Coarse Initial Orbit Determination for a Geostationary Satellite Using Single-Epoch GPS Measurements

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1 Sensos 15, 15, ; doi:1.339/s Aticle OPEN ACCESS sensos ISSN Coase Initial Obit Detemination fo a Geostationay Satellite Using Single-Epoch GPS Measuements Ghangho Kim, Chongwon Kim and Changdon Kee * School of Mechanical and Aeospace Engineeing and SNU-IAMD Seoul National Univesity, 1 Gwanak-o Gwanak-gu, Seoul 151-7, Koea; s: chew79@snu.ac.k (G.K.); nan77@snu.ac.k (C.K.) * Autho to whom coespondence should be addessed; kee@snu.ac.k; Tel.: ; Fax: Academic Edito: Anton de Ruite Received: Octobe 1 / Accepted: 7 Mach 15 / Published: 1 Apil 15 Abstact: A pactical algoithm is poposed fo detemining the obit of a geostationay obit (GEO) satellite using single-epoch measuements fom a Global Positioning System (GPS) eceive unde the spase visibility of the GPS satellites. The algoithm uses thee components of a state vecto to detemine the satellite s state, even when it is impossible to apply the classical single-point solutions (SPS). Though consideation of the chaacteistics of the GEO obital elements and GPS measuements, the components of the state vecto ae educed to thee. Howeve, the algoithm emains sufficiently accuate fo a GEO satellite. The developed algoithm was tested on simulated measuements fom two o thee GPS satellites, and the calculated maximum position eo was found to be less than appoximately km o even seveal kilometes within the geometic ange, even when the classical SPS solution was unattainable. In addition, extended Kalman filte (EKF) tests of a GEO satellite with the estimated initial state wee pefomed to validate the algoithm. In the EKF, a eliable dynamic model was adapted to educe the pobability of divegence that can be caused by lage eos in the initial state. Keywods: GEO; GPS; initial state; obit detemination; EKF; space GPS eceive

2 Sensos 15, Intoduction Many gound-tacking netwoks and facilities ae equied to tack the position of geostationay obit (GEO) satellites if a gound-tacking system is used [1,]. The Global Positioning System (GPS) eceive can povide high-accuacy position data at a low cost; thus, it is easonable to use GPS eceives in GEO satellites. Howeve, position accuacy, which is calculated using a single-point solution (SPS) algoithm with snapshot measuements, is low compaed with that of low Eath obit (LEO) satellites o gound uses; in cetain cases, no esult is poduced. These disadvantages occu because the GPS signal powe is lowe than that of the LEO satellite, and the geometical configuation of the GPS satellites elative to the GEO satellites is unfavoable. Theefoe, an obit detemination (OD) filte is needed to ovecome these unfavoable cicumstances [3]. Thee ae two main types of obit detemination (OD) filtes: post-pocessing and eal-time pocessing techniques. A eal-time OD filte is needed to opeate a geostationay obit (GEO) satellite instantly and popely; one such filte is the extended Kalman filte (EKF), which is well known fo its accuacy and efficiency. The EKF convegence time is detemined by the initial state and conditions []. Howeve, most studies do not conside how to detemine the initial state and its conditions [5]. Moe than fou GPS satellites ae obsevable in low Eath obit (LEO); thus, single-point solutions (SPS) can be applied at any time and the esult can be used as the initial states of the EKF with a high level of accuacy. Typically, fewe than fou GPS satellites ae obsevable at the GEO; theefoe, SPS ae not always applicable. In these situations, altenative solutions must be employed, such as the shot ac batch technique, which is a post-pocessing technique that equies measuements ove a long peiod of time; thus, this technique does not povide navigation data instantly [6,7]. In this pape, we developed a coase initial obit detemination algoithm to impove the accuacy of the initial EKF states unde the spase visibility of GPS satellites. The application of the poposed algoithm is illustated in Figue 1. We used the chaacteistics of GEOs to develop ou algoithm: GEO is an almost cicula obit, and its inclination angle is nealy zeo. We set the minimum numbe of state components to calculate the state of the satellite using snapshot measuements unde the spase visibility of GPS satellites. This algoithm can detemine the GEO satellite s state vecto even when fewe than fou GPS satellites ae visible, and it is vey pactical because it does not equie long-tem measuements. Figue 1. Application of the coase initial obit detemination algoithm.

