geo orbit time syncronization

Geostationary OrbitDetermination for TimeSynchronization usingAnalytical Dynamic Models

JAE-CHEOL YOONKorea Aerospace Research Institute

KEE-HOON LEEYonsei UniversityKorea

BYOUNG-SUN LEEElectronics and Telecommunications Research Institute

BANG-YEOP KIMKorea Aerospace Research Institute

KYU-HONG CHOIYonsei UniversityKorea

YOUNG-KEUN CHANGHankuk Aviation UniversityKorea

YONG-SIK CHUNKorea Aerospace Research Institute

SUNG-WOONG RAChung-Nam National UniversityKorea

Manuscript received October 17, 2001; revised June 3 and July 15,2004; released for publication July 15, 2004.

IEEE Log No. T-AES/40/4/839853.

Refereeing of this contribution was handled by M. Ruggieri.

0018-9251/04/$17.00 c 2004 IEEE

A real time analytical orbit determination method has been

developed for precision national time synchronization. The

one-way time transfer technique via a geostationary TV satellite

standard time and frequency signal (STFS) dissemination

system was considered. The differential method was also

applied for mitigating errors in geostationary satellite STFS

dissemination system. Analytical dynamic orbit determination

with extended Kalman filter (EKF) was implemented to

improve differential mode STFS (DSTFS) service accuracy by

acquiring better accuracy of a geostationary satellite position.

The perturbation force models applied for satellite dynamics

include the geopotential perturbation up to fifth degree and

order harmonics, luni-solar perturbations, and solar radiation

pressure. All of the perturbation effects were analyzed by secular,

short, and long period variations for equinoctial orbit elements

such as semimajor axis, eccentricity vector, inclination vector,

and mean right ascension of the geostationary satellite. The

reference stations for orbit determination were composed of

four calibrated stations. Simulations were performed to evaluate

the performance of real time analytical orbit determination in

Korea. The simulation results demonstrated that it is possible

to determine real time position of geostationary satellite with

the accuracy of 300 m rms. This performance implies that the

time accuracy is better than 25 ns all over the Korean peninsula.

The real time analytical orbit determination method developed

in this research can provide a reliable, extremely high accurate

time synchronization service through setting up domestic-only

benchmarks.

Authors’ addresses: J-C. Yoon and Y-S. Chun, KOMPSAT SystemsEngineering and Integration Dept., Korea Aerospace ResearchInstitute (KARI), 45 Eoeun-Dong, Youseong-Gu, Daejeon, 305-333,Korea, E-mail: ([email protected]); K-H. Lee, Dept. of Electrical& Electronics Engineering, Yonsei University, Shinchon-dong134, Seodaemun-gu, Seoul 120-749, Korea; B-S. Lee, DigitalBroadcasting Research Center, Electronics and TelecommunicationsResearch Institute (ETRI), 161, Gajeong-dong, Youseong-gu,Daejeon, 305-350, Korea; B-Y. Kim, Communication SatelliteSystems Dept., Korea Aerospace Research Institute (KARI), 45Eoeun-Dong, Youseong-Gu, Daejeon, 305-333, Korea; K-H.Choi, Dept. of Astronomy and Space Sciences, Yonsei University,Shinchon-dong 134, Seodaemun-gu, Seoul, 120-749, Korea;Y-K. Chang, Dept. of Aerospace Engineering, Hankuk AviationUniversity, 200-1, Hwajeon-dong, Deogyang-gu, Goyang-city,Geonggi-do, 412-791, Korea; S-W. Ra, Dept. of ElectronicsEngineering, Chung-Nam National University, 220, Gung-dong,Youseong-gu, Daejeon, 305-764, Korea.

1132 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 40, NO. 4 OCTOBER 2004

I. INTRODUCTION

Time and frequency synchronization techniqueis essential for the realization of the next generationhigh speed and broadband wired or wirelessintegrated network. It is also an infrastructuretechnique for the various fields like communications,broadcasting, navigations, and so on. For the purposeof synchronization of wired communication networkin Korea, multi-level broadcasting method hasbeen used. However, it undergoes degradation ofsynchronization accuracy and stability. There areseveral limitations inherent in the ground-basedradio methods characterized by subionosphericradio propagation. Most of these limitations on timeaccuracy can be overcome by using geostationarysatellite based dissemination. The frequencies usedfor geostationary satellite based dissemination arethe SHF or EHF (30/20-GHz) bands, for which theionosphere can be considered as virtually transparent.Especially, utilization of the EHF band frequencymakes not only the ionosphere effect lesser, but alsothe antenna gain higher, which allows the use ofsmall antenna station. It is possible for countrieshaving small territory such as Korea to provide adomestic highly accurate differential mode standardtime and frequency signal (DSTFS) service usingone-way mode via a geostationary satellite. However,since the position accuracy requirement for mostcommunication geostationary satellites is fitted byorbit control plan to allocate the satellite withinstation-keeping box, the accuracy of the satelliteorbit ephemeris provided by master orbit controlfacility is not good when tone ranging and anglemeasurement from a single tracking station isused. Especially, the orbit propagation accuracy isdegraded as time goes by. Thus a new concept of orbitdetermination is required for DSTFS service usingone-way mode via a geostationary satellite. In thisresearch, a new analytical dynamic orbit determinationmethod with extended Kalman filter (EKF) has beendeveloped to improve DSTFS accuracy. This methodmakes it possible to estimate accurate position of ageostationary satellite in real time by simple analyticalalgorithm. The geopotential perturbation was modeledup to fifth degree and order harmonics using Kaula’sformula [1]. Computer algebraic manipulation [2, 3]was used to develop algebraic expressions forthe calculation of perturbation due to luni-solargravitational effects. The solar radiation force wasalso modeled for three-axis stabilized geostationarysatellite. All of the perturbation effects were analyzedby secular, short, and long period variations forequinoctial orbit elements such as semimajor axis,eccentricity vector, inclination vector, and mean rightascension of the geostationary satellite. Simulationwas performed to verify whether the new analyticalorbit determination could improve the accuracy of

the domestic DSTFS service. It was assumed that thereference stations for real time orbit determinationconsisted of four stations with time synchronizedeach other. TWSTFT (two-way satellite time andfrequency transfer) system in which accurate positionsof satellite and Earth stations are not needed canbe recommended for time correction of referencestations. TWSTFT involves an exchange of timesignals between two Earth stations via a geostationarysatellite and provides time transfer with a theoreticalprecision of several hundred picoseconds [4]. In thisresearch, a measurement of time difference from tworeference stations is regarded as one observation forthe dynamic orbit determination and a set of threemeasurements from combination of four stations aregenerated at the same epoch. These measurementsare the same as those of the conventional trilaterationmethod using TDF2-satellite [5].

