Real-time kinematic positioning of LEO satellites using a single-frequency GPS receiver

Transcription

1 Real-time inematic positioning of LEO satellites using a single-fequency GPS eceive Pei Chen, Jian Zhang, and Xiucong Sun () School of Astonautics, Beihang Univesity, Beijing 1191, China Tel.: , sunxiucong@gmail.com Abstact Due to thei low cost and low powe consumption, single-fequency GPS eceives ae consideed suitable fo low-cost space applications such as small satellite missions. Recently, equiements have emeged fo eal-time accuate obit detemination at sub-mete level in ode to cay out onboad geocoding of high-esolution imagey, open-loop opeation of altimetes and adio occultation. This study poposes an impoved eal-time inematic positioning method fo LEO satellites using single-fequency eceives. The C/A code and L1 phase ae combined to eliminate ionospheic effects. The epoch-diffeenced caie phase measuements ae utilized to acquie eceive position changes which ae futhe used to smooth the absolute positions. A inematic Kalman filte is developed to implement inematic obit detemination. Actual flight data fom China s small satellite SJ-9A ae used to test the navigation pefomance. Results show that the poposed method outpefoms taditional inematic positioning method in tems of accuacy. A 3D position accuacy of.72 m and.79 m has been achieved using the pedicted potion of IGS ulta-apid poducts and boadcast ephemeides, espectively. Keywods: Real-time; Kinematic positioning; LEO; Single-fequency GPS eceive; Kinematic Kalman filte Intoduction The high accuacy, global coveage and thee-dimensional natue of the Global Positioning System (GPS) has led to a wide use of GPS eceives not only fo teestial applications but also fo space applications, such as obit detemination of low-eath-obiting (LEO) satellites. The GPS measuements suppot both gound-based high-pecision obit detemination and onboad 1

2 eal-time navigation tass (Zehentne and Maye-Gü 216; van den Ijssel et al. 215; Montenbuc et al. 213). In geneal, centimete-level accuacy is achieved using ionospheicfee caie phase obsevations fom dual-fequency eceives. With the impovement of measuement accuacy, single-fequency eceives have been poposed in ecent yeas fo mete o sub-mete level obit detemination (Wang et al. 215; Montenbuc et al. 212; Peng and Wu 212; Boc et al. 29; Hwang and Bon 25). The use of single-fequency eceives can save costs and educe enegy consumption. They ae pefeable fo small satellites o space missions equiing lowe obital accuacy. The ionospheic popagation delay is consideed a majo obstacle to single-fequency pecise obit detemination (POD). The GRAPHIC (GRoup And Phase Ionospheic Coection) method can be used to eliminate the fist-ode ionospheic effects (Yunc 1993), and sub-mete level position accuacies have been demonstated fo seveal satellites (Hwang et al. 211; Boc et al. 29). The GRAPHIC method uses the fact that the C/A code and L1 phase expeience the same ionospheic delay but with opposite sign. Unlie the ionospheic-fee P1/P2 pseudoange combination, the GRAPHIC obsevation epesents a heavily biased one-way ange which cannot be used fo single point positioning (Montenbuc 23). Actually, the GRAPHIC obsevation is simila to the ionospheic-fee L1/L2 caie phase combination but with lage obsevation noise due to the incopoation of C/A code measuements. This study investigates the use of the GRAPHIC pocessing methodology fo GPS singlefequency obit detemination of LEO satellites fo eal-time applications. The motivation oiginates fom an emeging equiement fo eal-time pecise obits with sub-mete accuacy in spacecaft opeations such as onboad geocoding of high-esolution imagey (Wemuth et al. 212), open-loop opeation of altimetes (Jayles et al. 21) and adio occultation (Montenbuc et al. 213). The single-fequency eceives ae compatible with onboad needs of these missions, povided that pecise GPS ephemeides ae available (Montenbuc et al. 212). A GRAPHIC-based dynamical filte fo eal-time single-fequency obit detemination was developed in Montenbuc and Ramos-Bosch (28). GPS measuements collected fom vaious LEO satellites including CHAMP, GRACE, TeaSAR-X, ICEsat, SAC-C, and MetOp wee used to test the algoithm. Position accuacies anging fom.48 m to.69 m (3D ms) wee achieved using boadcast ephemeides, and decimete level accuacies wee obtained using JPL eal-time ephemeides. The GRAPHIC filte algoithm was futhe adapted to be implemented 2