3 Sensos 15, The emainde of this pape is oganized as follows. In Section, the details of the coase initial obit detemination algoithm ae explained. In Section 3, the geneal EKF-based obit detemination scheme is discussed. In Section, simulations of the poposed algoithm and EKF ae pefomed to test the accuacy and the availability of the poposed algoithm. Finally, the esults of the developed algoithm and EKF ae discussed.. Coase Initial Obit Detemination Algoithm Fou unknown vaiables appea in the classical SPS algoithm using the GPS signal: position components ( xyz),, and eceive clock bias eo ( δ ). Thus, measuements fom at least fou GPS satellites ae equied. The use position can typically be calculated using the GPS at a LEO satellite at any time because moe than fou GPS satellites ae obseved at the LEO satellite s altitude. The eo of the calculated single point position of a LEO is less than seveal dozens of metes and can be used as the initial state of the EKF. Howeve, it is impossible to calculate the position using the classical SPS algoithm when fewe than fou GPS satellites ae visible at the GEO. Moe than fou GPS satellites ae infequently visible at the GEO satellite; thus, the point position is not always detemined. Thus, we must wait until moe than fou GPS satellites ae visible to obtain navigation data. We developed a coase initial obit detemination algoithm to calculate a point solution using measuements obtained fom the eceive with two o thee GPS satellites. The algoithm uses a minimum numbe of state vaiables, which wee selected by consideing the chaacteistics of the GEO. The classical obital elements of the ideal GEO ae shown in Figue. Figue. Simplified geostationay obit (GEO) model. a =,16 km and e= i=, e is the eccenticity and i is the inclination. The ight ascension of the ascending node (RAAN) of the ideal GEO cannot be defined because the inclination angle is zeo. The eccenticity is also zeo, and thus, the agument of peigee cannot be defined eithe. Thus, we can define the state of the GEO satellite using only its tue longitude value. The tue longitude is a useful tem when defining cicula and equatoial obits, and its equation is given by ˆ I cos( λ tue ) = Iˆ (1)

4 Sensos 15, If we assume that the satellite otates in the ideal GEO, the states of a satellite can be appoximately expessed by one element: the tue longitude. We included two additional vaiables (the clock bias and clock bias ate of the eceive) in the state vecto of the GEO satellite because these eo components ae included in the C/A code pseudoange and pseudoange ate. Thus, the state vecto of the GEO satellite is defined as T Kep = λ tue δ δ () X c c whee c is the speed of light, δ is the eceive s clock bias, and δ is the eceive s clock bias ate. The state vecto in Equation () can be conveted into Catesian coodinates and given by T X = cos( ) sin( ) sin( ) cos( ) cat a λ a λ a λ λ a λ λ cδ cδ tue tue tue tue tue tue (3) GM λ tue = M = n= 3 () a whee G is the gavitational constant and M is the mass of the Eath. The value of a was set to a constant, and thus, the diffeential of a is zeo. Then, the measuement vecto can be defined as [ ] t T Z = ρ1 ρ1 ρm ρm (5) ρ = t cδ t + cδ + n (6) ρ = eˆ ( vt v) cδ t + cδ + n (7) t eˆ = (8) whee M is the numbe of visible GPS satellites (two o thee in this pape); t and v t ae the position and velocity vectos, espectively, of the GPS satellite; and v ae the position and velocity vectos, espectively, of the GEO satellite; δ t and δ t ae the GPS satellite clock bias and clock bias ate, espectively; δ and δ ae the eceive clock bias and clock bias ate, espectively; ê is the unit diection vecto; ρ and ρ ae the pseudoange and pseudoange ate, espectively; and n is the measuement noise fom the assumed Gaussian distibution. The ionospheic eo is not included in Equations (6) and (7) because we assume that the ionospheic eo can be emoved by the dual-fequency GPS eceive. We use a least-squaes technique to calculate X Kep ; this technique minimizes the squae sum of the diffeence between Z and Z ef Z is the measuement vecto obtained fom the GPS eceive at the tue point X Kep, and Z ef is the calculated measuement vecto at the efeence point X ef. We assume that the GEO satellite s position is appoximately known when the GPS eceive stats its own signal pocessing; thus, X ef is chosen with an eo of 1 km in the fist calculation and is updated iteatively until convegence is eached. The function of Z ef is nonlinea, and thus, Z ef must by lineaizing at the point X. The least-squaes technique to detemine is given by ef X Kep Δ X Kep = XKep Xef (9)

