In (21) Δ119909 Δ119910 Δ119911
are relative positions of the second satellite in the orbit frameof the first
Here we shall outline the application of the secularapproximation 120579
119901from (7) when substituted into (21) This
is to enhance (21) to match the secular effects particularlyalong the track while considering all perturbations To do sowe replace the argument of latitude angle 120579
2for both satellites
in (21) by an appropriate 120579
119901 computed individually using (20)
and (7) as in Figure 6 for the main and secondary satellitesThese computed 120579
119901are then substituted into (21) as 120579
119898
and 120579
119904
respectively for the main and secondary satellite Now (21) isenhanced to match the secular effects particularly along thetrack of the full forcemodel Itmay be noted that in formationflying (21) is used for guidance and can also be used to derivethe initial conditions
6 International Journal of Aerospace Engineering
6 Conclusion
Approximate orbit model that captures the secular motion inthe argument of latitude while considering all perturbationforces is obtained Approximate state transition matrix isfirst derived for the orbital motion that matches the secularmotion with the effects of oblate earth This STM can alsoaccommodate the approximate orbit model with all forcesNumerical behavior of the state transition matrix and theorbit models has been provided Finally a direct applicationof the orbit model in relativemotion of two satellites has beenindicated The suggested approach is simpler for onboardimplementation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author acknowledges all the three anonymous refereesfor their comments that have enhanced the presentationTheauthor thanks Mr N S Gopinath Group Director and Dr SK Shivakumar Director from ISRO Satellite Centre for theirencouragements
References
[1] E Gill OMontenbruck andK Brieszlig ldquoGPS-based autonomousnavigation for the BIRD satelliterdquo in Proceedings of the 15thInternational Symposium on Spaceflight Dynamics BiarritzFrance June 2000
[2] S D Amico J-S Ardaens andOMontenbruck ldquoNavigation offormation flying spacecraft using GPS the PRISMA technologydemonstrationrdquo in Proceedings of the 22nd International Techni-cal Meeting of the Satellite Division of the Institute of Navigation(ION GNSS rsquo09) pp 1427ndash1441 Savannah Ga USA September2009
[3] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers Dordrecht The Netherlands2001
[4] K T Alfriend S R Vadali P Gurfil J P How and L S BregerSpacecraft Formation Flying Elsevier Oxford UK 2010
[5] H Schaub S R Vadali J L Junkins and K T Alfriend ldquoSpace-craft formation flying control using mean orbit elementsrdquo TheJournal of the Astronautical Sciences vol 48 no 1 pp 69ndash872001
[6] J S Shaver Formulation and evaluation of parallel algorithms fororbit determination problem [PhD thesis] Department of Aero-nautics Massachusetts Institute of Technology CambridgeMass USA 1980
[7] A P M Chiaradia H K K Kuga and A F B A PradoldquoComparison between two methods to calculate the transitionmatrix of orbit motionrdquoMathematical Problems in Engineeringvol 2012 Article ID 768973 12 pages 2012
[8] Y Tsuda ldquoState transitionmatrix approximation with geometrypreservation for general perturbed orbitsrdquo Acta Astronauticavol 68 no 7-8 pp 1051ndash1061 2011
[9] S R Vadali ldquoAn analytical solution of relative motion ofsatellitesrdquo in Proceedings of the Dynamics and Control of Systemsand Structures in Space Conference Cranfield UK July 2002
[10] C J Damaren ldquoAlmost periodic relative orbits under J2pertur-
bationsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 221 no 5 pp 767ndash774 2007
[11] A E Roy Orbital Motion Adam Hilger Bristol UK 1982[12] F L Markley ldquoApproximate Cartesian state transition matrixrdquo
The Journal of Astronautical Sciences vol 34 no 2 pp 161ndash1691986
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Shock and Vibration
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Acoustics and VibrationAdvances in
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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 3
z
x
y
Flight path
r
n
t
Figure 1 Orbit frame fromposition tangential and normal vectors
The acceleration is then obtained using the relations in (8)and is
nabla119880 = (minus
120583
119903
2) (119903
) minus (120583119877
2119869
2(
1
2
))(minus
15119911
2
119903
6) + (
3
119903
4) (119903
)
minus (120583119877
2119869
2(
1
2
))
6119911
119903
5 (119911
)
(11)
Equation (11) when substituted into (8) finally gives theaccelerations in the (119909 119910 119911) frame
119886
119909= minus
120583119909
(119903
3) [1 + (32) 119869
2(119877119903)
2(1 minus (5119911
2119903
2))]
119886
119910= (
119910
119909
) 119886
119909
119886
119911= minus
120583119911
(119903
3) [1 + (32) 119869
2(119877119903)
2(3 minus (5119911
2119903
2))]
(12)
