Centimeter-Level Precise Orbit Determination for the Luojia-1A Satellite Using Beidou Observations

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Article Centimeter-Level Precise Orbit Determination for the Luojia-1A Satellite Using BeiDou Observations

Lei Wang 1,2 , Beizhen Xu 1,* , Wenju Fu 1, Ruizhi Chen 1,2 , Tao Li 1 , Yi Han 1 and Haitao Zhou 1

1 State Key Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University, Wuhan 430079, China; [email protected] (L.W.); [email protected] (W.F.); [email protected] (R.C.); [email protected] (T.L.); [email protected] (Y.H.); [email protected] (H.Z.) 2 Collaborative Innovation Center for Geospatial Technology, Wuhan 430079, China * Correspondence: [email protected]; Tel.: +86-15527155902

Received: 1 June 2020; Accepted: 21 June 2020; Published: 26 June 2020

Abstract: Luojia-1A is a scientific experimental satellite operated by Wuhan University, which is the first low earth orbiter (LEO) navigation signal augmentation experimental satellite. The precise orbit is the prerequisite of augmenting existing Global Navigation Satellite System (GNSS) performance and improves users’ positioning accuracy. Meanwhile, LEO precise orbit determination (POD) with BeiDou-2 observations is particularly challenging since it only provides regional service. In this study, we investigated the method of precise orbit determination (POD) for Luojia-1A satellite with the onboard BeiDou observation to establish the high-precision spatial datum for the LEO navigation augmentation (LEO-NA) system. The multipath characteristic of the BeiDou System (BDS) observations from Luojia-1A satellite is analyzed, and the elevation-dependent BeiDou code bias is estimated with the LEO onboard observations. A weight reduction strategy is adopted to mitigate the negative effect of poor BeiDou-2 geostationary earth orbit (GEO) satellites orbit quality, and the Luojia-1A orbit precision can be improved from 6.3 cm to 2.3 cm with the GEO weighting strategy. The precision improvement of the radial direction, along-track, and out-of-plane directions are 53.47%, 47.29%, and 76.2%, respectively. Besides, tuning the pseudo-stochastic parameters is also beneficial for improving orbit precision. The experiment results indicate that about 2 cm overlapping orbit accuracy are achievable with BeiDou observations from Luojia-1A satellite if proper data processing strategies are applied.

Keywords: Luojia-1A satellite; BeiDou observations; precise orbit determination; reduce-dynamic; navigation augmentation

1. Introduction Low-Earth-Obiter (LEO) satellites have been used widely for space applications such as gravity ﬁeld measurement, high-resolution images, synthetic-aperture laser radar, and Global Navigation Satellite System (GNSS) radio occultation [1–3]. Apart from these traditional scientiﬁc missions, LEO satellites have also risen in the communication as well as navigation mission in recent years [4]. Recently, the LEO navigation augmentation (LEO-NA) technology has been considered as a promising technique to reduce the slow convergence issue in GNSS precise positioning [5,6]. The LEO satellite navigation augmentation technology includes the signal-based augmentation and information based augmentation [7]. The LEO satellite navigation signal augmentation method is achieved by generating and broadcasting the ranging signals to users, whereas the information augmentation method is achieved by transmitting the GNSS corrections information using the LEO