5 Sensos 15, Δ Z = Z Z ef (1) Zef = ρef 1 ρ ef 1 ρefm ρefm (11) ρ ef = t ef cδ t + cδ (1) T ρ ˆ ef = e ( v t v ef ) cδ t + cδ (13) Δ Z = HΔ X ef (1) T ( ) 1 Δ = Δ T Xef H H H Z H = H H cat Kep (15) The equations of the measuement matix X Kep = Xef +Δ Xef (16) H Kep and H cat ae given by H H X cat Kep = X (17) Kep cat Z = X (18) If we define the state vecto as Equation (), we can calculate the position of the GEO satellite using single-epoch measuements of two o thee GPS satellites. In the definition of the state vecto in Equation (), the inclination angle and eccenticity ae not included in the vaiables; howeve, these vaiables exist in the eal GEO obit and could incease the eo of the state vecto calculated iteatively. The geometic state eo estimated by the poposed algoithm can be defined as cat x cos( ν) y R3( ) R3( i) R3( ) sin( ) = Ω ω ν z ( x acos( )) ( y asin( )) ( z) ( c c ) (19) ε = λ est + λ est + + δ δ est () whee Ω is the ascending node, i is the inclination, ω is the agument of peigee, ν is the tue anomaly, is the adius, λ est is the estimated tue longitude, and cδ est is the estimated eceive clock bias. In Equation (), ( ) z is a constant bias tem that cannot be educed o emoved by the poposed algoithm; theefoe, the eo could be inceased if the z-component has geat value. Howeve, we can still detemine the state vecto as accuately as possible given the spase visibility of the GPS satellites. Afte calculating the state vecto using Equations (9) (18), the vecto can be used as the initial state value fo the EKF with a pactical level of accuacy.

6 Sensos 15, EKF Scheme The EKF is well known fo its accuacy and speed; thus, it is used in non-linea system applications, such as eal-time OD. We will not explain the entie algoithm in detail, as it is not the focus of this pape. Thus, the equations of the EKF algoithm scheme ae summaized in Table 1 [5,8]. Table 1. Extended Kalman filte (EKF) pocessing scheme using GPS measuements. Nonlinea Dynamics Model X k = φ k 1( Xk 1) + wk 1, wk N(, Q) Nonlinea Measuement Model Zk = hk( Xk) + vk, vk N(, R) State and Measuement T X = x y z x y z c c δ δ T Z = [ ρ1 ρ1 ρm ρm ] ρ = t + cδ cδt ρ = eˆ ( v t v ) + cδ cδ t Time update Pedicted state estimation: X ˆ k =φk 1( Xk 1) φk 1 Linea appoximation: Φk 1 X X= Xk 1 Pedicted covaiance matix: ˆ T Pk =Φk 1Pk 1Φ k 1 + Qk 1 Measuement update Pedicted Measuement: Zk = hk( Xk) hk Measuement matix lineaization: H k X X = Xk T T 1 Kalman gain computation: Kk = PH k k HkPH k k R + k State estimation: X ˆ k = Xk + K ( Z ) k k Zk Pˆ = I K H P Updating posteioi covaiance matix: [ ] k k k We used two-body gavity and othe petubations to pedict the state of the satellite. The equation of the geneal acceleations acting on a satellite is [5]: μ = + a 3 petubed (1) whee is the position vecto of the satellite in the Eath-centeed inetial (ECI) coodinate system, μ is the gavitational constant of the Eath and a petubed is the othe petubed acceleation. We included gavitational attaction by a nonspheical cental body, thid-body effects (Sun and Moon) and sola-adiation pessue in the petubed acceleations [5,9]. The gavitational attaction by the nonspheical cental body is expessed as: k