Partial differentiation with respect to the state variables(119909 119910 119911) in (12) yields the STM that includes both secular andperiodic components (see [7])
Here the term cos(2120579) in (10) is periodic and neglectedthe remaining term that is secular is retained In (2) we have
119880 (119903 120575) = (
120583
119903
) minus (
120583
119903
) (
119877
119903
)
2
119869
2(
1
2
) (
3
2
) sin2 (119894) minus 1
(13)
Note that the potential in (13) is independent of (119911) Thenet acceleration then is obtained as in (11) by denoting 120572 =
minus120583119877
2119869
2(32)(1minus (32)sin2(119894)) in Cartesian frame and is given
by
119886
119909= (
120572119909
119903
5) minus
120583119909
119903
3
119886
119910= (
120572119910
119903
5) minus
120583119910
119903
3
119886
119911= (
120572119911
119903
5) minus
120583119911
119903
3
(14)
Then
120597119886
119909
120597119909
= (minus
3120572
119903
5) + (
15120572119909
2
119903
7) +
3120583119909
2
119903
5minus
120583
119903
3
120597119886
119909
120597119910
= (
15120572119909119910
119903
7) +
3120583119909119910
119903
5
120597119886
119909
120597119911
= (
15120572119909119911
119903
7) +
3120583119909119911
119903
5
120597119886
119910
120597119909
=
120597119886
119909
120597119910
120597119886
119910
120597119910
= (minus
3120572
119903
5) + (
15120572119909
2
119903
7) +
3120583119910
2
119903
5minus
120583
119903
3
120597119886
119910
120597119911
= (
15120572119910119911
119903
7) +
3120583119910119911
119903
5
120597119886
119911
120597119909
=
120597119886
119909
120597119911
120597119886
119911
120597119910
=
120597119886
119910
120597119911
120597119886
119911
120597119909
= (minus
3120572
119903
5) + (
15120572119911
2
119903
7) +
3120583119911
2
119903
5minus
120583
119903
3
(15)
It may be noted that when 120579 is substituted into (8) by 120579
2in
(3a) and (3b) or 120579
119901in (7) the accelerations derived in (15)
are still valid This implies that the STM (to be derived in thesection) with the accelerations derivatives in (15) are valid for(1) without considering periodic effects
3 Approximate STM
Next use the total acceleration in (14) and derive the approxi-mate STM following Markley [12] The STM is then obtainedapproximately based on Taylor expansion
Φ(119905 119905
0) = (
Φ
119903119903120593
119903V
ΦV119903 120593VV) (16)
With a knowledge of initial states as in (3a) and (3b) at 1199050 the
matrixΦ can be used to obtain state at the subsequent instantldquo119905rdquo using
119883 (119905) = Φ (119905 119905
0)119883 (119905
0) (17)
4 International Journal of Aerospace EngineeringD
evia
tion
(km
)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 2 Deviation in position for Molniya orbit
Dev
iatio
n (k
m)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 3 Deviation in position for sun-synchronous orbit
where 119883(119905) is differential of the states (119909 119910 119911 119909
1015840 119910
1015840 119911
1015840) at 1199050
Discarding higher order terms in (16) we have
Φ
119903119903= 119868 +
(2G0+ G) (Δ119905)
2
6
Φ
119903V = 119868Δ119905 +
(2G0+ G) (Δ119905)
3
12
ΦV119903 =(G0+ G) (Δ119905)
2
Φ
119903V = 119868 +
(G0+ 2G) (Δ119905)
2
6
(18)
The gradient matrix is
G =
(
(
120597119886
119909
120597119909
120597119886
119909
120597119910
120597119886
119909
120597119911
120597119886
119910
120597119909
120597119886
119910
120597119910
120597119886
119910
120597119911
120597119886
119911
120597119909
120597119886
119911
120597119910
120597119886
119909
120597119911
)
)
(19)
The matrices G0and G denote G(119905
0) and G(119905) respectively
and Δ119905 is (119905 minus 119905
0)
4 Illustration
STM is not to be used as a propagator and is used between theupdates of the state over the duration of orbit determination
However an experiment of propagation is carried out hereover certain duration to ensure that the present modelefficiently captures the secular effects The selected orbit isMolniya orbit with 119886 = 26554 kms 119890 = 072 and 119894 = 634
degrees This orbit is more eccentric and the short periodiceffects are predominant The top dotted line in Figure 2 isthe result that depicts the absolute deviation in position(distance) from the STM that includes short and periodiceffects given in [7] with respect to (3a) and (3b) The bottomcurve is the absolute deviation between the proposed STM(15)ndash(19) with respect to (3a) and (3b) again as function oftime Both have the same initial conditions This illustratesthat the proposed model is closer to the mean or secularorbital motion given in (3a) and (3b) in the presence of 119869
2
effect neglecting the numerical error due to propagationThenumerical propagation error due to step size is common inboth This validates that the STM derived in Section 3 usingthe partial derivatives of accelerations in (15) contains seculareffects alone
Next a similar exercise as carried out in Figure 2 isconsidering a sun-synchronous polar orbit with 119886 = 71684119890 = 00011 and 119894 = 9851 The top dashed line in Figure 3is the result that depicts the deviation in position from theSTM that includes short and periodic effects The bottomcurve is the deviation in position from the proposed STMThe illustration confirms that the proposedmodel is closer to(3a) and (3b)