3 on the PROBA-2 micosatellite which caied a Phoenix-XNS single-fequency eceive fo ealtime navigation. An in-flight accuacy of 1.1 m (3D ms) was achieved using boadcast ephemeides, and the potential fo accuacy impovement to.7 m was also demonstated on gound with a efined filte tuning. The sub-mete level accuacies achieved fo eal-time single-fequency obit detemination mentioned above ae a esult of using dynamical filteing technique which educes the measuement noises and impoves obustness (Goldstein et al 21; Koes 26). In contast to GPS dynamic obit detemination, the inematic method estimates the satellite s position coodinates epoch-by-epoch and equies no a pioi nowledge of the spacecaft motion (Weinbach and Schön 213; Montenbuc 23; Bisnath and Langley 22). The inematic appoach can be applied to a wide ange of situations and is of paticula inteest fo maneuveing spacecaft and eenty vehicles due to its puely geometical natue. Moeove, the computational complexity is significantly educed compaed to the dynamical filteing technique. Howeve, the inematic appoach is sensitive to measuement outlies and bad viewing geomety. Thus its pactical use is esticted. The pesent study poposes an impoved inematic obit detemination method which utilizes epoch-diffeenced caie phase measuements to smooth GRAPHIC-deived positions fo ealtime single-fequency obit detemination of LEO satellites. The epoch-diffeencing of the caie phase eliminates the ambiguity tem as well as the majo pat of ionospheic effects and can be used to obtain highly accuate displacement infomation between two adjacent epochs. The displacement will constain the inematic positions estimated fom GRAPHIC obsevations, simila to the obital motion infomation in dynamic appoach. In addition, code outlies and caie phase cycle slips can be detected using epoch-diffeenced obsevations. Thus the obustness is enhanced compaed to the taditional inematic method. In the following chapte, the mathematical models of GPS obsevations and linea combinations fo single-fequency obit detemination ae descibed. The eal-time GPS obit and cloc poducts ae intoduced next, and the obsevation eos due to inaccuacies of the ealtime poducts ae analyzed. Aftewads, the detailed algoithm of the impoved inematic obit detemination method is pesented. The poposed method is finally tested with actual flight data fom the single-fequency GPS eceive onboad the China s SJ-9A (ShiJian-9A) low-cost small 3

4 satellite (Zhao et al. 213), whee both the boadcast ephemeides and the pedicted potion of the IGS ulta-apid ephemeides ae tested. GPS single-fequency obsevation model In the case of single-fequency GPS eceives, only the C/A code and L1 caie phase ae available. The simplified obsevation equations (in metes) ae given as follows (Leic et al. 215) whee Ct and Lt s s s C t t c t t t t I t t (1) C s s s the eceive-satellite geometic ange, L t t c t t t t I t A t (2) L L denote the C/A code and L1 phase at epoch t, s s s x t x t x t and t t ae eceive antenna phase cente s s s s position and cloc offset at the signal eception time, x t and t t antenna phase cente position and cloc offset at the signal tansmitting time, I t tavelling time, c is the vacuum speed of light, the non-intege ambiguity tem (unit m), and ae satellite s is the signal is the ionospheic popagation delay, t and t C L A L ae the andom themal noises. The elativistic effects, the phase wind-up, as well as the hadwae delays ae not discussed hee. The teatment of these eos could be found in Kouba (29) and Stele et al (215). The GRAPHIC combination eliminates the dominant eo souce in single-fequency GPS measuements, i.e., ionospheic popagation delay, and is defined as is 1 G t C t L t 2 t c t t t t A t s s s G (3) whee A epesents 2 L deviation of t is G A and t epesents the GRAPHIC obsevation noise. The standad G 4

5 G C L C (4) 2 2 whee C and L ae standad deviations of code and phase noises. The code noise is educed by a facto of 2 in the GRAPHIC obsevation. Howeve, a bias tem which equals half of the caie phase ambiguity is involved and must be solved along with eceive position and cloc offset. It should be noted that thee is a linea dependency between the eceive cloc offset and the GRAPHIC ambiguity. That is, only the change of cloc offset values with espect to the efeence epoch can be estimated. To addess this singulaity poblem, a pioi value of cloc offset at the efeence epoch will be used and fixed in the estimation pocess. Fo continuous GPS tacing, epoch-diffeenced obsevations can be fomed between two obsevations of the same type at two adjacent epochs as follows tt 1 2 tt 1 2 C C t C t 2 1 c t t I s s t1 t2 t1 t2 t1 t2 t1 t2 t1 t2 C L L t L t 2 1 c t t I s s t1 t2 t1 t2 t1 t2 t1 t2 t1 t2 L (5) (6) tt t t The ambiguity whee the epoch-diffeencing opeato is defined as tem in (2) cancel out if no cycle slips occu. The ionospheic eos ae highly coelated and the lagest pat of ionospheic path delay can also be eliminated. In the case of a shot epochdiffeencing inteval of 1 o 2 seconds, which coesponds to a sepaation distance of about 1 ilometes fo LEO satellites, the esidual tem tt I is on the ode of centimetes (Chen et al ; Koes 26). When the epoch-diffeencing inteval inceases to 5 o 1 seconds, the ionospheic path delay esidual could each decimete level. This will be discussed in the esults section. Let I denote the standad deviation of tt I. 1 2 Define C t1 t I 2 t1 t and 2 C The standad deviations of C and L espectively ae L t1 t I 2 t1 t. 2 L 2 (7) 2 2 C I C 5