7 Sensos 15, μ U = 1 + p sin( φ ) C cos( mλ ) + S sin( mλ ) n n R nm, [ sat ]{ nm, sat nm, sat } n= m= () whee R is the adius of the Eath, p is the associated Legende polynomials, φ sat is the latitude of the satellite, λsat is the longitude of the satellite, and C and S ae the gavitational coefficients. The equation of the thid-body effect is expessed as: sat sat3 3 μ sat = +μ3 3 3 (3) sat sat3 3 whee μ is the gavitational constant of the Eath, μ 3 is the gavitational constant of the thid body, sat is the position vecto pointing fom the Eath to the thid body, sat3 is the position vecto pointing fom the satellite to the thid body and 3 is the position vecto pointing fom the Eath to the thid body. The acceleation by sola adiation pessue is included and is one of the significant acceleations acting on GEO satellites. The sola adiation pessue is expessed as: PSRCR A sat asr = m () whee P SR is the sola pessue, sat C R is the eflectivity, A is the effective aea of the satellite and m is the mass of the satellite. We intoduced some eos into the dynamics model of the EKF filte compaed with the measuement dynamics model to bette appoximate a eal situation, and the eos ae listed in Table. The degee and ode of the geopotential wee loweed fom to 1, and the satellite aea fo the sola pessue model was adjusted to ceate a 1% eo []. The eos of the initial state wee assumed to be appoximately 1 km and.1 km in position and velocity, espectively. With these values, we set the initial covaiance as follows: P = diag σp1 σp1 σp1 σp σ P σp σp1 σp (5) whee σ = ( ) and ( ) P1 1 km σ P = 1 km s. The noise of the ange and that of the ange ate ae Gaussian distibutions with standad deviations of.1 km and.1 km/s, espectively, in the simulation. We set R as follows fo the case of two o thee GPS satellites, espectively: R = diag σr1 σr σr 1 σ R o diag σr1 σ R σr 1 σr σr 1 σr (6) whee σ = ( ) 8 and ( ) R1 1 km σ R = 1 km s.

8 Sensos 15, Table. COMS obital elements. Measuement Dynamics EKF Dynamics Eo Initial epoch time UTC :: 1 Januay 6 - Simulation time h - Geopotential model EGM-96 (Degee:, Ode: ) (Degee: 1, Ode: 1) Thid-body gavity Sun, Moon (DE5) - Sola pessue N/m - Coss-sectional aea m 17.6 m (1% eo) Satellite mass 157 kg - Numeical integation algoithm Runge-Kutta 68 - X 7, (km) - Y 31,685.5 (km) - Z (km) - V x.3981 (km/s) - V y.866 (km/s) - V z.19 (km/s) - The elements of the pocess noise covaiance must be appoximately 1 16, with the same level of eo as the dynamics model of the filte; howeve, we tuned these values with consideation fo the initial state eo and convegence time [1,11]. We detemined Q as follows: Q = diag σq1 σq1 σq1 σq σ Q σq σq1 σq (7) 1 whee σ = ( ) 16 and ( ) Q1 1 km σ Q = 1 km s.. Simulation and Results.1. Simulation Pocedue We chose the Communication, Ocean and Meteoological Satellite (COMS) launched by the Koea Aeospace Reseach Institute (KARI) as ou GEO satellite fo simulation. The COMS is located at 18. east, and COMS missions ae elated to Ka-band communication sevices, meteoological monitoing, and ocean obsevation []. We simulated the satellite s obit to validate and test the developed algoithm. Fist, we geneated the position and velocity data of COMS fo h using a numeical obit popagation algoithm. Then, the pseudoange and pseudoange ate, which wee obtained fom the eceive in the GEO satellite, wee geneated with simulated GPS signals. The developed algoithm was tested using the geneated measuements unde the condition that two o thee GPS satellites wee obsevable. Then, the calculated state vecto was set to the initial state of the EKF, and the EKF was pocessed. The oveall simulation pocedue is summaized in Figue 3.