Next example is about illustrating the modeling of (7)The satellite orbit has 119886 = 72443 119890 = 00003 and 119894 = 20
degrees The satellites placed in such orbits are mostly usedfor metrological purposesThe payload sensor usually coversa wide area as in microwave remote sensing applicationsTheargument of latitude 120579
2corresponding to (3a) and (3b) is first
derived The satellite motion is also obtained as solution of(3a) and (3b) while considering complete perturbations asmentioned in (1) The instantaneous argument of latitude is120579
119886 From (3a) and (3b) the argument of latitude denoted as
120579
2 is obtainedThe residue (120579
119886minus120579
2) is plotted against time in
Figure 4 It can be seen that this deviation is secular and haslarge effect along the track
Thedifference between the argument of latitude of the fullforce model 120579
119886and 120579
2over one orbit that is 102 minutes in
duration is then fit for the secular effect by a polynomial in aleast square sense For in this case it is
120579
119891= 120579
119887minus 003624 + 00012119905 minus 0000013119905
2+ 0000000046119905
3
(20)
The same fit is proposed here to extend over successive orbitsHowever at the end of each orbital period the ordinate valueof difference 120579
119887is used for the next orbital period Here for
example the value of 120579
119887is minus0113 at the start This approach
of orbit model representation can be considered as analternative to the existing methods for onboard applications[12]
The difference between the argument latitude of the fullforce model 120579
119886and 120579
119891in (7) is computed This residue is
illustrated in Figure 5 over four orbits and is observed tobe periodic over each orbital period The initial states for
International Journal of Aerospace Engineering 5
Time (min)
(deg
)
09
07
05
03
01
minus01
minus03
0 102 204 306 408 510 612 714 816 918 1020
Figure 4 Difference in argument of latitude
Time (min)0 102 204 306 408
01
0
minus01
minus02
minus03Resid
ue (d
eg)
Residue in the argument of latitude
Figure 5 Residue with respect to the polynomial fit
the STM are obtained from the proposed orbit model whileusing (3a) and (3b) along with the correction in (20) Theabsolute deviation is with respect to the position of full forcesecular propagation model The plot in Figure 6 is similarto that in Figure 1 or Figure 2 The deviation in positionin the plot uses the STM in Section 3 and using (15) canbe seen to accommodate secular effects in the presence allperturbations
5 Application
Equations (3a) and (3b) describe the dynamics of the orbitingsatellite when oblate earth effect is considered The relativemotion based on geometric approach is given in [9] Theyindependently make use of the dynamics in (3a) and (3b)for both satellites The two satellites have identical (119886 119890 119894) in(3a) and distinct (119908Ω119872) in (3b) The relative motion asreproduced from [9] is
Δ119909 (Δ119905) = minus1 + 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (120575120579 minus 120575Ω)
+ 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (2120579
1+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119888 (120575120579) minus 119888 (2120579
119898+ 120575120579)]
Dev
iatio
n (k
m)
Time (s)
45
30
15
0
0 100 200 300 400 500 600 700 800 900 1000 1100
STM using (14)ndash(18)
Figure 6 Deviation in position
Δ119910 (Δ119905) = 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (120575120579 minus 120575Ω)
minus 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119904 (120575120579) + 119904 (2120579
119898+ 120575120579)]
Δ119911 (Δ119905) = minus119904 (119894
119898) 119904 (120575Ω) 119888 (120579
119904)
minus [119904 (
119894
119898
2
) 119888 (119894
119904) 119888 (120575Ω) minus 119888 (119894
119898) 119904 (119894
119904)] 119904 (120579
119904)
(21)
Here 120579
119898is the instantaneous argument of latitude that is 120579
2
of the main or chief satellite Similarly 120579
119904is the instantaneous
argument of latitude of the second or follower satellite andtheir difference (120579
119898minus 120579
119904) which is denoted as 120575120579 The
differences in the instantaneous longitude of the ascendingnode between them is 120575Ω The 119888 and 119904 functions denotecos and sin functions and 119894
119898and 119894
119904denote the inclinations
of the main and follower respectively Also Δ119905 denotes theincremental time as used in (3a) and (3b) In (21) Δ119909 Δ119910 Δ119911
are relative positions of the second satellite in the orbit frameof the first
Here we shall outline the application of the secularapproximation 120579
119901from (7) when substituted into (21) This
is to enhance (21) to match the secular effects particularlyalong the track while considering all perturbations To do sowe replace the argument of latitude angle 120579