6 2 (8) 2 2 L I L The noise level of tt L 1 2 is much lowe than that of C. tt Lage code outlies o significant 1 2 caie phase cycle slips can be easily detected fom the following geomety-fee diffeence between epoch-diffeenced code and phase obsevations C L (9) t1 t2 t1 t2 C L whee only the noise tems emain afte the diffeencing opeation. The citeion fo the detection is given as follows C L 3 (1) 2 2 t1 t2 t1 t2 C L with a tail pobability of.3%. In the obit detemination section, the GRAPHIC model (3) will be used to estimate the eceive s absolute position and cloc offset, wheeas the epoch-diffeenced caie phase model (6) will be used to estimate the vaiations of eceive position and cloc offset between two adjacent epochs. Real-time obit and cloc poducts With ionospheic effects eliminated though the GRAPHIC combination, the GPS obit and cloc infomation becomes the ey facto influencing the accuacy of eal-time single-fequency navigation. In geneal, thee ae two sets of GPS ephemeides available fo obit detemination, the boadcast ephemeides and the pecise ephemeides. The boadcast ephemeides ae geneated and contolled by the GPS opeational contol segment (OCS) and tansmitted in the navigation message (van Dieendonc et al 1978). The pecise ephemeides ae povided by the Intenational GNSS Sevice (IGS) and its analysis centes, including the final, the apid, and the ulta-apid poducts (Dow et al. 29). In ode to satisfy eal-time uses with moe pecise poducts, the IGS now povides eal-time sevice (RTS) poducts, which ae steamed ove the Intenet using Netwoed Tanspot of RTCM via Intenet Potocol (Hadas and Bosy 215; Elsobeiey and Al-Habi 216). 6

7 Among these obit and cloc poducts, only the boadcast ephemeides, the pedicted potion of the IGS ulta-apid poducts, and the IGS RTS ae available fo eal-time uses. The RTS poducts ae cuently only steamed ove the Intenet. Continuous uploading of RTS poducts to an obiting LEO satellite equies a netwo of globally distibuted gound stations and bings additional budens to satellite communication and mission opeation. In contast, the ulta-apid poducts ae updated fou times pe day, at 3:, 9:, 15:, and 21: UTC. The obseved obits have an initial latency of 3 hous at the elease time. Fom elease to uploading, an additional time anging fom a few minutes to one obit peiod (~1.5 hous) will be equied, which depends on the elative position between the LEO use satellite and the gound station at the elease time. Thus a scheduled uploading with an aveage inteval of 6 hous would be sufficient fo eal-time obit detemination. Within this study, only the boadcast ephemeides and the pedicted potion of the IGS ulta-apid poducts ae used fo eal-time navigation. The inaccuacy of eal-time ephemeides will intoduce navigation eos. The Signal-In- Space Range Eo (SISRE) descibes a global oot-mean-squae (ms) value of ange eos due to GPS obit and cloc eos (Waen and Raquet 23; Montenbuc et al 25). In ode to investigate the time-vaying chaacteistics of GPS obit and cloc eos, this study intoduces the concept of Equivalent Range Eo (ERE) which descibes location dependent, instantaneous pojected eos. The ERE is computed by ERE s s s s whee x RT t and trt t s s s s t x t x t x t xrt t s s s s c trt t t t s s s s povided by eal-time ephemeides, x t and t t (11) ae satellite antenna phase cente and cloc coection ae tue values. It is noted that the IGS poducts efe to satellite s cente of mass and need to be coected with absolute phase cente offsets. Define x x x and t s t s t s. A Taylo seies expansion of (11) s s s RT RT tuncated at the fist-ode tem is given as follows whee s s s EREt e x c t (12) s e is the line-of-sight vecto fom the eceive to the GPS satellite. RT 7