II. CONVENTIONAL DIFFERENTIAL MODE STFS

India has been broadcasting standard timeand frequency signal (STFS) on one of the radionetworking channels of INSAT since 1988. Currently,the accuracy of INSAT DSTFS service is better than1 ¹s and frequency stability is 1 part in 1011 all overthe Indian subcontinent [6, 7]. In Korea, the one-waytime dissemination system for STFS disseminationvia KOREASAT satellite was proposed. Since theaccuracy requirement for time synchronization ofcode-division multiple access (CDMA) network inKorea is better than 100 ns, DSTFS disseminationsystem using geostationary satellite will be areasonable choice to provide time synchronizationservice economically. Korea Telecomm, which is thelargest communication service company in Korea, hasbeen developing such a system since 1999 in order tostart time synchronization service in 2001.

A transmit station and a reference stationsimultaneously transmit STFS to acquire correctiondata. Whenever the satellite orbit information isreceived from master orbit control facility onceevery two or three weeks, the error induced bydifference between the predicted ephemeris andthe actual position of the satellite is calculated andtransmitted to the reference station. The correctionaccuracy is primarily affected by the inaccuracy ofthe satellite orbit modeling. The distance betweenthe ground stations and the geostationary satelliteis nearly 40,000 km compared with a maximum ofabout 500 km between two Earth stations in Korea.Thus, if the residual recorded at the specified Earthstation is subtracted from the error residuals recordedat a reference station, most of this common modeinaccuracy can be cancelled out. Transmit stationuplinks these error data to the satellite periodically.At any receiving station, the correction data canthen be decoded and used for on-line correction of

YOON ET AL.: GEOSTATIONARY ORBIT DETERMINATION FOR TIME SYNCHRONIZATION 1133

Fig. 1. Propagation delay by difference between predicted ephemeris and actual position.

the propagation delay. This calibration techniqueis differential correction known as DSTFS and thebasic method is also being used with good successin common view GPS technique for minimizing timeand frequency errors including ephemeris error andatmospheric effect.Fig. 1 shows the occurrence of propagation delay

by difference between predicted ephemeris and actualposition of geostationary satellite. Most of the errorsin the DSTFS time dissemination are due to a satelliteposition vector error ¢S. The differential errorbetween two stations, A (receiving) and R (reference)is given by [7]

¢TAR =¢TTA ¢TTR =1c(°A °R) ¢S (1)

where °A and °R are unit vectors in the directionof the ray path to the satellite. It is easily inferredthat ¢TAR becomes smaller if the two stations Aand R are closer or the prediction accuracy ofgeostationary satellite position is more improved.However, since the position accuracy requirementfor most communication geostationary satellites isoptimized by station-keeping boundary, the accuracyof the satellite orbit ephemeris received from masterorbit control facility is not good for DSTFS servicepurpose when tone ranging and angle measurementfrom a single tracking station is used. It was predictedthat the KOREASAT orbit error computed using rangeand angle measurement from a single tracking stationis about 4.2 km through covariance analysis usingthe ORAN [8] program [9]. Assuming ¢Sx = 2 km,¢Sy = 2 km, and ¢Sz = 2 km, DSTFS service canprovide users in south Korea with residual error of50 ns or better (Fig. 2).

III. A NEW ANALYTICAL ORBIT DETERMINATION

A new analytical dynamic orbit determinationmethod with EKF has been developed to reduce the

Fig. 2. Error contour of differential STFS over Korea.¢Sx = 2 km, ¢Sy = 2 km, ¢Sz = 2 km. Kumsan is the reference

station.

prediction error of geostationary satellite position, ¢Sof (1). A measurement of time difference betweentwo stations is regarded as one observation forthe analytical dynamic orbit determination and themeasurements from combination of four stations areused. These measurements are the same as those ofthe conventional trilateration method, which performscommon view transfer between remote clocks usingFrench TV geostationary satellite TDF2 [5]. Thismeasurement system uses a specific type of signal asan absolute time marker. The time of arrival (TOA)of this marker at each involved station will be datedrelative to each local time scale. The measured TOAdifference between pairs of stations includes the valueof difference in UTC (universal time coordinated) and


the delays RAB , which includes satellite position error,atmospheric, and relativistic effects

TOA[A] TOA[B] = UTC[A] UTC[B] +RAB:

(2)

In addition, EKF technique is implemented toestimate the position and velocity of geostationarysatellite in real time using the above measurementsand analytical orbit propagator that has beendeveloped with general perturbation theory thatsolves Lagrange equations of orbital elements usinganalytical series expansions applied to provide thereference orbit in the EKF algorithm. Since thegeostationary orbit eccentricity and inclination arealmost zero, the equinoctial orbit elements shouldbe used to avoid the singularity of the classical orbitelement in the process of deriving the equations. Theequinoctial orbit elements used in this research are a,ec = ecos(!+), es = esin(!+), Wc = sin icos,Ws = sin isin, and ¸= !++M , where a, e, i, ,!, and M are semimajor axis, eccentricity, inclination,right ascension of ascending node, argument ofperigee, and mean anomaly, respectively, and ¸ ismean right ascension of the satellite.The equations of motion of geostationary satellite

are expressed such as (3) in equinoctial elements asa general form in terms of the disturbing function R[10]

da

dt= 2

na

GM

@R

@M

decdt=

na

GM1 e2

@R

@es

na

GM

es1 e2

tani

2@R

@i

na

GM

ec 1 e2

(1+ 1 e2)

@R

@M

desdt=

na

GM1 e2

@R

@ec+

na

GM

ec

1 e2tan

i

2@R

@i

na

GM

es 1 e2

(1+ 1 e2)

@R

@M

dWcdt

=nacos i

GM 1 e2@R

@Ws(3)

na

GM

Wc cos i

1 e2(1+ cos i)

@R

@!+@R

@M

dWsdt

=nacos i

GM 1 e2@R

@Wc

na

GM

Ws cos i

1 e2(1+ cos i)