9 Sensos 15, Figue 3. Diagam of the simulation pocedue. 8 7 Numbe of visible GPS satellites 3 db-hz 5 db-hz 6 5 Satellites Time (hou) Figue. Numbe of visible GPS satellites ove h with 5 db-hz and 3 db-hz thesholds fo signal acquisition and tacking; 3 db-hz is the minimum value that conventional eceives can tack... Geneation of Measuements The fist simulation step was to popagate the GEO satellite s obit. We popagated the GEO using Cowell s method, which popagates the position and velocity of the satellite by integating the acceleations caused by petubations at each time step [5]. We included the geopotential, sola pessue, and thid-body gavity (the Sun and Moon). We chose the EGM-96 model as the geopotential, and the degee and ode wee both set to. The Runge-Kutta 68 algoithm was chosen as the numeical integato, and the integal step was set to 1 s. The initial obital elements fo popagation ae listed in Table.

10 Sensos 15, The GPS satellites position and velocity wee geneated evey 1 s fo h using the Almanac data. Afte geneating the positions and velocities of the GEO and GPS satellites, we checked whethe the GPS satellites wee obsevable at each epoch. A GPS satellite was only visible when it was not blocked by the Eath and its signal powe was sufficiently stong to be pocessed by the GPS eceive. Afte detemining the visibility of the GPS satellites, the C/A code pseudoange and pseudoange ate wee calculated. The C/A code pseudoange is given by [1] ρ= TX t( ) RX ( ) + cσ( RX) cσ t( TX) + n (8) = ( RX τ) ( RX ) + cσ ( RX ) cσ ( RX τ ) + n t t t ( ) ( ) σ = b + b1 t t σ = a + a t t 1 whee ρ is the C/A code pseudoange in L1, RX is the signal eception time, TX is the signal emission time, σ is the eceive clock bias, σ t is the GPS satellite clock bias, n is noise, τ is the time delay, t is the efeence time, a and a 1 ae the polynomial coefficients of the GPS satellite clock bias, and b and b1 ae the polynomial coefficients of the eceive clock bias. The geometic distance fom the GPS satellite to the eceive was calculated using TX t ( ) and ( RX) ; we did not use TX ( ) [13]. The GPS signal tavels though space at the speed of light, and thus, the eceive does not instantly eceive the signal emitted fom the GPS satellite. Theefoe, the signal eception time is late than the signal emission time. The equation of the elapsed time fom the emission to eception is given by TX t ( ) RX ( ) τ= c (3) t ( RX τ) ( RX) = c τ appeas on both sides of Equation (3). Thus, an iteation technique was used to calculate the pope τ. Fist, t ( RX) was used instead of t ( RX τ) to calculate the tempoay τ on the left side of the equation, Then, the tempoay τ was used on the ight side of Equation (3) to update the tempoay τ, and the iteations continued until τ conveged [13,1]. The equation fo the pseudoange ate is simila to that fo the pseudoange and is given by ρ = eˆ [ vt( TX) v( RX) ] + cδ ( RX) cδ t( TX) + n (31) whee δ and δ t ae the clock bias ates of the eceive and GPS satellite, espectively. These vaiables ae calculated fom the deivatives of σ and..3. Simulation Results σ t The algoithm was tested using data fom the fou points (A, B, C and D) selected fom the h simulated GEO obit. We selected fou points at 6 h intevals to igoously validate the algoithm. The simulation times of the selected points A, B, C and D wee UTC ::, 6::, 1:: and 18::, espectively, on 1 Januay 6. The position and velocity vecto of each point is given in Table 3. (9)