8 The time-vaying chaacteistics of EREs fo the eal-time obit and cloc poducts ae illustated in Figue 1. A subset of GPS data fom the SJ-9A satellite have been pocessed and the GPS satellite PRN 12 (Bloc IIR-M) is investigated. The IGS final poducts ae used as efeence. The absolute phase cente offsets have been coected fo IGS obits. In ode to account fo time scale biases fo diffeent ephemeis poducts, an ensemble cloc diffeence is computed at each epoch fom the aveage cloc diffeences of all the satellites, and the individual cloc offset diffeences ae coected fo this ensemble aveage (Montenbuc et al. 215). The boadcast ephemeides contain lage obit and cloc coection eos. The coesponding ERE cuve (blue solid line) can be decomposed by a difting bias and andom noise. Howeve, the pedicted potion of IGS ulta-apid poducts ae mainly govened by cloc coection eos. The coesponding ERE cuve (ed dashed line) esembles a constant bias contaminated by andom noise. Simila phenomena have been found fo the othe GPS satellites, although not pesented hee. The constant pat of ERE will be absobed by the ambiguity tem and the eceive cloc offset. In ode to incopoate the ERE s andom noise pat, the GRAPHIC ambiguity is modeled as the following stochastic pocess A t A t v (13) 1 whee v is assumed to be zeo-mean Gaussian noise, and its standad deviation can be pedetemined fom statistical analysis. ERE (Boadcast), m Boadcast IGS ulta-apid -1. Octobe 25th, 212 7:2: to 8:5: (GPST) , 1,5 2, 2,5 Elapsed time, s ERE (IGS ulta-apid), m Fig. 1 Time-vaying EREs of boadcast ephemeides (blue solid) and the pedicted potion of IGS ultaapid poducts (ed dashed) of PRN 12. The IGS final poducts ae used as efeence 8

9 The effects of obit and cloc coection eos on epoch-diffeenced obsevations can be evaluated by the epoch-diffeenced ERE tt 12 t t ERE ERE ERE 2 1 s s s s s t 1t e 2 xrt t1 e t1 t1t x 2 RT c t1t t 2 (14) whee e t12 t s is the vaiation of the line-of-sight vecto. In the case of shot time inteval of 1 o s 1 m 4 2 seconds, t t e The time diffeences of 26 m s x RT and c s t ae usually quite small due to the smoothness of the pedicted GPS obits and clocs, which ae on the odes of millimetes. Thus the standad deviation of ERE tt 12 can be estimated to be on the ode of millimetes. This has been validated by the actual values of time diffeences of the ERE data which wee used fo plotting Figue 1. The ms values of ERE tt 12 fo the boadcast ephemeides and the IGS ulta-apid (pedicted) poduct ae.934 mm and.923 mm, espectively. The ms of The magnitude of ERE tt can be used as the standad deviation of the ambiguity noise. 12 ERE tt is much smalle than the epoch-diffeenced caie phase noise level 12 and will be neglected in the estimation of vaiations of eceive position and cloc offset. Impoved inematic eal-time navigation In taditional inematic navigation, the epoch-wise eceive positions, the epoch-wise eceive cloc offsets, as well as the constant ambiguity tems which ae associated with continuous, uninteupted acs of caie phase tacing ae estimated by a ecusive least-squaes filte in a sequential manne. Without dynamical constaints o data pepocessing, the estimated positions will be sensitive to obsevation noise, outlies and cycle slips, especially fo single-fequency navigation whee code measuements ae used. In ode to educe the obsevation noise and enhance the obustness at the pesence of code outlies o caie phase cycle slips, we intoduce epoch-diffeenced obsevations fo inematic navigation and develop a inematic Kalman filte to implement the state estimation. Figue 2 illustates the pocedue of the poposed filte fo eal-time single-fequency inematic obit detemination, whee the + supescipt denotes a posteioi estimation and the supescipt 9

10 denotes a pioi estimation. The filte in one cycle consists of a time-update stage and a measuement-update stage. At the time-update stage, a chec of visible GPS satellite change is fist caied out to addess the cases whee new satellites ae available and/o old satellites disappea. Next, the epoch-diffeenced obsevations ae pocessed to detect outlies and cycle slips, see (1). Afte emoval of eoneous measuements, the epoch-diffeenced caie phases ae then used to estimate the vaiations of eceive position and cloc offset fom pevious epoch to cuent epoch, and the ambiguities ae popagated accoding to (13). At the measuementupdate stage, the obseved-minus-computed GRAPHIC measuements ae used to estimate the absolute eceive position, cloc offset, and ambiguities. ˆ A 1 xˆ ˆ, 1 t, 1 t1 t t1 t C L Reodeing if visible satellites change Time-update stage A 1 Estimation of vaiations of position and cloc offset Outlie and/o cycle slip detection & emoval Popagation of ambiguities ˆ A xˆ ˆ, t, Real-time ephemeides Measuement-update stage Computation of GRAPHIC obsevation Linea coection GRAPHIC measuements ˆ A xˆ ˆ, t, Fig. 2 Bloc diagam of one cycle in the inematic Kalman filte Estimation of vaiations of eceive position and cloc offset The epoch-diffeenced caie phase measuements contain the time-vaying infomation of the igoous pseudoange, which efes to topocentic satellite distance plus cloc offset tems, fom which the vaiations of eceive position and cloc offset can be estimated with at least fou continuous obseved GPS satellites in the absence of cycle slips. Conside a LEO satellite continuously tacs n ( n 4 ) GPS satellites fom epoch t 1 to t. The epoch-diffeenced caie phase obsevations coected with GPS satellite cloc coections can be cast into the following vectoial model 1