Remote Sens. 2020, 12, 2063; doi:10.3390/rs12122063 www.mdpi.com/journal/remotesensing Remote Sens. 2020, 12, 2063 2 of 17 satellite communication channels. Some research has demonstrated that the LEO satellite signal augmentation method can dramatically reduce the convergence time of Precise Point Positioning (PPP) [8] as well as long-baseline real-time kinematic (RTK) positioning by taking the advantages of the rapid geometric change of LEO satellites [9,10]. Moreover, it also can improve the usability and reliability of the present Global Navigation Satellite System (GNSS) and supporting the Positioning Navigation Timing (PNT) systems as well as the space-based real-time Positioning, Navigation, Timing, Remote-sensing, and Communication (PNTRC) information service systems [11,12]. Luojia-1A satellite is operated by Wuhan University, which was launched in June 2018 to carry out the LEO-based navigation signal augmentation experiment. The experiment was successful, and its initial performance has been assessed from a different perspective [5,13]. As for the LEO navigation signal augmentation system, it is critical to maintaining the precise orbit and time systems as well as operating the precise and stable ranging signals to users. Therefore, LEO precise orbit determination (POD) is of great importance to the LEO navigation signal augmentation system. Presently, the reduced-dynamic POD method using GPS observations has been extensively studied and widely used in many LEO satellites. Since demonstrated centimeter-level orbit precision of the TOPEX/POSEIDON satellite with onboard GPS observations [14], various LEO missions have achieved high precision with the GPS-based POD, such as CHAMP, GRACE, GOCE, and SWARM [15–18]. The BeiDou system consists of geostationary earth orbit (GEO) satellites, inclined geostationary (IGSO) satellites, and medium earth orbit (MEO) satellites, which is going to provide the global navigation services in 2020 [19,20]. With the emerging BeiDou system, there is only a few LEO satellites carried BeiDou receiver for orbit determination. The LINGQIAO satellite, a mobile communication experimental satellite in China, equipped with a Beidou system (BDS) receiver, demonstrated the initial performance of BDS based orbit determination [21]. The major challenge for BeiDou-based LEO POD is the poor precision of the GEO satellite precise orbit. The Chinese meteorological satellite FengYun-3C achieved centimeters-level POD with the BeiDou system by excluding the GEOs observations [22]. Zhang, et al. [23] improved the BDS based precise orbit result of FengYun-3C with the calibration of orbit biases in BeiDou GEO satellites. However, the strategies of BDS-based LEO POD are still far from fully understood. In this study, we make use of BDS dual-frequency observations from the Luojia-1A satellite to investigate the BDS based LEO POD methods. Firstly, the Luojia-1A satellite is briefly introduced, including the satellite-body-fixed (SBF) system. Secondly, the visibility and continuity of onboard BDS measurements are analyzed. The BDS elevation-dependent code biases are modeled with the Luojia-1A observations. Then, the optimal weighting strategy for BeiDou-based LEO POD as well as the optimal precise orbit determination strategy is investigated.

2. Luojia-1A Satellite Platform Luojia-1A is a small scientiﬁc experimental satellite with night light remote sensing and LEO signal navigation augmentation functions. The weight of the satellite is 19.8 kg, and its size is 520 870 390 mm. Luojia-1A satellite is operated in a sun-synchronous orbit at 645 km. The outlook × × of Luojia-1A satellite is illustrated in Figure1, three GNSS antennas were installed: two of them are used to receive GPS/BDS dual-frequency signals in the opposite direction, and one antenna is used to transmit the navigation augmentation signals to users. Besides, a night light camera and a satellite communication antenna are installed on the +Z side of the satellite. Luojia-1A satellite can receive the GPS/BDS dual-frequency signals and provide autonomous orbit and timing information onboard and transmit dual-frequency ranging signals to the users. Remote Sens. 2020, 12, 2063 3 of 17 Remote Sens. 2018, 10, x FOR PEER REVIEW 3 of 17

Figure 1. The appearance of the Luojia-1A satellite.