@R

@!+@R

@M

d¸

dt=

2narGM

R+ (1 1 e2)@(!+)@t

+2 1 e2 sin2i

2@

@t

TABLE IPerturbations Applied for Analytical Orbit Propagator

Perturbations Descriptions

Geopotential 5 5 harmonics coefficients of JGM-3model are applied

Luni-solar gravity Ephemeris of Moon and Sun arecalculated by analytical method

Solar radiationpressure

Constant cross section area isconsidered

where, n is mean orbital angular velocity, G isgravitational constant, M is mass of the Earth,and ! is longitude of perigee (+!). R is thedisturbing function, which consists of secular,short, and long period variations by the Earthgravitational perturbation, luni-solar perturbations,and solar radiation pressure. Computer algebraicmanipulation was used to derive algebraic expressionsfor the calculation of perturbation due to luni-solargravitational effects [2, 3]. The derived equations areincluded in the Appendix. The detailed characteristicsfor the secular, short, and long period variationsof the equinoctial orbit elements were describedin [11]. The computed osculating orbit elements atan instance are the sum of the secular variations,short-period variations, long-period variations, andthe initial mean orbit elements. Table I summarizesthe perturbations applied for the analytical orbitpropagator. Although it is difficult to derive analyticalmodels, analytical propagator has a major advantage,that is, its processing time is faster than numericalpropagator which implements high order numericalintegration for many equations of motion. And alsoit can secure a proper performance for geostationaryorbit of which perturbations is less critical than lowEarth orbit, though its accuracy during short time isslightly worse than that of numerical propagator.

The nonlinear systems with dynamics andmeasurement are described by the following equations[12],

_x(t) = f(x(t), t) +!(t)

y(t) = h(x(t)) + À(t)(4)

where x(t) is the state vectors, f is the variation ofequinoctial orbit elements, and h is the measurementequation of the TOA difference between pair stations.Although this model is restrictive in assuming addeduncorrelated white noise processes ! and À, themodel is stereotyped to the systems found in orbitdetermination applications. The predicted state at timetk+1, x (tk+1) is computed by integrating the variationof equinoctial orbit elements with reference state xat tk. The state transition matrix Á from tk to tk+1 iscalculated by applying the numerical finite differencemethod. The covariance matrix of the state, P at thetime tk+1 is predicted using the state transition matrix


Fig. 3. Position error of analytical orbit propagator with respectto numerical orbit propagator.

and the covariance of the state noise, Q at the time tk

Á(tk) =@x(tk+1)@x(tk) x(tk)=x(tk)

P (tk+1) = Á(tk)P(tk)Á(tk)T +Q(tk):

(5)

The covariance of the state noise waspredesignated depending on the accuracy of theanalytical orbit propagator that was determined bycomparing with the precise numerical orbit models.In the numerical orbit propagator, the satellite’sequations of motion were numerically integratedusing Adams-Cowell 11th-order predictor-correctormethod [13]. The perturbations such as geopotential,the gravity of the Sun and Moon, solar radiationpressure, solid Earth and ocean tides, and Earth’sdynamic polar motion have been modeled. Thegeopotential model was EGM-96 360 360 [14],which is widely considered as one of the bestgeopotential models currently available for satelliteorbit determination. The solid Earth and ocean tidesmodels, described by Colombo [15], that performprecise tidal analysis of tracking and altimetry datawere used. The ephemeris of the Sun and the Moonwas computed by Chebyshev interpolation from JPLplanetary ephemeris DE403 [16]. The macro model[17], which models the satellite as a combinationof flat plates arranged in the shape of a box andthe connected solar arrays, has been used for theanalysis of solar radiation pressure on a satellite.And also the attitude mode of the spacecraft wasapplied. It had been verified that the accuracy ofthese numerical orbit dynamic models including theMSISE-90 atmospheric density model [18] is about10 m on low Earth orbit [19]. Fig. 3 and Fig. 4 showthat the maximum position and velocity error of theanalytical orbit propagator with respect to the precisenumerical orbit propagator during 4 days is about1000 m and 0.07 m/s, respectively. The propagationerror in the measurement time interval was considered

Fig. 4. Velocity error of analytical orbit propagator with respectto numerical orbit propagator.

Fig. 5. Relative position error of analytical orbit propagator withrespect to numerical orbit propagator during 60 s.

Fig. 6. Relative velocity error of analytical orbit propagator withrespect to numerical orbit propagator during 60 s.

to determine the covariance of the state noise. In thisresearch, the measurement time interval was set to60 s. Fig. 5 and Fig. 6 describe the relative positionand velocity error of the analytical orbit propagatorwith respect to the numerical orbit propagator during


Fig. 7. Tracking loop clock performance: Allan variance.

60 s. The covariance of the state noise can be setwith 2 m, 2 m, 2 m, 1:5 10 4 m/s, 1:5 10 4 m/s,1:5 10 4 m/s at the position and velocity diagonalcomponents. The initial states uncertainty can bededuced from low accurate position requirementfor most of communication geostationary satellitewhich is optimized by station-keeping boundary.Thus, the initial a priori covariance of states wasset with 10000 m, 10000 m, 10000 m, 3.0 m/s,3.0 m/s, 3.0 m/s at the position and velocity diagonalcomponents.The measurement partial derivative H at time tk+1

is calculated using the predicted state x (tk+1)

H(tk+1) =@y(tk+1)@x(tk+1) x(tk+1)=x (tk+1)

: (6)

As the real measurement y(tk+1) at the time tk+1is processed with the covariance of the measurementnoise R, the state and the covariance matrix areestimated and updated by the following equation

x(tk+1) = x (tk+1)+K(tk+1)[y(tk+1) y(tk+1)]

P(tk+1) = [I K(tk+1)H(tk+1)]P (tk+1)

[I K(tk+1)H(tk+1)]T

+K(tk+1)R(tk+1)K(tk+1)T:

(7)

The covariance of the measurement noise betweentwo stations was set with 10 ns. This noise levelwas based on experiment that measured the Allanvariance of satellite loop back clock received withno correction in single station and compared it withthat of master clock. The results of Allan variancemeasurements are depicted in Fig. 7 [20] and showthat the error must be smaller than 10 ns.The Kalman gain matrix K is computed as

K(tk+1) = P (tk+1)H(tk+1)T[R(tk+1)

+H(tk+1)P (tk+1)H(tk+1)T] 1: (8)

TABLE IIPairs of Station for Domestic-Only Baseline Setup

Earth-Centered-Fixed CoordinatesStations X (km) Y (km) Z (km)

1. Seoul 3044:845 4044.031 3867.1142. Jejudo 3168:735 4278.329 3500.4373. Busan 3290:803 4056.776 3647.4844. Kangnung 3170:440 3929.676 3883.930

Pair I: 1–2, Pair II: 2–3, Pair III: 1–4

Fig. 8. Position error of real time orbit determination when forcemodels of simulation and estimation are same. Error of each

component for Earth-centered fixed coordinates.