11 y L c t 1 1 t1t t1t RT L c t 2 2 t1t t1t RT L c t n n t1t t1t RT 1 1 xˆ ˆ, 1 t 1t x xrt, x, 1 x RT, 1 c 2 2 xˆ ˆ, 1 t 1t x xrt, x, 1 x RT, 1 c t ε 1 n n xˆ, 1 t 1t x x ˆ c RT, x, 1 xrt, 1 t t L (15) whee x ˆ, 1 eception time, t denote the a posteioi estimated eceive position at i x RT, 1 i and t 1 efeing to signal x RT, i 1, 2,..., n denote the ith GPS satellite s positions at efeing to signal tansmitting time and ae povided in the eal-time ephemeides, and i t i 1, 2,..., n is the eal-time cloc coection of the ith GPS satellite. RT ε L t 1 and is the obsevation noise vecto and its covaiance matix is denoted by R, I, whee matix. L n n L I n n is an identity The unnowns in (15) include thee inematic eceive position changes 1 t t x and one eceive cloc offset change t. 1t t The estimation eo of x ˆ, 1, denoted as x,, 1 will affect the equation by the elation e x which is easily obtained by lineaization of s, t1 t, 1 (15) aound x ˆ, 1. This eo tem is many times smalle than the obsevation noise. As mentioned ealie, the epoch-diffeenced ERE is on the ode of millimetes. Thus the inaccuacies of eal-time ephemeides can also be neglected. The nonlinea equation (15) can be solved using an iteated least-squaes estimato. The iteation will involve the patial deivatives of y with espect to the unnowns, which ae given as follows 11

12 H, e e e 1 T, 2 T, n T,,,, c c c n 4 (16) The covaiance matix of the estimates is T 1 1 P H R H (17),, L,, P, will be used fo the measuement update of eceive position and cloc offset. Absolute position, cloc offset and ambiguity estimation A pioi estimates of the eceive position, cloc offset and ambiguities at epoch obtained fom the following state equations, t can be xˆ xˆ x ˆ (18),, 1 t1 t tˆ tˆ tˆ (19),, 1 t1 t ˆ A A ˆ (2) 1 whee 1 t t x ˆ and tˆ ae the estimated vaiations of position and cloc offset fom t1 t t 1 to. t The associated covaiance of these a pioi estimates ae given by P P, 4n P 1 n 4 QA (21) whee Q A is the covaiance of the ambiguity andom-wal noise. In the case of newly obseved satellites, the caie-minus-code obsevations can be used to povide a pioi estimates fo the newly added ambiguities. At the measuement-update stage, the GRAPHIC obsevations ae fist lineaized aound at these a pioi estimates 12

13 z G ˆ ˆ x, xrt, ct, ct RT G ˆ ˆ x, xrt, ct, ct RT Aˆ n n n G ˆ ˆ x, xrt, ct, ct RT x, H, G t A ε, (22) whee z compises the obseved-minus-computed GRAPHIC obsevations, x,, t, and A epesent small eos, and ε G epesent the GRAPHIC noise vecto and its covaiance matix is denoted by RG, I n n G. Define H H,, I n n. The estimates of the eceive position, cloc offset as well as ambiguities and the associated covaiance can be updated as follows xˆ xˆ ˆ ˆ A A,, tˆ ˆ, t, Kz (23) whee K T, P I K H P I K H K R K (24) is the filte gain and is given by T G T 1, K P H H P H R (25) T G So fa, all of the unnown paametes have been estimated sequentially. The inematic Kalman filte offes a bette way fo eal-time single-fequency navigation than taditional inematic method though using epoch-diffeenced caie phases to povide pseudo dynamic infomation. It should be pointed out that the inematic Kalman filte cannot output velocity estimation and cannot deal with data outages, since no eal-wold dynamic models ae used. A summay of similaities and diffeences between the poposed and taditional techniques fo single-fequency inematic obit detemination is povided in Table 1. 13

14 Table 1 Compaison between the poposed and taditional inematic single-fequency obit detemination methods Item Taditional method Poposed method GPS measuement model Ionospheic-fee L1 code-caie combination (GRAPHIC) Ionospheic-fee L1 code-caie combination (GRAPHIC) Epoch-diffeenced caie phase Boadcast ephemeides Constant ambiguity Boadcast ephemeides/igs ulta-apid (pedicted) poducts Stochastic ambiguity Obital dynamic model Not incopoated Not incopoated Ionospheic effects Not consideed Not consideed fo GRAPHIC Consideed fo epochdiffeenced caie phase Refeence fame WGS84 WGS84/ITRF Estimation technique Recusive least-squaes filte Kinematic Kalman filte Infomation fo outlie/cycle slip detection Measuement esidual Geomety-fee diffeence between epoch-diffeenced code and phase SJ-9A flight data pocessing esults The inematic Kalman filte descibed above is implemented in an obit detemination softwae (MATLAB language) developed by the Spacecaft Simulation Technology (SST) laboatoy at Beihang Univesity. The eal GPS data collected fom the SJ-9A satellite have been used to study the navigation pefomance in an offline envionment. Mission and data set desciption The SJ-9 mission objective is to test China s new space technologies such as dual-spacecaft fomation flying, high-pefomance small camea, and electonic populsion system. The SJ- 9A/B satellites wee launched on Octobe 14, 212 into a sun synchonous obit. SJ-9A is based on the matue CAST-2 platfom and the satellite mass is about 79 g. SJ-9B is based on the 14