A satellite-body-fixedsatellite-body-fixed (SBF)(SBF) coordinatecoordinate system system of of Luojia-1A Luojia-1A is is established established for for LEO LEO POD, POD, which which is theis the right-handed right-handed coordinate coordinate system. system. The The origin origin of of the theSBF SBF systemsystem isis atat thethe centercenter ofof mass (COM) of the satellite. satellite. The The +X+ axisX axis is aligned is aligned to the to thesatellit satellitee velocity velocity direction. direction. The +Z The axis+ Zpoints axis pointsto the optical to the opticalcamera, camera, and the and+Y axis the is+ Yperpendicular axis is perpendicular to the XO toZ plane. the XOZ Moreover, plane. Moreover, two receive two antennas receive of antennas Luojia- of1A Luojia-1A are in the are +Z in and the +−ZZ anddirection,Z direction, respectively. respectively. The +Z Theaxis+ pointZ axis to point the toearth the earthduring during the data the − datatransferring, transferring, and the and solar the panel solar panelis kept is facing kept facingthe sun the in sunstandby in standby mode. The mode. small The size small of the size satellite of the satellitebody makes body the makes offset the between offset between the GNSS the antennas GNSS antennas and the and satellite the satellite COM at COM the atdecimeter the decimeter level, level,which which is smaller is smaller than thanmost most satellites. satellites. The The satellit satellitee attitude attitude mode mode of ofthe the Luoj Luojia-1Aia-1A satellite satellite is is also special since the satellite keeps its solar panel facing the sun to receive more solar power, and it turns its ++ZZ side side facing facing to to the the earth earth during the mission pe period.riod. Most Most navigation navigation satellites always keep their ++ZZ side toward thethe earthearth sincesince theythey continuouslycontinuously transmittransmit navigationnavigation signals.signals. Hence, the Luojia-1A satellite is more complicated, whichwhich alsoalso aaffectsffects thethe BeiDouBeiDou signalsignal tracking.tracking.

3. BeiDou Multipath Modeling with Luojia-1A Observations The high-qualityhigh-quality BeiDouBeiDou observations observations are are the the prerequisites prerequisites of of LEO LEO POD. POD. Hence, Hence, we we analyzed analyzed the BeiDouthe BeiDou observations observations from thefrom Luojia-1A the Luojia-1A satellite beforesatellite starting before the starting orbit determination the orbit determination procedure. procedure. 3.1. Data Collection 3.1. DataIn this Collection study, the Luojia-1A onboard BDS data of 2018 DOY 159 were collected with a 1-s sample rate. The onboard BDS data include both B1I and B2I frequency data. To demonstrate the effectiveness of In this study, the Luojia-1A onboard BDS data of 2018 DOY 159 were collected with a 1-s sample BDS based orbit determination, the onboard BDS observations of Luojia-1A are analyzed in this section. rate. The onboard BDS data include both B1I and B2I frequency data. To demonstrate the The statistical details of BDS observations are shown in Figure2. The left panel shows the number effectiveness of BDS based orbit determination, the onboard BDS observations of Luojia-1A are of observations tracked during the observation period, and the right panel shows the percentage of analyzed in this section. three types of BeiDou satellites. The observation condition of the Luojia-1A satellite depends on the The statistical details of BDS observations are shown in Figure 2. The left panel shows the altitude of the satellite. The figure indicates that the Luojia-1A tracks the most observations from the number of observations tracked during the observation period, and the right panel shows the C03 and the C07 satellites. Regarding the observation ratio, it concludes that the observations from the percentage of three types of BeiDou satellites. The observation condition of the Luojia-1A satellite GEO satellite contribute to 42.18%, which is nearly the same as IGSO satellites. It should be noticed depends on the altitude of the satellite. The figure indicates that the Luojia-1A tracks the most that GEO and IGSO satellites contribute to more than 80% observations, while these satellites are only observations from the C03 and the C07 satellites. Regarding the observation ratio, it concludes that available in Asia–Pacific and surrounding regions. The MEO satellites contribute fewer observations the observations from the GEO satellite contribute to 42.18%, which is nearly the same as IGSO due to fewer MEO satellites and faster change of relative geometry between MEO and LEO satellites. satellites. It should be noticed that GEO and IGSO satellites contribute to more than 80% observations, while these satellites are only available in Asia–Pacific and surrounding regions. The MEO satellites contribute fewer observations due to fewer MEO satellites and faster change of relative geometry between MEO and LEO satellites.