2for both satellites
in (21) by an appropriate 120579
119901 computed individually using (20)
and (7) as in Figure 6 for the main and secondary satellitesThese computed 120579
119901are then substituted into (21) as 120579
119898
and 120579
119904
respectively for the main and secondary satellite Now (21) isenhanced to match the secular effects particularly along thetrack of the full forcemodel Itmay be noted that in formationflying (21) is used for guidance and can also be used to derivethe initial conditions
6 International Journal of Aerospace Engineering
6 Conclusion
Approximate orbit model that captures the secular motion inthe argument of latitude while considering all perturbationforces is obtained Approximate state transition matrix isfirst derived for the orbital motion that matches the secularmotion with the effects of oblate earth This STM can alsoaccommodate the approximate orbit model with all forcesNumerical behavior of the state transition matrix and theorbit models has been provided Finally a direct applicationof the orbit model in relativemotion of two satellites has beenindicated The suggested approach is simpler for onboardimplementation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author acknowledges all the three anonymous refereesfor their comments that have enhanced the presentationTheauthor thanks Mr N S Gopinath Group Director and Dr SK Shivakumar Director from ISRO Satellite Centre for theirencouragements
References
[1] E Gill OMontenbruck andK Brieszlig ldquoGPS-based autonomousnavigation for the BIRD satelliterdquo in Proceedings of the 15thInternational Symposium on Spaceflight Dynamics BiarritzFrance June 2000
[2] S D Amico J-S Ardaens andOMontenbruck ldquoNavigation offormation flying spacecraft using GPS the PRISMA technologydemonstrationrdquo in Proceedings of the 22nd International Techni-cal Meeting of the Satellite Division of the Institute of Navigation(ION GNSS rsquo09) pp 1427ndash1441 Savannah Ga USA September2009
[3] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers Dordrecht The Netherlands2001
[4] K T Alfriend S R Vadali P Gurfil J P How and L S BregerSpacecraft Formation Flying Elsevier Oxford UK 2010
[5] H Schaub S R Vadali J L Junkins and K T Alfriend ldquoSpace-craft formation flying control using mean orbit elementsrdquo TheJournal of the Astronautical Sciences vol 48 no 1 pp 69ndash872001
[6] J S Shaver Formulation and evaluation of parallel algorithms fororbit determination problem [PhD thesis] Department of Aero-nautics Massachusetts Institute of Technology CambridgeMass USA 1980
[7] A P M Chiaradia H K K Kuga and A F B A PradoldquoComparison between two methods to calculate the transitionmatrix of orbit motionrdquoMathematical Problems in Engineeringvol 2012 Article ID 768973 12 pages 2012
[8] Y Tsuda ldquoState transitionmatrix approximation with geometrypreservation for general perturbed orbitsrdquo Acta Astronauticavol 68 no 7-8 pp 1051ndash1061 2011
[9] S R Vadali ldquoAn analytical solution of relative motion ofsatellitesrdquo in Proceedings of the Dynamics and Control of Systemsand Structures in Space Conference Cranfield UK July 2002
[10] C J Damaren ldquoAlmost periodic relative orbits under J2pertur-
bationsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 221 no 5 pp 767ndash774 2007
[11] A E Roy Orbital Motion Adam Hilger Bristol UK 1982[12] F L Markley ldquoApproximate Cartesian state transition matrixrdquo
The Journal of Astronautical Sciences vol 34 no 2 pp 161ndash1691986
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DistributedSensor Networks
International Journal of
4 International Journal of Aerospace EngineeringD
evia
tion
(km
)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 2 Deviation in position for Molniya orbit
Dev
iatio
n (k
m)
Time (s)
Deviation in position
Present secular STMUsing STM in [7]
6
4
2
0
0 120 240 360 480 600 720 840
Figure 3 Deviation in position for sun-synchronous orbit
where 119883(119905) is differential of the states (119909 119910 119911 119909
1015840 119910
1015840 119911
1015840) at 1199050
Discarding higher order terms in (16) we have
Φ
119903119903= 119868 +
(2G0+ G) (Δ119905)
2
6
Φ
119903V = 119868Δ119905 +
(2G0+ G) (Δ119905)
3
12
ΦV119903 =(G0+ G) (Δ119905)
2
Φ
119903V = 119868 +
(G0+ 2G) (Δ119905)
2
6
(18)
The gradient matrix is
G =
(
(
120597119886
119909
120597119909
120597119886
119909
120597119910
120597119886
119909
120597119911
120597119886
119910
120597119909
120597119886
119910
120597119910
120597119886
119910
120597119911
120597119886
119911
120597119909
120597119886
119911
120597119910
120597119886
119909
120597119911
)
)
(19)
The matrices G0and G denote G(119905
0) and G(119905) respectively
and Δ119905 is (119905 minus 119905
0)
4 Illustration
STM is not to be used as a propagator and is used between theupdates of the state over the duration of orbit determination