11 Sensos 15, Table 3. Position and velocity of the COMS at the fou selected points. Point A B C D UTC time :: 6:: 1:: 18:: x (km) 7, ,79.8 7, ,.5366 y (km) 31, , ,9.159 z (km) v x (km/s) v y (km/s) v z (km/s) Semi-majo axis (km),165.9, ,165.11, Eccenticity Inclination (deg) Ascending node (deg) Agument of peigee (deg) Tue anomaly (deg) The obsevable GPS satellites ove h ae depicted in Figue ; howeve, the simulations wee pefomed unde contolled conditions in which only two o thee GPS satellites wee visible. The initial eo in the efeence position was set to 1 km at each of the fou points, and the efeence eceive clock bias was set to 1 km. The noise tems in Equations (8) and (31) wee selected fom Gaussian distibutions with standad deviations of.1 km and.1 km/s, espectively. The positions of the COMS and its visible GPS satellites at each point ae shown in Figues 5 1. The ed spot epesents the COMS, and the yellow spots epesent GPS satellites. At each point, we poduced scenaios such that two o thee GPS satellites wee visible at the COMS. Thus, we intentionally chose GPS satellites among those visible to contol the numbe of visible satellites if moe than thee GPS satellites wee visible. Fo consistency, we simply emoved one satellite fom the scenaio whee thee GPS satellites wee visible such that only two GPS satellites wee visible. As shown in the figues, the visible GPS satellites ae located behind the Eath and ae located closely togethe. x 1 z axis (km) - - GEO x 1 y axis (km) x axis (km) x 1 Figue 5. Point A with two visible GPS satellites.

12 Sensos 15, x 1 z axis (km) - - GEO x 1 y axis (km) x axis (km) x 1 Figue 6. Point A with thee visible GPS satellites. x 1 z axis (km) - GEO - x 1 y axis (km) x axis (km) x 1 Figue 7. Point B with two visible GPS satellites. x 1 z axis (km) - GEO - x 1 y axis (km) x axis (km) x 1 Figue 8. Point B with thee visible GPS satellites.

13 Sensos 15, x 1 GEO z axis (km) - - x 1 y axis (km) x axis (km) x 1 Figue 9. Point C with two visible GPS satellites. x 1 GEO z axis (km) - - x 1 y axis (km) x axis (km) x 1 Figue 1. Point C with thee visible GPS satellites. x 1 z axis (km) - GEO - x 1 y axis (km) x axis (km) x 1 Figue 11. Point D with two visible GPS satellites.

14 Sensos 15, x 1 z axis (km) - - GEO D D1 D3 x 1 y axis (km) x axis (km) x 1 Figue 1. Point D with thee visible GPS satellites. We tested the developed algoithm using a single-epoch measuement at each point. The calculated state vectos of the COMS wee compaed to the tue values, and the diffeences of each ae summaized in Tables and 5. We also tested the influence of the position of the thid GPS satellite at point D; moe than thee GPS satellites ae obsevable at point D. We an simulations with two fixed GPS satellites and a thid satellite placed at seveal diffeent positions. The esults ae summaized in Table 6. The esidual efes to the diffeence between the calculated and tue values. The esults indicate that the esidual of the calculated position is less than km in ange, and thee of the fou points have esiduals of less than seveal kilometes in ange when using thee visible GPS satellites. Table. Estimated eo when using two visible GPS satellites. Residuals A B C D x (km) y (km) z (km) V x (km/s) V y (km/s) V z (km/s) clock bias (km) clock bias ate (km/s) The esiduals incease as the z-component of obital state inceases. The maximum eo occus when the z-component of the obital state is geatest, and the eos ae small when the z-components of the obital state ae small. This elationship occus because the developed algoithm does not include the z-component in the state vecto, and thus, the z-component in the eal obit influences the x- and y-components in the state vecto. The z-component value inceases with the inclination and the elationship between the inclination and eo at point D ae pesented in Figue 13. Based on Figue 13, we can conclude that the accuacy level of the poposed algoithm is high when the GEO satellite s inclination is small.