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Figure 2. Statistics of BDS observations. (a) The number of observations for BDS satellites. (b) The Figure 2. 2. StatisticsStatistics of BDS of BDS observations. observations. (a) The (a )number The number of observations of observations for BDS for satellites. BDS satellites. (b) The percentage of different BeiDou satellite types. percentage(b) The percentage of different of di BeiDoufferent BeiDousatellite satellitetypes. types. The onboard BDS receiver equipped on the Luojia-1A satellite supports 12 channels for each The onboard BDS receiver equipped on the Luoj Luojia-1Aia-1A satellite supports 12 channels for each frequency, which is abundant for Beidou-2 signal tracking. As the BeiDou-3 satellite deployment, frequency,frequency, which is abundant for Beidou-2 signal tracking.tracking. As As the the BeiDou-3 BeiDou-3 satellite satellite deployment, deployment, more channels are necessary for future receivers capable of tracking BeiDou-2 and BeiDou-3 signals. more channels are necessary for for future future receivers receivers capable capable of of tracking tracking BeiDou-2 BeiDou-2 and and BeiDou-3 BeiDou-3 signals. signals. The time series of observed satellite number of the epoch and its statistic results are shown in Figure The time series of observed satellitesatellite numbernumber ofof thethe epochepoch and and its its statistic statistic results results are are shown shown in in Figure Figure3. 3. The figure indicates that the Luojia-1A suffers mild intermittent interruption during BDS satellite 3.The The figure figure indicates indicates that that the the Luojia-1A Luojia-1A su suffersffers mild mild intermittent intermittent interruptioninterruption duringduring BDS satellite signals tracking, and the reason is worth further investigation. The right panel indicates that the signals tracking, and and the reason is worth furtherfurther investigation.investigation. The The right right panel panel indicates that the Luojia-1A can track five or seven satellites per epoch in most cases. There is only a 57.8% chance for Luojia-1A can track five five or seven satellites per epoc epochh in most cases. There There is is only only a a 57.8% chance for for Luojia-1A to track more than four BDS satellites simultaneously. This is mainly because most BeiDou Luojia-1A to track more than four BDS BDS satellites satellites simultaneously. simultaneously. This is mainly mainly because because most most BeiDou BeiDou satellites are located above the Asia-Pacific region, which is also challenging for obtaining continuous satellites are located located above above the the Asia-Pacific Asia-Pacific region, region, which is also also challenging challenging for for obtaining obtaining continuous continuous and precise orbit with only Beidou-2 observations. and precise orbit with only Beidou-2 observations.

Figure 3. The numberFigure of 3. observedThe number satellites of observed (left panel) satellites and the (a) andstatistic the statisticresult (right result panel). (b). Figure 3. The number of observed satellites (left panel) and the statistic result (right panel). TheThe sky sky view view of of GEO GEO satellites, satellites, IGSO IGSO satellites, satellites, and and MEO MEO satellites satellites from from the Luojia-1A is shownshown in The sky view of GEO satellites, IGSO satellites, and MEO satellites from the Luojia-1A is shown in FigureFigure4 .4. In In the the figure, figure, di ffdifferenterent colors colors mean mean diff erentdifferent satellite satellite types. types. The skyThe view sky ofview the BDSof the satellites BDS in Figure 4. In the figure, different colors mean different satellite types. The sky view of the BDS satellitesis relative is relative to the antenna to the antenna reference reference frame (ARF) frame of (ARF) the Luojia-1A of the Luojia-1A satellite. satellite. The azimuth The azimuth angle is angle started satellites is relative to the antenna reference frame (ARF) of the Luojia-1A satellite. The azimuth angle is fromstarted the from+X the axis +X in axis the in SBF. the Referring SBF. Referring to the to sky the view sky view of C02 of C02 and and C03 C03 satellites, satellites, it is itapparent is apparent that is started from the +X axis in the SBF. Referring to the sky view of C02 and C03 satellites, it is apparent thatthe the Luojia-1A Luojia-1A has has a better a better observational observational condition condition for GEOfor GEO satellites satellites without without a noticeable a noticeable data breach.data that the Luojia-1A has a better observational condition for GEO satellites without a noticeable data breach.As for As the for C08 the satellite, C08 satellite, the azimuth the azimuth varies from varies 0◦ to from 180 ◦0°, and to 180°, the observations and the observations are more concentrated. are more breach. As for the C08 satellite, the azimuth varies from 0° to 180°, and the observations are more concentrated.Moreover, there Moreover, are data there outages are data of the outages C10 satellite of theobservation C10 satellite arc. observat As forion the arc. MEO As satellites for the MEO shown concentrated. Moreover, there are data outages of the C10 satellite observation arc. As for the MEO satellitesin the figure, shown the in observationthe figure, the arc observation is shorter than arc is that shorter of GEO than and that IGSO of GEO satellites, and IGSO and thesatellites, continuity and is satellites shown in the figure, the observation arc is shorter than that of GEO and IGSO satellites, and thenot continuity as good asis not that as of good GEO as and that IGSO of GEO satellites. and IGSO satellites. the continuity is not as good as that of GEO and IGSO satellites.