However an experiment of propagation is carried out hereover certain duration to ensure that the present modelefficiently captures the secular effects The selected orbit isMolniya orbit with 119886 = 26554 kms 119890 = 072 and 119894 = 634
degrees This orbit is more eccentric and the short periodiceffects are predominant The top dotted line in Figure 2 isthe result that depicts the absolute deviation in position(distance) from the STM that includes short and periodiceffects given in [7] with respect to (3a) and (3b) The bottomcurve is the absolute deviation between the proposed STM(15)ndash(19) with respect to (3a) and (3b) again as function oftime Both have the same initial conditions This illustratesthat the proposed model is closer to the mean or secularorbital motion given in (3a) and (3b) in the presence of 119869
2
effect neglecting the numerical error due to propagationThenumerical propagation error due to step size is common inboth This validates that the STM derived in Section 3 usingthe partial derivatives of accelerations in (15) contains seculareffects alone
Next a similar exercise as carried out in Figure 2 isconsidering a sun-synchronous polar orbit with 119886 = 71684119890 = 00011 and 119894 = 9851 The top dashed line in Figure 3is the result that depicts the deviation in position from theSTM that includes short and periodic effects The bottomcurve is the deviation in position from the proposed STMThe illustration confirms that the proposedmodel is closer to(3a) and (3b)
Next example is about illustrating the modeling of (7)The satellite orbit has 119886 = 72443 119890 = 00003 and 119894 = 20
degrees The satellites placed in such orbits are mostly usedfor metrological purposesThe payload sensor usually coversa wide area as in microwave remote sensing applicationsTheargument of latitude 120579
2corresponding to (3a) and (3b) is first
derived The satellite motion is also obtained as solution of(3a) and (3b) while considering complete perturbations asmentioned in (1) The instantaneous argument of latitude is120579
119886 From (3a) and (3b) the argument of latitude denoted as
120579
2 is obtainedThe residue (120579
119886minus120579
2) is plotted against time in
Figure 4 It can be seen that this deviation is secular and haslarge effect along the track
Thedifference between the argument of latitude of the fullforce model 120579
119886and 120579
2over one orbit that is 102 minutes in
duration is then fit for the secular effect by a polynomial in aleast square sense For in this case it is
120579
119891= 120579
119887minus 003624 + 00012119905 minus 0000013119905
2+ 0000000046119905
3
(20)
The same fit is proposed here to extend over successive orbitsHowever at the end of each orbital period the ordinate valueof difference 120579
119887is used for the next orbital period Here for
example the value of 120579
119887is minus0113 at the start This approach
of orbit model representation can be considered as analternative to the existing methods for onboard applications[12]
The difference between the argument latitude of the fullforce model 120579
119886and 120579
119891in (7) is computed This residue is
illustrated in Figure 5 over four orbits and is observed tobe periodic over each orbital period The initial states for
International Journal of Aerospace Engineering 5
Time (min)
(deg
)
09
07
05
03
01
minus01
minus03
0 102 204 306 408 510 612 714 816 918 1020
Figure 4 Difference in argument of latitude
Time (min)0 102 204 306 408
01
0
minus01
minus02
minus03Resid
ue (d
eg)
Residue in the argument of latitude
Figure 5 Residue with respect to the polynomial fit
the STM are obtained from the proposed orbit model whileusing (3a) and (3b) along with the correction in (20) Theabsolute deviation is with respect to the position of full forcesecular propagation model The plot in Figure 6 is similarto that in Figure 1 or Figure 2 The deviation in positionin the plot uses the STM in Section 3 and using (15) canbe seen to accommodate secular effects in the presence allperturbations
5 Application
Equations (3a) and (3b) describe the dynamics of the orbitingsatellite when oblate earth effect is considered The relativemotion based on geometric approach is given in [9] Theyindependently make use of the dynamics in (3a) and (3b)for both satellites The two satellites have identical (119886 119890 119894) in(3a) and distinct (119908Ω119872) in (3b) The relative motion asreproduced from [9] is
Δ119909 (Δ119905) = minus1 + 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (120575120579 minus 120575Ω)
+ 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (2120579
1+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119888 (120575120579) minus 119888 (2120579
119898+ 120575120579)]
Dev
iatio
n (k
m)
Time (s)
45
30
15
0
0 100 200 300 400 500 600 700 800 900 1000 1100
STM using (14)ndash(18)
Figure 6 Deviation in position
Δ119910 (Δ119905) = 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (120575120579 minus 120575Ω)
minus 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119904 (120575120579) + 119904 (2120579