IV. SIMULATION RESULTS

For the purpose of verifying the performanceimprovement of time synchronization using theanalytical dynamic orbit determination technique,a simulation was implemented with measurementfrom four stations and a one geostationary satellite.Table II shows the simulated pairs of station fordomestic-only baseline setup in Korea. It is assumedthat the times among four stations are synchronizedeach other. One geostationary satellite is assumed asKOREASAT-3 at 113 east longitude. If estimationprocess was implemented using the same modelsas were used to simulate the measurements, theresult would be a solution essentially identical tothe truth orbit, assuming no data noise was applied.However, if estimation process is performed withslightly difference models, the result will differ fromthe truth orbit. Then the cause of the differences willbe the differences in models used. The process usingsimulated data shows how well the filter can estimatethe original model parameters using erroneous models.In this research, the orbit ephemeris generation ofgeostationary satellite for measurement simulationwas performed with high accurate numerical orbitpropagator that has been described in Section III, andthe ephemeris is considered as the truth orbit.

Fig. 8 describes the accuracy of orbitdetermination when force models of simulation and


Fig. 9. Position error of real time analytical dynamic orbitdetermination when force models of simulation and estimation are

different. Error of each component for Earth-centered fixedcoordinates.

estimation are same and show the error of eachcomponent for Earth-centered-fixed coordinates.After the filter convergence period, 5 h, the positioncomponent errors of (x,y,z) were 59 m, 216 m, and20 m rms and did not exceed 140 m, 610 m, and52 m, respectively. The major effect on the aboveaccuracy is considered as the measurement noise inthe identified observation geometry because the sameforce models between simulation and estimation wereused.Fig. 9 describes the accuracy of analytical dynamic

orbit determination using the different force modelsfor both simulation and estimation. After the filterconvergence period, 5 h, the position componenterrors of (x,y,z) were 93 m, 307 m, and 40 m rmsand did not exceed 238 m, 960 m, and 129 m,respectively. The result of Fig. 9 is worse than that ofFig. 8 and its major cause is erroneous force modelsof analytical orbit propagator. However, it describesmore realistic performance of analytical dynamicorbit determination. Although the domestic-onlybaseline setup is applied and also the geometrybetween geostationary satellite and four stations ispoor, the analytical dynamic orbit modeling andstatistic estimation technique can improve the positionaccuracy of geostationary satellite. Fig. 10 and Fig. 11describe the error contour of time synchronizationusing real time analytical dynamic determinedorbit. In the Korean peninsula, time synchronizationerror does not exceed 25 ns applying the maximumestimated position error. Time is synchronized within8 ns using rms estimated position error.

V. CONCLUSIONS

A real time analytical orbit determination methodhas been developed for further enhancement to theDSTFS dissemination system. All of the perturbation

Fig. 10. Error contour of differential STFS over Korea withgeostationary orbit determined by analytical dynamic. In case of

maximum position error, ¢Sx = 238 m, ¢Sy = 960 m,

¢Sz = 129 m. Kumsan is the reference station.

Fig. 11. Error contour of differential STFS over Korea withgeostationary orbit determined by analytical dynamic. In case ofrms position error, ¢Sx = 93 m, ¢Sy = 307 m, ¢Sz = 40 m.

Kumsan is the reference station.

effects were analyzed and formulated in terms ofsecular, short-period, and long-period variationsfor equinoctial elements. Computer algebraicmanipulations were used for developing algebraicexpressions for the calculation of perturbation dueto luni-solar gravitational effects. This analytical


orbit solution was used for the generation ofreference orbit in EKF filter process. This methodmakes it possible to estimate accurate position of ageostationary satellite in real time by simple analyticalalgorithm. The simulation results showed that thereal time analytical orbit determination using threetime difference measurements from four calibratedstations makes it possible to estimate the positionof KOREASAT within maximum 1 km accuracy,which can improve the DSTFS time accuracydown to maximum 25 ns in Korea peninsular. Thisproposed new method successfully can provide clocksynchronization with performance at least six timesbetter than that of conventional DSTFS system inKorea and have an important role as a backupof GPS.

APPENDIX

A. Variation of Semimajor Axis

Secular variations due to the tesseral and sectorialterms of the Earth up to 5 5

(¢a)sec =GM

a3

R

a

2

a 3(1+cos i)2(S22 cos(2Lo) C22 sin(2Lo))+11:25R

a(1+cos i)3(S33 cos(3Lo) C33 sin(3Lo))

+R

a(1:875sin2 i(1+3cos i) 1:5(1+cos i))(S31 cos(Lo) C31 sin(Lo))+52:5

R

a

2

(1+cos i)4(S44 cos(4Lo) C44 sin(4Lo))+R

a

2

(52:5sin2 icos i(1+cos i) 7:5(1+cos i)2)

(S42 cos(2Lo) C42 sin(2Lo))+R

a

3472516

(1+cos i)5 (S55 cos(5L0) C55 sin(5L0))

+286232

sin2 i (5cos3 i+9cos2 i+3cos i 1)3158

(1+cos i)3 (S53 cos(3L0) C53 sin(3L0))

+31564

sin4 i (5cos i+1)10516

sin2 i (3cos i+1)+158

(1+cos i)

(S51 cos(L0) C51 sin(L0)) t (9)

where Lo is mean longitude of satellite, t is elapsedtime, C22, C31, C33, C42, C44, C51, C53, C55, S22, S31,S33, S42, S44, S51, S53, S55 are tesseral and sectorialharmonics of the Earth, R is equatorial radius of theEarth.Short-period variations due to the J2 term of the

Earth

(¢a)sp j2 =32J2R

a

R

sin2 icos(2¸ 2): (10)

Short-period variations due to the Sun

(¢a)sp =+434:26cos(2¸ 2LH)+ 37:32cos(2¸)

+ 25:44cos(2¸ 2LH MH)

3:63cos(2¸ 2LH MH) (11)

where LH is mean ecliptic longitude of the Sun, andMH is mean anomaly of the Sun.