15 new CAST-1 platfom and the mass is about 26 g. The pictues of the two satellites ae pesented in Zhao et al. (213). Each satellite is equipped with two 12-channel single-fequency GPS eceives fo absolute eal-time navigation and opeation contol. The eceives wee manufactued by the China Academy of Space Technology (CAST) and povide C/A code and L1 caie phase measuements. The eceives on the same satellite ae connected to a common antenna to pefom zeo-baseline test. In addition, caie phase diffeential GPS measuements fom the two satellites ae used to povide high accuacy inte-satellite baseline solutions (Chen et al 215). Howeve, the absolute navigation pefomance is the only concen in this study. A 7-hou GPS data set fom the main eceive onboad the SJ-9A satellite coveing the time fom 6:: (GPS time) to 13:: (GPS time) on Octobe 25, 212 has been used fo the test. The GPS data fom SJ-9B ae not used because its antenna position is not povided and pecise efeence obits ae difficult to obtain. The obital height of SJ-9A on this day was about 65 m. This indicates that the satellite expeiences elatively smalle ionospheic path delays than those at lowe obits, such as CHAMP and GRACE. The aw GPS measuements ae stoed in a binay file at a sampling inteval of 1 s. Zeo-baseline test has shown a code noise of 6 cm and a caie phase noise of 1 mm. This implies a noise level of 3 cm fo GRAPHIC obsevations. It is noted that multipath o goup delay vaiations which cannot be detected fom zeo-baseline test may degade the accuacy of measuements. In addition to the mandatoy GPS measuements, the spacecaft attitude data as well as the antenna position ae also povided fo tansfomation between antenna phase cente and cente of mass. The eal-time ephemeides including boadcast ephemeides and the IGS ulta-apid poducts ae downloaded via anonymous ftp fom ftp://gane.ucsd.edu/. The IGS final poducts ae also downloaded in ode to obtain pecise efeence obits fo accuacy compaison. An elevation cut-off angle of 5 has been applied to eject satellites in poo geomety. The numbe of taced GPS satellites above the elevation mas and the position dilution of pecision (PDOP) values ae shown in Figue 3. The numbe of available GPS satellites anges fom 6 to 12, and the PDOP value anges fom 1.2 to 3.4. The PDOP value always inceases with dop in numbe of GPS satellites. 15

16 Numbe of satellites PDOP value Octobe 25, 212 Time, hou Fig. 3 Numbe of taced GPS satellites above 5 elevation mas and PDOP values Pecise efeence obits and SPP solutions Befoe the pesentation of inematic obit detemination esults, dynamic obit detemination solutions using a batch least-squaes estimato which pocesses GRAPHIC obsevations and IGS final poducts ae povided as pecise efeence fo accuacy evaluation. In addition, single point position (SPP) fixes using C/A code and boadcast ephemeides without ionospheic model coections ae also pesented fo compaison. The obit dynamic models employed in the batch estimato ae summaized in Table 2. The EGM28 gavity model tuncated at degee and ode 1 is used to obtain the acceleations due to the eath s static gavity field. The solid eath tides and ocean tides ae modeled to account fo tidal effects. The thid-body attactions ae computed using analytical expansions of sola and luna coodinates. The maximum aea-to-mass atio of SJ-9A is about.16 m 2 /g. The acceleations due to atmospheic dag and sola adiation pessue ae less than m/s 2. Thus the non-gavitational acceleations ae not modeled. Within this study, piecewise constant empiical acceleations with a 15 min time inteval and a constaint of m/s 2 ae used to compensate fo unmodeled foces. The GPS measuements ae esampled at intevals of 1 s. 16

17 Although not depicted hee, obit ovelapping analysis indicates a 3D position accuacy of.2 m fo the batch obit detemination solutions. Table 2 Summay of dynamic models used in batch obit detemination of SJ-9A Item SJ-9A pecise obit detemination Gavitational foces EGM28 model (1 1) Solid eath and ocean tides Sola and luna gavitational attactions (analytical fomulas) Non-gavitational foces No atmospheic dag and sola adiation pessue model Piecewise constant empiical acceleation at 15 min inteval, constant: m/s 2 The single point positioning algoithm maes use of only C/A code measuements. No empiical models ae applied to eliminate the ionospheic effects. The position diffeences between SPP solutions and the pecise efeence obits ae shown in Figue 4. The oot-meansquae (ms) values of the adial, along-tac, coss-tac, and 3D position eos ae 2.81 m,.95 m,.87 m, and 3.1 m, espectively. Instant 3-dimensional position eos could each 9 m at some epochs. The lagest pat of SPP position eos ae attibuted to ionospheic delays. 17