119898+ 120575120579)]
Δ119911 (Δ119905) = minus119904 (119894
119898) 119904 (120575Ω) 119888 (120579
119904)
minus [119904 (
119894
119898
2
) 119888 (119894
119904) 119888 (120575Ω) minus 119888 (119894
119898) 119904 (119894
119904)] 119904 (120579
119904)
(21)
Here 120579
119898is the instantaneous argument of latitude that is 120579
2
of the main or chief satellite Similarly 120579
119904is the instantaneous
argument of latitude of the second or follower satellite andtheir difference (120579
119898minus 120579
119904) which is denoted as 120575120579 The
differences in the instantaneous longitude of the ascendingnode between them is 120575Ω The 119888 and 119904 functions denotecos and sin functions and 119894
119898and 119894
119904denote the inclinations
of the main and follower respectively Also Δ119905 denotes theincremental time as used in (3a) and (3b) In (21) Δ119909 Δ119910 Δ119911
are relative positions of the second satellite in the orbit frameof the first
Here we shall outline the application of the secularapproximation 120579
119901from (7) when substituted into (21) This
is to enhance (21) to match the secular effects particularlyalong the track while considering all perturbations To do sowe replace the argument of latitude angle 120579
2for both satellites
in (21) by an appropriate 120579
119901 computed individually using (20)
and (7) as in Figure 6 for the main and secondary satellitesThese computed 120579
119901are then substituted into (21) as 120579
119898
and 120579
119904
respectively for the main and secondary satellite Now (21) isenhanced to match the secular effects particularly along thetrack of the full forcemodel Itmay be noted that in formationflying (21) is used for guidance and can also be used to derivethe initial conditions
6 International Journal of Aerospace Engineering
6 Conclusion
Approximate orbit model that captures the secular motion inthe argument of latitude while considering all perturbationforces is obtained Approximate state transition matrix isfirst derived for the orbital motion that matches the secularmotion with the effects of oblate earth This STM can alsoaccommodate the approximate orbit model with all forcesNumerical behavior of the state transition matrix and theorbit models has been provided Finally a direct applicationof the orbit model in relativemotion of two satellites has beenindicated The suggested approach is simpler for onboardimplementation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author acknowledges all the three anonymous refereesfor their comments that have enhanced the presentationTheauthor thanks Mr N S Gopinath Group Director and Dr SK Shivakumar Director from ISRO Satellite Centre for theirencouragements
References
[1] E Gill OMontenbruck andK Brieszlig ldquoGPS-based autonomousnavigation for the BIRD satelliterdquo in Proceedings of the 15thInternational Symposium on Spaceflight Dynamics BiarritzFrance June 2000
[2] S D Amico J-S Ardaens andOMontenbruck ldquoNavigation offormation flying spacecraft using GPS the PRISMA technologydemonstrationrdquo in Proceedings of the 22nd International Techni-cal Meeting of the Satellite Division of the Institute of Navigation(ION GNSS rsquo09) pp 1427ndash1441 Savannah Ga USA September2009
[3] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers Dordrecht The Netherlands2001
[4] K T Alfriend S R Vadali P Gurfil J P How and L S BregerSpacecraft Formation Flying Elsevier Oxford UK 2010
[5] H Schaub S R Vadali J L Junkins and K T Alfriend ldquoSpace-craft formation flying control using mean orbit elementsrdquo TheJournal of the Astronautical Sciences vol 48 no 1 pp 69ndash872001
[6] J S Shaver Formulation and evaluation of parallel algorithms fororbit determination problem [PhD thesis] Department of Aero-nautics Massachusetts Institute of Technology CambridgeMass USA 1980
[7] A P M Chiaradia H K K Kuga and A F B A PradoldquoComparison between two methods to calculate the transitionmatrix of orbit motionrdquoMathematical Problems in Engineeringvol 2012 Article ID 768973 12 pages 2012
[8] Y Tsuda ldquoState transitionmatrix approximation with geometrypreservation for general perturbed orbitsrdquo Acta Astronauticavol 68 no 7-8 pp 1051ndash1061 2011
[9] S R Vadali ldquoAn analytical solution of relative motion ofsatellitesrdquo in Proceedings of the Dynamics and Control of Systemsand Structures in Space Conference Cranfield UK July 2002
[10] C J Damaren ldquoAlmost periodic relative orbits under J2pertur-
bationsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 221 no 5 pp 767ndash774 2007
[11] A E Roy Orbital Motion Adam Hilger Bristol UK 1982[12] F L Markley ldquoApproximate Cartesian state transition matrixrdquo
The Journal of Astronautical Sciences vol 34 no 2 pp 161ndash1691986
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 5
Time (min)
(deg
)
09
07
05
03
01
minus01
minus03
0 102 204 306 408 510 612 714 816 918 1020