Short-period variations due to the Moon

(¢a)sp m =+967:67cos(2¸ 2LM)

+ 188:83cos(2¸ 2LM MM )

+ 80:61cos(2¸) 36:08cos(2¸ 2LM FM)

+ 35:82cos(2¸ 2LM +MM 2D)

+ 35:12cos(2¸ LM +FM)

+ 30:67cos(2¸ 2LM 2D)

26:84cos(2¸ 2LM +MM)

+ 25:47cos(2¸ 2LM 2MM )

+ 7:56cos(2¸ 2LM MM 2D)

7:04cos(2¸ LM MM FM)

7:03cos(2¸ 2LM MM +2D)

+ 6:72cos(2¸ MM) +6:44cos(2¸+MM)

+ 3:79cos(2¸ 2LM +2FM)

3:12cos(2¸ 2LM MH)

+ 3:08cos(2¸ 2LM +MH)

+ 2:92cos(2¸ LM MM +FM)


+2:83cos(2¸ LM +MM +FM)

2:78cos(2¸ 2LM 2MM +2D)

+ 2:49cos(2¸ 2LM 3MM)

+ 2:08cos(2¸ 2LM 2D+MH)

+ 1:79cos(2¸ 2LM MM +MH)

+ 1:67cos(2¸ 2LM +MM 2D+MH)

+ 1:67cos(2¸+2LM) +1:56cos(2¸ 2D)

1:56cos(2¸ 2LM MM MH)

1:52cos(2¸+LM FM)

+ 1:44cos(2¸+LM +FM )

+ 1:32cos(2¸+MM 2D)

+ 1:20cos(2¸ MM +2D)

+ 83:68cos(3¸ 3LM)+ 33:79cos(¸ LM)

+ 23:26cos(3¸ 2LM MM)

+ 11:69cos(¸ 3LM)

+ 11:47cos(¸+LM)+ 10:63cos(3¸ LM )

+ 5:74cos(¸ LM MM) 4:99cos(¸ FM)

4:68cos(3¸ 2LM FM)

4:68cos(3¸ 3LM +MM)

+ 4:63cos(¸ 2LM FM) +4:64cos(¸+FM)

+ 4:35cos(3¸ 3LM +MM 2D)

4:37cos(¸ 2LM +FM)

+ 4:66cos(3¸ 2LM +FM)

+ 4:09cos(3¸ 3LM 2D)

+ 4:07cos(3¸ 3LM 2MM)

+ 3:35cos(¸ 3LM MM)

+ 1:81cos(¸+LM +MM)

+ 1:76cos(3¸ LM MM)

+ 1:78cos(¸ LM +MM)

1:29cos(3¸ 2LM MM FM)

+ 1:25cos(3¸ 3LM MM 2D)

1:16cos(3¸ 3LM MM +2D)

+ 7:56cos(4¸ 4LM) +2:74cos(4¸ 4LM MM)

+ 2:24cos(2¸ 2LM)

+ 1:46cos(2¸)+ 1:35cos(2¸ 4LM)

+ 1:30cos(2¸ 2LM) (12)

where LM is mean longitude of the Moon, MM ismean anomaly of the Moon, FM is mean argument oflatitude of the Moon, and D is mean elongation of theMoon from the Sun.

B. Variations of Inclination Vector

Secular variations due to the Sun and the Moon

(¢Wc)sec

= [( 0:6359+0:0121cosM)sinM +0:0021sin2!M]t

10 5=day(13)

(¢Ws)sec

= [4:1074+0:4667cosM 0:0019cos2!M]t

10 5=day:

where M is longitude of ascending node of the Moonand !M is mean longitude of perigee of lunarorbit.

Short period variations due to J2 of the Earth

(¢Wc)sp j2

=32J2

R

a

2

(1+ cos i)(sin icos cos2¸

+sin isin sin2¸)

sin3 icos3 cos2¸+sin3 isin3 sin2¸1+cos i

(14)(¢Ws)sp j2

=32J2

R

a

2

(1+ cos i)(sin icos sin2¸

sin isin cos2¸)

+sin3 icos3 sin2¸ sin3 isin3 cos2¸

1+cos i:

Short-period variations due to the Sun

(¢Wc)sp =+0:1069 10 5 cos(2¸ 2LH)

0:1020 10 5 cos(2¸)

(¢Ws)sp =+0:1069 10 5 sin(2¸ 2LH)

0:1020 10 5 sin(2¸):

(15)

Short-period variations due to the Moon

(¢Wc)sp m =+0:2381 10 5 cos(2¸ 2LM )

0:2204 10 5 cos(2¸)+ 0:4650

10 6 cos(2¸ 2LM MM )

+0:4500 10 6 cos(2¸ LM FM)

0:4330 10 6 cos(2¸ LM +FM)

0:1820 10 6 cos(2¸ MM )

+0:7330 10 6 cos(¸ 3LM)

0:6510 10 6 cos(¸+LM)


(¢Ws)sp m =+0:2381 10 5 sin(2¸ 2LM)

0:2204 10 5 sin(2¸)

+0:4650 10 6 sin(2¸ 2LM MM)

+0:4500 10 6 sin(2¸ LM FM)

+0:4330 10 6 sin(2¸ LM +FM)

0:1820 10 6 sin(2¸ MM )

+0:1218 10 5 sin(¸ LM )

0:6170 10 6 sin(¸ 3LM )

0:5530 10 6 sin(¸+LM ): (16)

Long-period variations due to Luni-solarperturbations

(¢Wc)lp m = 1:3126sin !H sinLH +[ 40:4397+0:5951cosM 0:1789cos2!M] cos2LH

+[0:5951sinM 0:1789sin2!M] sin2LH +( 1:5781cos !H) cos3LH +( 1:5781sin !H) sin3LH

+[(0:3678+0:0780cosM)cos !M +0:5238sinM sin !M] cosLM

+[( 0:3789 0:0701cosM)sin !M 0:3755sinM cos !M] sinLM +( 6:6684 1:3791cosM) cos2LM

+( 1:3791sinM) sin2LM +( 0:8591cos !M) cos3LM +( 0:8591sin !M) sin3LM 10 5

(17)(¢Ws)lp m = 3:1435sin !H cosLH +4:3684cos !H sinLH +( 0:4412sinM 0:2559sin2!M)cos2LH

+( 37:0630+0:4412cosM +0:2559cos2!M)sin2LH +(1:4468sin !H)cos3LH +( 1:4468cos !H)sin3LH

+[( 1:6854+0:4049cos2LH 0:2805cosM)sin !M +( 0:4049sin2LH 0:0519sinM)cos !M]cosLM

+[(2:3703+0:4049cos2LH +0:3901cosM)cos !M +(0:4049sin2LH +0:0576sinM) sin !M]sinLM

+(1:0111sinM 0:811sin2LH)cos2LM +( 6:0928 1:0111cosM +0:1811cos2LH) sin2LM 10 5

where !H is mean longitude of perihelion.