18 Radial, m Along-tac, m Coss-tac, m Octobe 25, 212 Time, hou Fig. 4 Accuacy of SPP positions elative to pecise efeence obit in adial, along-tac, and coss-tac diections. Ionospheic effects ae not addessed in SPP Kinematic obit detemination esults Both the taditional and the impoved inematic obit detemination methods have been tested. The boadcast ephemeides and the IGS ulta-apid poducts ae used fo eal-time navigation pefomance evaluation, wheeas the IGS final poducts ae only used fo accuacy compaison. The ey paametes of the inematic Kalman filte ae set as follows. The sampling ate is 1 Hz. The standad deviation of GRAPHIC obsevation noise is set to 3 cm. No elevation dependent weighting of GPS data is applied. The standad deviation of the epoch-diffeenced ionospheic esidual is set to 2 cm. The standad deviations of the ambiguity noise when using boadcast ephemeides and IGS ulta-apid poducts ae both set to 1 mm. A constant ambiguity model is used when using IGS final poducts. The initial position estimates ae obtained fom the SPP solutions. The position accuacies of inematic obit detemination using IGS final poducts, IGS ultaapid poducts, and boadcast ephemeides ae illustated in Figues 5-7, espectively. The pecise obit obtained fom batch dynamic obit detemination has been used as efeence. It is seen that the position solutions obtained fom the poposed method ae always smoothe than those obtained fom the taditional method, no matte what ind of ephemeis poducts ae used. 18

19 Howeve, the convegence speed can be baely impoved. The eason is that the inematic Kalman filte only constains the position vaiations but cannot povide velocity o acceleation infomation to acceleate the convegence pocess. Radial, m Along-tac, m Coss-tac, m Taditional Impoved Octobe 25, 212 Time, hou Fig. 5 Time-vaying position eos of taditional and impoved inematic obit detemination using IGS final poducts Radial, m Along-tac, m Coss-tac, m Octobe 25, 212 Time, hou Taditional Impoved Fig. 6 Time-vaying position eos of taditional and impoved inematic obit detemination using IGS ulta-apid poducts 19

20 Radial, m Along-tac, m Coss-tac, m Octobe 25, 212 Time, hou Taditional Impoved Fig. 7 Time-vaying position eos of taditional and impoved inematic obit detemination using boadcast ephemeides The statistics of position eos afte convegence ae summaized in Table 3. Fist of all, compaed with SPP solutions, all the 6 sets of inematic solutions have bette accuacies, since the ionospheic effects ae eliminated and the C/A code noise is educed by a facto of 2 in the GRAPHIC obsevations. Second, the inematic Kalman filte effectively educes the position eos. As stated ealie, the epoch-diffeenced caie phase obsevations ae used in the filte to estimate position vaiations. The estimation eos of position vaiations ae shown in Figue 8. The ms value of estimation eos is.1 m. The estimated position vaiations constain the absolute position estimates and povide moe infomation fo obit detemination. The inematic Kalman filte behaves lie a low-pass filte and significantly suppesses the noise. As epoted fom Table 3, compaed with taditional method, the inematic Kalman filte educes the 3D position eos by 38.6%, 31.%, and 49.% when using IGS final poducts, IGS ulta-apid poducts, and boadcast ephemeides, espectively. Howeve, the moe slowly vaying, timecoelated eos ae baely suppessed by the poposed method. The eason lies in that systematic measuement eos, such as multi-path effects, phase cente vaiations, and higheode ionospheic effects, cannot be completely emoved in the epoch-diffeenced obsevations. The accumulation of esidual eos would lead to slowly vaying estimation eos. Finally, as 2

21 expected, the inematic obit detemination using IGS final poducts achieves the best accuacy, the IGS ulta-apid poducts, and the boadcast ephemeides wost. Table 3 Statistics of position eos of inematic obit detemination esults Ephemeides Radial (m) Along-tac Coss-tac 3D ms 3D max Pecentage (m) (m) (m) (m) of 3D eo > 1 m, % IGS final poducts IGS ultaapid poducts Boadcast ephemeides Taditional.33 ± ± ± Impoved.35 ± ± ± Taditional.7 ± ± ± Impoved.22 ± ± ± Taditional.53 ± ± ± Impoved.63 ± ± ± RMS:.1 m Estimation eo, m Octobe 25, 212 Time, hou Fig. 8 Estimation eos of position vaiations The ms values of 3D position eos obtained fom the inematic Kalman filte ae all below 1 m, and the maximum 3D position eos ae all below 1.5 m. The pecentages of points having a 3D position eo above 1 m ae.28%, 7.6%, and 16.1%. These indicatos imply the possibility of the impoved inematic obit detemination method fo sub-mete accuacy ealtime spacecaft navigation. 21

