Research Article Approximate State Transition Matrix and ...downloads.hindawi.com/journals/ijae/2015/475742.pdf · e state transition matrix (STM) is a part of the onboard orbit determination

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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VLSI Design

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

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