15 Sensos 15, Table 5. Estimated eo when using thee visible GPS satellites. Residuals A B C D1 x (km) y (km) z (km) V x (km/s) V y (km/s) V z (km/s) clock bias (km) clock bias ate (km/s) Table 6. Estimated eos when using two fixed GPS satellites and a thid satellite at vaious positions at point D. Residuals D1 D D3 x (km) y (km) z (km) V x (km/s) V y (km/s) V z (km/s) clock bias (km) clock bias ate (km/s) No significant diffeence occus when using measuements fom two o thee visible GPS satellites, except fo point B, whee the esidual deceases when using the measuements fom thee GPS satellites. Futhemoe, the geometic elationship among the GPS satellites and the GEO also affected the accuacy of the algoithm, as demonstated by the simulation esults fo point D. 6 Inclination vs. Eo 5 Eo Nom (km) Inclination (deg) Figue 13. Relationship between inclination and the estimated eo at point D.

16 Sensos 15, The EKF was tested using the initial state vecto calculated by the developed algoithm at each point. The time update was pocessed using the Runge-Kutta method. The simulation esults of the EKF ae shown in Figues The time fo filte convegence vaied acoss the simulation points and the eo of the initial state; howeve, the filte conveged within 1 min with an accuacy of 1 m in all simulations. The eos expessed in the RIC fame afte the filte is stabilized ae shown in Figues 19 and, and the eo nom is bounded at 3 m when using thee GPS satellites. These esults wee quite acceptable because the EKF filte fo a GEO conveges vey slowly due to the obit s chaacteistics. Fo example, the convegence time of the EKF filte fo a GEO is appoximately one o two hous unde the spase visibility of GPS satellites [15]. The convegence ate of the filte depends on the geometic location of the GPS satellites and the changing visibility of the GPS satellites, and thus, the convegence ate vaies even though the accuacies of the initial conditions ae not significantly diffeent. 1 9 GPS SV 3 GPS SV 8 7 Range Eo (km) Time (min) Figue 1. Test esult of the EKF simulation stated at point A GPS SV 3 GPS SV 5 Range Eo (km) Time (min) Figue 15. Test esult of EKF simulation stated at point B.

17 Sensos 15, GPS SV 3 GPS SV 8 Range Eo (km) Time (min) Figue 16. Test esult of the EKF simulation stated at point C. 35 GPS SV 3 GPS SV 3 Range Eo (km) Time (min) Figue 17. Test esult of the EKF simulation stated at point D fo GPS satellite position D GPS, D1 3 GPS, D 3 GPS, D3 3.5 Range Eo (km) Time (min) Figue 18. Test esult of the EKF simulations stated at point D fo vaious positions of the thid GPS satellite. The Y axis is zoomed fo convenience.

18 Sensos 15, Eo of stabilized EKF using GPS SV in RIC fame A B C D C (m) I (m) 6 R (m) Figue 19. The stabilized EKF eos in a RIC fame when using two GPS satellites. (R is adial, I is along-tack and C is coss-tack diection). Eo of stabilized EKF using 3 GPS SV in RIC fame A B C D C (m) I (m) 6 R (m) 5. Conclusions Figue. The stabilized EKF eos in a RIC fame when using thee GPS satellites. The main goal of the algoithm is to calculate the state of the GEO satellite using a single-epoch measuement unde the spase visibility of the GPS satellites, which is usually impossible, even when applying the classical SPS algoithm. The poposed algoithm can calculate the position, velocity and eceive s clock bias using only a single-epoch measuement of two o thee GPS satellites without data fom extenal souces. Theefoe, the calculated esult can be used as the initial state as soon as the measuements ae geneated by the eceive. The esulting maximum ange eo is less than km, and when using the esult as the initial state of the EKF, which uses a vey accuate dynamic model, the filte conveges to an eo of 1 m within 1 min in the wost case. The maximum eo nom of the EKF afte filte stabilization is bounded at 3 m when thee GPS satellites ae obsevable.