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Figure 4. The sky view of the BeiDou satellites from the ARF of Luojia-1A satellite. The red, green,

and blue lines represent the geostationary earth orbit (GEO), inclined geostationary (IGSO), and medium earthFigure orbit 4. (MEO) The sky satellites, view of respectively.the BeiDou satellites from the ARF of Luojia-1A satellite. The red, green, and blue lines represent the geostationary earth orbit (GEO), inclined geostationary (IGSO), and 3.2. Modelingmedium the earth Multipath orbit (MEO) of BeiDou satellites, Observations respectively. The pseudorange observations of BeiDou are reported as suffering from the elevation-dependent 3.2. Modeling the Multipath of BeiDou Observations multipath biases, which may lead to an adversary impact on positioning accuracy [24]. Currently, the elevation-dependentThe pseudorange code observations variations ofof BDS-2BeiDou IGSO are and reported MEO satellites as suffering are based from on thethe data elevation- from grounddependent stations. multipath The ground biases, stations which aremay not lead able to toan capture adversary the elevation-dependentimpact on positioning biases accuracy for the[24]. GEOCurrently, satellites, the due elevation-dependent to a stable geometry code relationship. variations The of BDS-2 LEO onboard IGSO and BDS MEO observations satellites canare bebased used on tothe capture data thefrom geometric ground changestations. of The the GEOground satellites, stations which are not can able be used to capture as a complementary the elevation-dependent approach forbiases the multipath for the biasesGEO satellites, modeling. due The multipathto a stable error geometry can be estimatedrelationship. with The the linearLEO combinationonboard BDS ofobservations code and carrier-phase can be used observations, to capture the which geometric can be ch expressedange of the as GEO follows satellites, [25,26]: which can be used as a complementary approach for the multipath biases modeling. The multipath error can be estimated 2 + 2 2 2 + 2 2 with the linear combinationfi fjof code and2 fj carrier-phasefi observations,fj 2 f jwhich can be expressed as MPi = Ci λiϕi + λjϕj + λiNi λjNj Dc (1) follows [25,26]: − f 2 f 2 f 2 f 2 f 2 f 2 − f 2 f 2 − i − j i − j i − j i − j ff22++22 f 2 ff 22 f 2 where C andMϕPCrepresent=− theij codeλϕ and carrier-phase + j λ observables, ϕ + ij respectively. λ N − The jf isλ the N frequency, − D λ ii22−− ii 22 jj 22 − ii 22 − jjc (1) is the wavelength, and Nffijrepresents the carrier ff ij phase ambiguity. ff ij The subscripts ff iji, j (i , j) are used to D specify different frequencies;ϕ c is the constant part of hardware delays. whereThe BDSC and code bias represent was analyzed the code based and on carrier-phase Luojia-1A onboard observables, measurements. respectively. In thisThe section, f is the the MP1 and MP2λ represent the code multipath of B1 and B2 frequency. The MP1 and the elevation frequency, is the wavelength, and N represents the carrier phase ambiguity. The subscripts i, j (i of several BDS satellites are shown in Figure5, which indicates that most BeiDou satellites present ≠ j) are used to specify different frequencies; Dc is the constant part of hardware delays. systematic elevation-dependent variations in multipath combination. According to the GEO satellites, The BDS code bias was analyzed based on Luojia-1A onboard measurements. In this section, the such as the C02 and C05, it is obvious that the multipath is decreased as the elevation increases. As for MP1 and MP2 represent the code multipath of B1 and B2 frequency. The MP1 and the elevation of the IGSO satellite, e.g., C07, the trend is similar to the GEO satellites but with a more dramatic variation. several BDS satellites are shown in Figure 5, which indicates that most BeiDou satellites present Another IGSO satellite, the C10 satellite, has a low elevation, so it has a larger multipath bias without systematic elevation-dependent variations in multipath combination. According to the GEO satellites, an obvious trend. Considering the MEO satellites, the multipath is linearly increasing as the elevation such as the C02 and C05, it is obvious that the multipath is decreased as the elevation increases. As angle decreases, which is shown in panel (c). The multipath error of the C14 satellite is generally for the IGSO satellite, e.g., C07, the trend is similar to the GEO satellites but with a more dramatic smaller than 2 m without significant elevation-dependent variation. variation. Another IGSO satellite, the C10 satellite, has a low elevation, so it has a larger multipath bias without an obvious trend. Considering the MEO satellites, the multipath is linearly increasing as the elevation angle decreases, which is shown in panel (c). The multipath error of the C14 satellite is generally smaller than 2 m without significant elevation-dependent variation.