Figure 4 Difference in argument of latitude
Time (min)0 102 204 306 408
01
0
minus01
minus02
minus03Resid
ue (d
eg)
Residue in the argument of latitude
Figure 5 Residue with respect to the polynomial fit
the STM are obtained from the proposed orbit model whileusing (3a) and (3b) along with the correction in (20) Theabsolute deviation is with respect to the position of full forcesecular propagation model The plot in Figure 6 is similarto that in Figure 1 or Figure 2 The deviation in positionin the plot uses the STM in Section 3 and using (15) canbe seen to accommodate secular effects in the presence allperturbations
5 Application
Equations (3a) and (3b) describe the dynamics of the orbitingsatellite when oblate earth effect is considered The relativemotion based on geometric approach is given in [9] Theyindependently make use of the dynamics in (3a) and (3b)for both satellites The two satellites have identical (119886 119890 119894) in(3a) and distinct (119908Ω119872) in (3b) The relative motion asreproduced from [9] is
Δ119909 (Δ119905) = minus1 + 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (120575120579 minus 120575Ω)
+ 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119888 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119888 (2120579
1+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119888 (120575120579) minus 119888 (2120579
119898+ 120575120579)]
Dev
iatio
n (k
m)
Time (s)
45
30
15
0
0 100 200 300 400 500 600 700 800 900 1000 1100
STM using (14)ndash(18)
Figure 6 Deviation in position
Δ119910 (Δ119905) = 119888
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (120575120579 + 120575Ω)
+ 119904
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (120575120579 minus 120575Ω)
minus 119904
2(
119894
119898
2
) 119888
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 + 120575Ω)
+ 119888
2(
119894
119898
2
) 119904
2(
119894
119904
2
) 119904 (2120579
119898+ 120575120579 minus 120575Ω)
+
1
2119904 (119894
119898) 119904 (119894
119904) [119904 (120575120579) + 119904 (2120579
119898+ 120575120579)]
Δ119911 (Δ119905) = minus119904 (119894
119898) 119904 (120575Ω) 119888 (120579
119904)
minus [119904 (
119894
119898
2
) 119888 (119894
119904) 119888 (120575Ω) minus 119888 (119894
119898) 119904 (119894
119904)] 119904 (120579
119904)
(21)
Here 120579
119898is the instantaneous argument of latitude that is 120579
2
of the main or chief satellite Similarly 120579
119904is the instantaneous
argument of latitude of the second or follower satellite andtheir difference (120579
119898minus 120579
119904) which is denoted as 120575120579 The
differences in the instantaneous longitude of the ascendingnode between them is 120575Ω The 119888 and 119904 functions denotecos and sin functions and 119894
119898and 119894
119904denote the inclinations
of the main and follower respectively Also Δ119905 denotes theincremental time as used in (3a) and (3b) In (21) Δ119909 Δ119910 Δ119911
are relative positions of the second satellite in the orbit frameof the first
Here we shall outline the application of the secularapproximation 120579
119901from (7) when substituted into (21) This
is to enhance (21) to match the secular effects particularlyalong the track while considering all perturbations To do sowe replace the argument of latitude angle 120579
2for both satellites
in (21) by an appropriate 120579
119901 computed individually using (20)
and (7) as in Figure 6 for the main and secondary satellitesThese computed 120579
119901are then substituted into (21) as 120579
119898
and 120579
119904
respectively for the main and secondary satellite Now (21) isenhanced to match the secular effects particularly along thetrack of the full forcemodel Itmay be noted that in formationflying (21) is used for guidance and can also be used to derivethe initial conditions
6 International Journal of Aerospace Engineering
6 Conclusion
Approximate orbit model that captures the secular motion inthe argument of latitude while considering all perturbationforces is obtained Approximate state transition matrix isfirst derived for the orbital motion that matches the secularmotion with the effects of oblate earth This STM can alsoaccommodate the approximate orbit model with all forcesNumerical behavior of the state transition matrix and theorbit models has been provided Finally a direct applicationof the orbit model in relativemotion of two satellites has beenindicated The suggested approach is simpler for onboardimplementation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author acknowledges all the three anonymous refereesfor their comments that have enhanced the presentationTheauthor thanks Mr N S Gopinath Group Director and Dr SK Shivakumar Director from ISRO Satellite Centre for theirencouragements
References
[1] E Gill OMontenbruck andK Brieszlig ldquoGPS-based autonomousnavigation for the BIRD satelliterdquo in Proceedings of the 15thInternational Symposium on Spaceflight Dynamics BiarritzFrance June 2000