19 Sensos 15, Given this acceptable esult and the benefits of the algoithm, we expect that ou algoithm is sufficiently accuate, obust, efficient, and pactical fo detemining the initial obit of GEO satellites. Acknowledgments This eseach was suppoted by the National Space Laboatoy (NSL) pogam though the National Reseach Foundation of Koea funded by the Ministy of Education, Science and Technology (NRF ), contacted though the Institute of Advanced Aeospace Technology at Seoul National Univesity. Autho Contibutions G. K. and C. Kee conceived of and designed the eseach, G. K. and C. Kim pefomed the eseach, G. K. and C. Kee analyzed the data, and G. K. wote the pape. Conflicts of Inteest The authos declae no conflict of inteest. Refeences 1. Choi, J.; Choi, Y.-J.; Yim, H.-S.; Jo, J.H.; Han, W. Two-site optical obsevation and initial obit detemination fo geostationay eath obit satellites. J. Aston. Space Sci. 1, 7, Hwang, Y.; Lee, B.-S.; Kim, H.-Y.; Kim, H.; Kim, J. Obit detemination accuacy impovement fo geostationay satellite with single station antenna tacking data. Eti J. 8, 3, Moeau, M.C.; Axelad, P.; Gaison, J.L.; Long, A. GPS eceive achitectue and expected pefomance fo autonomous navigation in high eath obits. Navigation, 7, Padal, P.; Kuga, H.; de Moaes, R.V. Robustness assessment between sigma point and extended Kalman filte fo obit detemination. J. Aeosp. Eng. 11, 3, Vallado, D.A.; McClain, W.D. Fundamentals of Astodynamics and Applications, 3d ed.; Spinge, NY, USA, Zhou, N. Onboad Obit Detemination Using GPS Measuements fo Low Eath Obit Satellites. PhD Thesis, Queensland Univesity of Technology, Bisbane, Austlia,. 7. Vallado, D.A.; Cate, S.S. Accuate obit detemination fom shot-ac dense obsevational data. J. Astonaut. Sci. 1998, 6, Gewal, M.S.; Andews, A.P. Kalman Filteing: Theoy and Pactice Using Matlab, 3d ed.; Wiley: Hoboken, NJ, USA, Montenbuck, O.; Gill, E. Satellite Obits; Spinge: Amstedam, The Nethelands, Bolandi, H.; Laki, M.H.A.; Abedi, M.; Esmailzade, M. GPS based onboad obit detemination system poviding fault management featues fo a LEO satellite. J. Navig. 13, 66, Vette, J.R. Fifty yeas of obit detemination: Development of moden astodynamics methods. J. Hopkins Appl. Tech. D 7, 7, Hofmann-Wellenhof, B.; Lichtenegge, H.; Collins, J. GPS Theoy and Pactice, 5th ed.; Spinge-Velag: New Yok, NY, USA, 1.

20 Sensos 15, Chiaadia, A.P.M.; Kuga, H.K.; Pado, A.F.B.D. Onboad and eal-time atificial satellite obit detemination using GPS. Math. Pobl. Eng. 13, 13, 53516: :8. 1. Dong, L. IF GPS Signal Simulato Development and Veification. Maste s Thesis, Univesity of Calgay, AB, Canada, Mehlen, C.; Lauichesse, D. In eal-time GEO obit detemination using TOPSTAR 3 GPS eceive. In Poceedings of the ION GPS, Salt Lake City, UT, USA, 19 Septembe ; pp by the authos; licensee MDPI, Basel, Switzeland. This aticle is an open access aticle distibuted unde the tems and conditions of the Ceative Commons Attibution license (