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FigureFigure 5. The 5. The relationship relationship between between multipath multipath and antennaand antenna elevation elevation angle angle for di forﬀerent different BDS satellites:BDS (a) C02, (b) C07, (c) C11, (d) C05, (e) C10, (f) C14.satellites.

To mitigatemitigate the impactimpact ofof multipathmultipath biases,biases, an elevation-dependentelevation-dependent modelmodel is established with the BeiDou observationsobservations fromfrom thethe Luojia-1ALuojia-1A satellite.satellite. The elevation-dependentelevation-dependent modeling method proposed by Wanninger andand BeerBeer [[24]24] is adopted in this study for the multipath (MP) series. To To define define the piecewise piecewise linear linear models models for for MP MP series, series, the the elevations elevations interval, interval, which which varies varies from from 0° to 0 90°,◦ to is 90 split◦, is bysplit 10 bynodes 10 nodes with the with spacing the spacing of 10°. Therefore, of 10◦. Therefore, the values the of values 10 nodes of 10are nodes estimated are estimatedfor the elevation- for the dependentelevation-dependent model, and model, the restriction and the restrictionthat the aver thatage the of averagemodel values of model is zero. values The isMP zero. observables The MP withobservables the elevation with the e between elevation ek eandbetween ek+1 canek beand expressedek+1 can beas follows: expressed as follows: ! ! e −−e e e  sss ( ) = eekkks + eek s s+ MMPPef()e =+−+− P Pf ,k+1+ 11 − P f P,k vi v (2)(2) f ek+1 −−ek fk,1− ek+1 ek  fk , i eekk++11− ee kk−  where f represents the frequency and s is the type of the BeiDou satellites. There is a 10 difference where f represents the frequency and s is the type of the BeiDou satellites. There is a 10°◦ difference between e and e . Ps and Ps are the MP values of the contiguous grid point, which are estimated k k+1 Pf ,ks f ,k+P1s betweenusing the e methodsk and ek+1 of. leastf ,k and square f estimation.,1k + are the MP values of the contiguous grid point, which are estimatedFigure using6 shows the methods the elevation-dependent of least square estimation. MP models, which are built respectively for the GEO, IGSO,Figure and MEO 6 shows satellites the elevation-dependent for both B1 and B2 frequencies. MP models, The which results are from built Wanninger respectively and for Beer the [24 GEO,] are IGSO,also presented and MEO in satellites the figure for for both comparison. B1 and B2 The frequencies. upper panels The of results the figure from represent Wanninger the and model Beer for [24] the areB1 signal.also presented It is obvious in the that figure the MP1 for modelscomparison. of the Th Luojia-1Ae upper satellitepanels of are the elevation-dependent, figure represent the which model is formarginal the B1 belowsignal. the It is 40 obvious◦ and gradually that the MP1 decreased models as of the the elevation Luojia-1A increase satellite after are 40elevation-dependent,◦. According to the panelswhich (b)is andmarginal (c), the below MP1 modelsthe 40° of and the Luojia-1Agradually satellite decreased are similaras the toelevation that of the increase Wanninger after model 40°. Accordingabove 40◦, to especially the panels the (b) model and (c), of the the MEO MP1 satellites.models of However,the Luojia-1A since satellite the results are similar of the Luojia-1Ato that of thesatellite Wanninger also include model the above satellite-induced 40°, especially MP the eff ects,model there of the are stillMEO minor satellites. differences. However, On since the other the resultshand, asof shownthe Luojia-1A in the lower satellite panels, also include there is the no elevationsatellite-induced dependence MP effects, for the there MP2 are models still minor of the differences.Luojia-1A satellite, On the whichother showshand, as a slowshown decrease in the afterlower 60 panels,◦. The obviousthere is dinoff erenceselevation between dependence MP1 andfor theMP2 MP2 models models may beof attributedthe Luojia-1A to the satellite, Luojia-1A which receiver shows that cana slow smooth decrease the BeiDou after B260°. signal. The obvious The MP differencesmodels built between from the MP1 Luojia-1A and MP2 onboard models observations may be attributed are not asto smooththe Luojia-1A as the Wanninger receiver that model, can whichsmooth may the beBeiDou because B2 thesignal. Wanninger The MP modelmodels adopts built from much the longer Luojia-1A observed onboard time. observations The MP models are cannot asbe smooth used to as correct the Wanninger the pseudorange model, measurements which may be directlybecause for the the Wannin single-frequencyger model adopts orbit determination much longer observedof LEO satellites. time. The MP models can be used to correct the pseudorange measurements directly for the single-frequency orbit determination of LEO satellites.