C. Variations of Eccentricity Vector

Secular variations due to the Moon and solarradiation pressure

(¢ec)sec = +1:73 10 7=day sin !Mt

+0:0102(Aeff=msat)cosLH(18)

(¢es)sec = 4:07 10 7=day cos !Mt

+0:0102(Aeff=msat) sinLH

where Aeff=msat is effective area to mass ratio of thesatellite.

Short period variations due to J2 of the Earth

(¢ec)sp j2

=32J2

R

a

2

(1 32 sin

2 i) sin¸+ 14 sin

2 i(cos2 cos¸+sin2 sin¸)

+ 712 sin

2 i(cos2 cos3¸+sin2 sin3¸)

(19)(¢es)sp j2

=32J2

R

a

2

(1 32 sin

2 i) sin¸ 14 sin

2 i(cos2 sin¸ sin2 cos¸)

+ 712 sin

2 i(cos2 sin3¸ sin2 cos3¸) :

Short period variations due to the Sun

(¢ec)sp =+0:1504 10 4 cos(¸ 2LH)

+ 0:1715 10 5 cos(3¸ 2LH)

0:1516 10 5 cos(¸)

+ 0:8826 10 6 cos(¸ 2LH MH)

0:4113 10 6 cos(¸+2LH) (20)

(¢es)sp = 0:1594 10 4 sin(¸ 2LH) 0:4171

10 5 sin(¸) +0:1715

10 5 sin(3¸ 2LH)

0:9347 10 6 sin(¸ 2LH MH)

0:4680 10 6 sin(¸+2LH):


Short period variations due to the Moon

(¢ec)sp m =+0:3478 10 4 cos(¸ 2LM )

+0:6932 10 5 cos(¸ 2LM MM)

+0:3784 10 5 cos(3¸ 2LM)

0:3272 10 5 cos(¸)

0:1763 10 5 cos(¸ LM FM)

+0:1648 10 5 cos(¸ LM +FM)

+0:1310 10 5 cos(¸ 2LM +MM 2D)

+0:1147 10 5 cos(¸ 2LM 2D)

+0:9594 10 6 cos(¸ 2LM 2MM)

0:9440 10 6 cos(¸ 2LM +MM)

0:8296 10 6 cos(¸+2LM)

+0:7325 10 6 cos(3¸ 2LM MM)

0:3451 10 6 cos(¸ LM MM FM)

+0:2243 10 5 cos(2¸ 3LM)

+0:6280 10 6 cos(2¸ 3LM MM)

+0:3687 10 6 cos(4¸ 3LM)

+0:1871 10 6 cos(2¸ LM)

+0:1816 10 6 cos(3¸ 4LM)

(¢es)sp m = 0:3680 10 4 sin(¸ 2LM ) (21)

0:9007 10 5 sin(¸)

0:7341 10 5 sin(¸ 2LM MM)

+0:3784 10 5 sin(3¸ 2LM)

0:1398 10 5 sin(¸ 2LM +MM 2D)

0:1215 10 5 sin(¸ 2LM 2D)

0:1016 10 5 sin(¸ 2LM 2MM)

+0:9993 10 6 sin(¸ 2LM +MM)

0:9448 10 6 sin(¸+2LM)

+0:9090 10 6 sin(¸ LM FM)

0:8506 10 6 sin(¸ LM +FM)

0:7649 10 6 sin(¸ MM)

+0:7325 10 6 sin(3¸ 2LM MM)

0:7115 10 6 sin(¸+MM)

+0:4522 10 6 sin(¸+LM FM)

0:4175 10 6 sin(¸+LM +FM)

+0:3186 10 6 sin(3¸)

0:2309 10 5 sin(2¸ 3LM)

0:6463 10 6 sin(2¸ 3LM MM)

0:3837 10 6 sin(2¸ LM)

+0:3687 10 6 sin(4¸ 3LM)

+0:1282 10 6 sin(2¸ 3LM +MM)

0:1816 10 6 sin(3¸ 4LM)

0:1475 10 6 sin(¸ 2LM):

Long period variations due to the Moon

(¢ec)lp m = ( 1:5117+1:0737cosM) 10 5 cosLM

+0:4564 10 5 sinM sinLM

+( 0:2725 0:1022cosM) 10 5 cos3LM

(¢es)lp m =+0:3700 10 5 sinM cosLM (22)

+ ( 3:4267+0:3724cosM) 10 5 sinLM

+0:3031 10 5 sin !M cos2LM

+( 0:2531cos !M ) 10 5 sin2LM

+0:1230 10 5 sinM cos3LM

+( 0:2970 0:1230cosM) 10 5 sin3LM:

D. Variations of Mean Satellite Longitude

Secular variation due to harmonics of the Earth

_ =GM

a3R

a

2

(4:5(1+cos i)2 1:5sin2 i)(C22 cos(2L0) +S22 sin(2L0))

+R

a(15(1+cos i)3 5:625sin2 i (1+ cos i))

(C33 cos(3L0) +S33 sin(3L0))+R

a

1:5+16:5cos i 7:5cos2 i 22:5cos3 i+sin2 i1+cos i

( 2:0625+1:875cos i+8:4375cos2 i)

(C31 cos(L0)+ S31 sin(L0))+R

a

2

(65:625(1+ cos i)4

26:25sin2 i(1+ cos i)2)(C44 cos(4L0) +S44 sin(4L0))

+R

a

2

(5:25sin2 icos2 i 9:375sin2 i

13:125sin2 icos i+131:25sin2 icos i(1+cos i)