[2] S D Amico J-S Ardaens andOMontenbruck ldquoNavigation offormation flying spacecraft using GPS the PRISMA technologydemonstrationrdquo in Proceedings of the 22nd International Techni-cal Meeting of the Satellite Division of the Institute of Navigation(ION GNSS rsquo09) pp 1427ndash1441 Savannah Ga USA September2009
[3] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers Dordrecht The Netherlands2001
[4] K T Alfriend S R Vadali P Gurfil J P How and L S BregerSpacecraft Formation Flying Elsevier Oxford UK 2010
[5] H Schaub S R Vadali J L Junkins and K T Alfriend ldquoSpace-craft formation flying control using mean orbit elementsrdquo TheJournal of the Astronautical Sciences vol 48 no 1 pp 69ndash872001
[6] J S Shaver Formulation and evaluation of parallel algorithms fororbit determination problem [PhD thesis] Department of Aero-nautics Massachusetts Institute of Technology CambridgeMass USA 1980
[7] A P M Chiaradia H K K Kuga and A F B A PradoldquoComparison between two methods to calculate the transitionmatrix of orbit motionrdquoMathematical Problems in Engineeringvol 2012 Article ID 768973 12 pages 2012
[8] Y Tsuda ldquoState transitionmatrix approximation with geometrypreservation for general perturbed orbitsrdquo Acta Astronauticavol 68 no 7-8 pp 1051ndash1061 2011
[9] S R Vadali ldquoAn analytical solution of relative motion ofsatellitesrdquo in Proceedings of the Dynamics and Control of Systemsand Structures in Space Conference Cranfield UK July 2002
[10] C J Damaren ldquoAlmost periodic relative orbits under J2pertur-
bationsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 221 no 5 pp 767ndash774 2007
[11] A E Roy Orbital Motion Adam Hilger Bristol UK 1982[12] F L Markley ldquoApproximate Cartesian state transition matrixrdquo
The Journal of Astronautical Sciences vol 34 no 2 pp 161ndash1691986
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
6 International Journal of Aerospace Engineering
6 Conclusion
Approximate orbit model that captures the secular motion inthe argument of latitude while considering all perturbationforces is obtained Approximate state transition matrix isfirst derived for the orbital motion that matches the secularmotion with the effects of oblate earth This STM can alsoaccommodate the approximate orbit model with all forcesNumerical behavior of the state transition matrix and theorbit models has been provided Finally a direct applicationof the orbit model in relativemotion of two satellites has beenindicated The suggested approach is simpler for onboardimplementation
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The author acknowledges all the three anonymous refereesfor their comments that have enhanced the presentationTheauthor thanks Mr N S Gopinath Group Director and Dr SK Shivakumar Director from ISRO Satellite Centre for theirencouragements
References
[1] E Gill OMontenbruck andK Brieszlig ldquoGPS-based autonomousnavigation for the BIRD satelliterdquo in Proceedings of the 15thInternational Symposium on Spaceflight Dynamics BiarritzFrance June 2000
[2] S D Amico J-S Ardaens andOMontenbruck ldquoNavigation offormation flying spacecraft using GPS the PRISMA technologydemonstrationrdquo in Proceedings of the 22nd International Techni-cal Meeting of the Satellite Division of the Institute of Navigation(ION GNSS rsquo09) pp 1427ndash1441 Savannah Ga USA September2009
[3] D A Vallado Fundamentals of Astrodynamics and ApplicationsKluwer Academic Publishers Dordrecht The Netherlands2001
[4] K T Alfriend S R Vadali P Gurfil J P How and L S BregerSpacecraft Formation Flying Elsevier Oxford UK 2010
[5] H Schaub S R Vadali J L Junkins and K T Alfriend ldquoSpace-craft formation flying control using mean orbit elementsrdquo TheJournal of the Astronautical Sciences vol 48 no 1 pp 69ndash872001
[6] J S Shaver Formulation and evaluation of parallel algorithms fororbit determination problem [PhD thesis] Department of Aero-nautics Massachusetts Institute of Technology CambridgeMass USA 1980
[7] A P M Chiaradia H K K Kuga and A F B A PradoldquoComparison between two methods to calculate the transitionmatrix of orbit motionrdquoMathematical Problems in Engineeringvol 2012 Article ID 768973 12 pages 2012
[8] Y Tsuda ldquoState transitionmatrix approximation with geometrypreservation for general perturbed orbitsrdquo Acta Astronauticavol 68 no 7-8 pp 1051ndash1061 2011
[9] S R Vadali ldquoAn analytical solution of relative motion ofsatellitesrdquo in Proceedings of the Dynamics and Control of Systemsand Structures in Space Conference Cranfield UK July 2002
[10] C J Damaren ldquoAlmost periodic relative orbits under J2pertur-
bationsrdquo Proceedings of the Institution of Mechanical EngineersPart G Journal of Aerospace Engineering vol 221 no 5 pp 767ndash774 2007
[11] A E Roy Orbital Motion Adam Hilger Bristol UK 1982[12] F L Markley ldquoApproximate Cartesian state transition matrixrdquo
The Journal of Astronautical Sciences vol 34 no 2 pp 161ndash1691986
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