22 In ode to investigate the effects of sampling intevals on the pefomance of the inematic Kalman filte, thee diffeent sampling intevals, i.e., 2 s, 5 s, and 1 s, have also been tested. It is noted that the standad deviation of epoch-diffeenced ionospheic esidual vaies with the sampling inteval. Within this study, the coesponding standad deviations ae set to 4 cm, 1 cm, and 2 cm, espectively. The statistics of position eos ae listed in Table 4. With inceasing sampling inteval, the ionospheic esiduals emaining in epoch-diffeenced obsevations will get lage. Thus the estimation accuacy of position vaiations deceases. Accodingly, the absolute position accuacy deceases. As seen fom Table 4, the 3D ms eos using a sampling inteval of 2 s ae slightly lage than those using a 1 s inteval. When the sampling inteval is inceased to 1 s which coesponds to about 6 m sepaation distance fo LEO satellites, the advantage of the poposed method is no longe notable. Theefoe, fo LEO satellites and vehicles having lage velocity, the poposed method is only valid when high sampling ates (>.2 Hz) ae used. Table 4 Statistics of position eos with diffeent sampling intevals Sampling inteval, s Radial (m) Along-tac Coss-tac 3D ms 3D max Pecentage (m) (m) (m) (m) of 3D eo > 1 m, % IGS final poducts 2.24 ± ± ± ± ± ± ± ± ± IGS ultaapid poducts Boadcast ephemeides 2.9 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Conclusions With the impovement of eliability and accuacy, single-fequency GPS eceives have been poposed in ecent yeas fo onboad satellite obit detemination. Although the dynamical filteing technique dominates eal-time single-fequency navigation, the inematic method 22

23 equies lowe computational complexity and can be applied fo maneuveing spacecaft. In this study, the epoch-diffeenced caie phase obsevations ae utilized to impove inematic obit detemination accuacy. Based on the expeimental esults obtained using actual flight data fom SJ-9A, it can be concluded that the poposed method effectively educes position eos. Specifically, lage andom estimation eos ae emoved, and most of the position eos ae lowe than 1 m. With a sampling inteval of 1 s, the 3D ms position eos ae.72 m and.79 m fo IGS ulta-apid poducts and boadcast ephemeides, espectively. These accuacies ae compaable to those using dynamic filteing techniques. The single-fequency GPS eceives have low cost and simple achitectue. The use of singlefequency eceives will offe advantages of notable savings in mass, volume, and powe fo satellite missions and guaantee mete o sub-mete level navigation accuacy at the same time. Acnowledgements This eseach was suppoted by the National Natual Science Foundation of China though coopeative ageement No and has been funded in pat by Ministy of Science and Technology of China though coopeative ageement No. 214CB Refeences Bisnath SB, Langley RB (22) High-pecision, inematic positioning with a single GPS eceive. Navigation 49(3): doi:1.12/j tb265.x Boc H, Jäggi A, Dach R, Schae S, Beutle G (29) GPS single-fequency obit detemination fo low eath obiting satellites. Adv Space Res 43(5): doi:1.116/j.as Chen P, Shu L, Ding R, Han C (215) Kinematic single-fequency elative positioning fo LEO fomation flying mission. GPS Solut 19(4): doi:1.17/s Dow JM, Neilan RE, Rizos C (29) The Intenational GNSS Sevice in a changing landscape of Global Navigation Satellite Systems. J Geod 83(3): doi:1.17/s Elsobeiey M, Al-Habi S (216) Pefomance of eal-time Pecise Point Positioning using IGS eal-time sevice. GPS Solut 2(3): doi:1.17/s z 23

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27 Autho Biogaphies Pei Chen eceived his Ph.D. degee in aeospace engineeing fom Beihang Univesity, Beijing, in 28. He is cuently an associate pofesso at the school of astonautics, Beihang Univesity. His cuent eseach activities compise spacecaft navigation, GNSS application, and astodynamics and simulation. Jian Zhang eceived his B.S. degees in aeospace engineeing fom Beihang Univesity, Beijing, in 215. He is a maste student at the school of astonautics, Beihang Univesity. The focus of his cuent eseach lies in GNSS navigation. Xiucong Sun eceived his B.S. degee in aeospace engineeing fom Beihang Univesity, Beijing, in 21. He is a Ph.D. student at the school of astonautics, Beihang Univesity. The focus of his cuent eseach mainly lies in GNSS navigation, spacecaft obit and attitude detemination. 27