18:75(1+ cos i)2)(C42 cos(2L0)+ S42 sin(2L0)) : (23)


Drift rate of the satellite longitude _L= _ _µE , where_µE is the rotation rate of the Earth.The acceleration component of the satellite

longitude is expressed as

L= M = _n =dn

da

da

dt: (24)

Short period variations due to the Sun

(¢L)sp =+0:1804 10 4 sin(2¸ 2LH)

0:1549 10 5 sin(2¸)

+ 0:1058 10 5 sin(2¸ 2LH MH):

(25)

Short period variations due to the Moon

(¢L)sp m =+0:4089 10 4 sin(2¸ 2LM)

+ 0:8040 10 5 sin(2¸ 2LM MM )

+ 0:3336 10 5 sin(2¸)

+ 0:1534 10 5 sin(2¸ 2LM +MM 2D)

0:1525 10 5 sin(2¸ LM FM )

+ 0:1445 10 5 sin(2¸ LM +FM)

+ 0:1319 10 5 sin(2¸ 2LM 2D)

0:1135 10 5 sin(2¸ 2LM +MM )

+ 0:1098 10 5 sin(2¸ 2LM 2MM )

+ 0:3406 10 5 sin(¸ LM)

+ 0:3045 10 5 sin(3¸ 3LM)

+ 0:1312 10 5 sin(¸ 3LM)

+ 0:1201 10 5 sin(¸+LM ): (26)

Long period variations due to the Moon

(¢L)lp m

=+[(5:6338 0:9639cos2LH 0:5924cosM) sin !M

+(0:9639sin2LH 0:1159sinM )cos !M]

10 5 cosLM

+[( 5:0432 0:9639cos2LH

+0:8332cosM)cos !M

+( 0:9639sin2LH +0:1250sinM)sin !M ]

10 5 sinLM

+[2:2297sinM +0:4766sin2LH

+0:2612sin2!M] 10 5 cos2LM

+[ 5:4008 2:2297cosM

0:4766cos2LH +0:2612cos2!M]

10 5 sin2LM

+[(0:6988+0:2766cosM)sin !M

+0:2766sinM cos !M] 10 5 cos3LM

+[( 0:6988 0:2766cosM)cos !M

+0:2766sinM sin !M] 10 5 sin3LM: (27)

REFERENCES

[1] Kaula, W. M. (1996)Theory of Satellite Geodesy.London: Blaisdell Publishing Company, 1996, 1–11.

[2] Dasenbrock, R. R. (1973)Algebraic manipulation by computer.Technical report 7564, Naval Research Laboratory,Washington, D.C., June 1973.

[3] Dasenbrock, R. R. (1982)Algebraic manipulation by computer.Technical report 8611, Naval Research Laboratory,Washington, D.C., June 1982.

[4] Kirchner, D., Ressler, H., Grudler, P., Baumont, F., Veillet,C., Lewandowski, W. Hanson, W., Klepczynski, W. J., andUhrich, P. (1993)Comparison of GPS common-view and two-way satellitetime transfer over a baseline of 800 km.Metrologia, 30 (1993), 183–192.

[5] Meyer, F. (1995)One-way time transfer using geostationary satellite TDF2.IEEE Transactions on Instrumentation and Measurements,44, 2 (Apr. 1995), 103–106.

[6] Gupta, A. S., Hanjura, A. K., and Mathur, B. S. (1991)Satellite broadcasting of time and frequency signals.In Proceedings of the IEEE, 79, 7 (July 1991), 973–982.

[7] Gupta, A. S., and Mathur, B. S. (1997)Standard time and frequency signal broadcast viaINSAT-accuracy improvements using differential mode.IEEE Transactions on Instrumentation and Measurements,46, 2 (Apr. 1997), 212–215.

[8] McCarthy, J. (1995)The Operational Manual for the ORAN Multi-SatelliteError Analysis Program.Maryland: Hughes STX, 1995, 1–52.

[9] Lee, B-S., Lee, J-S., and Choi, K-H. (1999)Analysis of a station-keeping maneuver strategy forcollocation of three geostationary satellites.Control Engineering Practice, 7 (1999), 1153–1161.

[10] Brouwer, D., and Clemence, G. M. (1961)Methods of Celestial Mechanics.New York: Academic, 1961.

[11] Lee, B-S., Lee, J-S., Yoon, J-C., and Choi, K-H. (1997)A new analytical ephemeris solution for the geostationarysatellite and its application to KOREASAT.Space Technology, 17, 5–6 (1997), 299–309.

[12] Gelb, A., Kasper, J. F., Jr., Nash, R. A., Jr., Price, C. F., andSutherland, A. A., Jr. (1974)Applied optimal Estimation, Cambridge, MA: The MITPress, 1974, 119–124.

[13] Mauty, J. L., and Brodsky, G. D. (1969)Cowell type numerical integration as applied to satelliteorbit computations.Technical report GSFC X-553-69-46, Goddard SpaceFlight Center, Greenbelt, Maryland, Dec. 1969.

[14] Lemoine, F. G., Smith, D. E., Kunz, L., Smith, R., Pavlis,E. C., Pavlis, N. K., Klosko, S. M., Chinn, D. S., Torrence,M. H., Williamson, R. G., Cox, C. M., Rachlin, K. E.,Wang, Y. M., Kenyon, S. C., Salman, R., Trimmer, R., Rapp,R. H., and Nerem, R. S. (1997)The development of the NASA GSFC and NIMA jointgeopotential model.In The International Symposium on Gravity, Geoid andMarine Geodesy, 117 (1997), 461–469.


[15] Colombo, O. L. (1984)Altimetry orbits and tides.Technical memorandum NASA-TM-86180, NASA, Nov.1, 1984.

[16] Standish, E. M., Newhall, X. X., Williams, J. G., andFolkner, W. M. (1995)JPL planetary and lunar ephemerides: DE403/LE403.Interoffice Memorandum IOM 314.10-127, Jet PropulsionLaboratory, May 22, 1995.

[17] Marshall, J. A., and Luthcke, S. B. (1994)Modeling radiation forces acting on Topex/Poseidon forprecision orbit determination.Journal of Spacecraft and Rockets, 31, 1 (1994), 99–105.

Jae-Cheol Yoon was born in Busan, Korea on November 14, 1970. He receivedhis B.S., M.S., and Ph.D. degrees in astronomy and space sciences from YonseiUniversity, Seoul, Korea, in 1995, 1997, and 2002, respectively.In 2003, he began to work at Korea Aerospace Research Institute (KARI) as

a senior research staff of the KOMPSAT Systems Engineering and IntegrationDepartment. His job at KARI has focused on developing a precise attitudedetermination system using a gyro and two star trackers, and also concentratedon the verification of the GPS based precision orbit determination system ofKOMPSAT-2 satellite.Dr. Yoon is a member of the Korean Space Science Society and Korea Society

for Aeronautical and Space Sciences.

Kee-Hoon Lee was born in Boseong, Korea, in 1965. He received the B.S.degree in aeronautical engineering from the ROK Air Force (ROKAF) Academy,Cheong-ju, Korea, in 1987 and the M.S. degree from University of Dayton,OH, in 1996 and the Ph.D. degree in electrical and electronics engineering fromYonsei University, Seoul, Korea, in 2004.Since joining the ROKAF in 1987, he has been working as an information

and communication officer. His current research interests are the areas of digitalmodems, satellite communication systems, and standard time and frequencydissemination systems.

[18] Hedin, A. E. (1991)Extension of the MSIS thermospheric model into themiddle and lower atmosphere.Journal of Geophysical Research, 96 (1991).

[19] Yoon, J-C., Lee, B-S., and Choi, K-H. (2000)Spacecraft orbit determination using GPS navigationsolutions.Aerospace Science and Technology, 4, 3 (2000), 215–221.

[20] Lee, J. S., and Lee, M. J. (2001)Implementation of the trial system for timing/frequencysynchronization using the Koreasat satellite.In Proceedings of KICS, 23, 2 (July 2001), 2482–2485.


Byoung-Sun Lee was born in Seoul, Korea on May 1, 1963. He received theB.S., M.S., and Ph.D. degrees in astronomy and space sciences from YonseiUniversity, Seoul, Korea in 1986, 1988, and 2001, respectively.He joined Electronics and Telecommunications Research Institute (ETRI) in

1989, where he was involved in developing the KOREASAT project. From 1992to 1994, he had been an OJT engineer in Lockheed-Martin Astrospace, U.S. andMartra-Marconi Space, U.K. for the KOREASAT project. From 1995 to 1999, hehad participated in the KOMPSAT-1 Ground Mission Control project as a seniormember of research staff in Mission Analysis and Planning Subsystem. He is nowworking for the KOMPSAT-2 and COMS-1 Ground Mission Control project asa principal member of research staff. His research interests are tracking and orbitdetermination of the satellite, and station-keeping maneuvers of the collocatedgeostationary satellites.Dr. Lee is a member of the American Astronautical Society, Korean Space

Science Society, Korea Society for Aeronautical and Space Sciences, and theInstitute of Control, Automation and Systems Engineer, Korea. He is a memberof the editorial board of the Journal of Astronomy and Space Sciences.

Bang-Yeop Kim was born in Daegu, Korea on 1967. He received the B.S., M.S.,and Ph.D. degrees in astronomy and space sciences from Yonsei University,Seoul, Korea in 1993, 1995, and 2002, respectively.Since 1995, he has been working as a researcher in the field of aerospace

engineering. He is currently a senior researcher of the Korea Aerospace ResearchInstitute (KARI) in Daejeon, Korea. Most of his works are related to the systemsengineering for the development of a geostationary satellite. So far, his researchefforts were given to the mission analysis and the developments of satelliteoperation software. Currently, his research interests are mainly in the precise orbitestimation and on-board star tracker.Dr. Kim is a member of the American Institute of Aeronautics and

Astronautics, American Astronautical Society, Korean Space Science Society, andKorea Society for Aeronautical and Space Sciences.

Kyu-Hong Choi received the B.S. degree in astronomy form Seoul NationalUniversity, Seoul, Korea, in 1972, and Ph.D. in astrophysics from University ofPennsylvania, Philadelphia, in 1980.From 1980 to 1981, he was with COMSAT, Washington, D.C. Since 1981, he

has been on the faculty of the Department of Astronomy and Space Sciences atYonsei University, Seoul, Korea, where he is currently a professor. His currentresearch interests are in Astrodynamics and satellite attitude control.Dr. Choi also served a consultant to government and private industry in the

areas of space technology and satellite communication.


Young-Keun Chang received his M.S. and Ph.D. degrees in aerospaceengineering from Virginia Polytechnic Institute & State University, Blacksburg,VA and University of Tennessee, Knoxville, respectively.He worked for Korea Aerospace Research Institute (KARI) between 1992

and 2000 as a system manager. He was with General Electric Astro Division,NJ, as a visiting engineer for 3 years for the development of Koreasat 1 and 2.He is currently working as a professor at the School of Aerospace Engineeringin Hankuk Aviation University. He is also working as a director of Space SystemResearch Lab, which is one of the National Research Labs awarded by Ministryof Science and Technology. His concern is on satellite system engineering,product assurance, and applications of satellite communications.Dr. Chang is a member of the American Astronautical Society and Korea

Society for Aeronautical and Space Sciences.

Yong-Sik Chun was born in Seoul, Korea on October 17, 1963. He receivedthe B.S. and M.S. degrees in electrical engineering from Dankook University,Seoul, Korea, in 1985 and 1987, respectively. He has been in the Ph.D. programat Chung-Nam National University, Daejeon, Korea, since 2001.He served in the army as a communication officer until 1991, and has been

working on satellite system engineering in Korea Aerospace Research Institute(KARI) from 1991 as a principal researcher. He is interested in satellite systemdesign and signal processing with a lossless image compression coding method.Mr. Chun is a member of the Korea Institute of Communication Sciences,

Korea Electromagnetic Engineering Society, and Institute of Electronics Engineersof Korea.

Sung-Woong Ra received the B.S. degree in electrical engineering from SeoulNational University, Seoul, Korea, and the M.S and Ph.D. degrees in electricalengineering from KAIST, Daejeon, Korea, in 1976, 1978, and 1992, respectively.He joined the faculty at the Chung-Nam National University, Daejeon, Korea,

where he is currently a processor in the Department of Electronics Engineering.His research interests are in the area of image signal processing and digitalcommunication systems.Dr. Ra is a member of the Korean Institute of Communication Sciences.


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