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Figure 6. The elevation-dependent multipath models for the GEO, IGSO, and MEO satellites. The upper Figure 6. The elevation-dependent multipath models for the GEO, IGSO, and MEO satellites. The and the bottom panels represent models for B1 and B2 respectively. upper and the bottom panels represent models for B1 and B2 respectively. 4. Methodologies for the Beidou-Based LEO POD 4. Methodologies for the Beidou-Based LEO POD There are a few challenges in BeiDou-based LEO POD, including the BeiDou GEO orbit error handlingThere and are tuning a few the challenges pseudo-stochastic in BeiDou-based parameters LEO due POD, to discontinuity including the of theBeiDou BeiDou GEO observations, orbit error whichhandling are discussedand tuning in this the study. pseudo In this-stochastic section, theparameters theory of due LEO to POD discontinuity is briefly introduced. of the BeiDou observations, which are discussed in this study. In this section, the theory of LEO POD is briefly 4.1.introduced. Theory of Reduce-Dynamic Orbit Determination The reduced-dynamic model for the LEO POD has been the mainstream method, which combined 4.1. Theory of Reduce-Dynamic Orbit Determination the GNSS observations and the dynamic models for LEO POD. To reduce the computation complexity, a fewThe empirical reduced-dynamic forces or pseudo-stochastic model for the parameters LEO POD are has introduced been the into mainstream the system tomethod, assimilate which the ecombinedffect of unmodeled the GNSS forces.observations Hence, and the the pseudo-stochastic dynamic models parameters for LEO POD. are To critical reduce for the the computation LEO POD. Thecomplexity, precise point a few positioning empirical (PPP)forces modelor pseudo-stochas is used for involvingtic parameters the GNSS are introduced observations, into while the system the LEO to observationsassimilate the are effect not perturbedof unmodeled by the forces. tropospheric Hence, th delay.e pseudo-stochastic The motion equation paramete of thers are LEO critical satellite for can the beLEO expressed POD. The as follows:precise point positioning (PPP) model is used for involving the GNSS observations, while the LEO observations are not perturbed by the tropospheric delay. The motion equation of the .. r . . LEO satellite can be expressedr = asGM follows:+ f t, r, r, Q1, ... , Qd, P1, ... , Ps = f (3) − r3 1 r GM =− +()  = r where the is the productrfrrf of theGM constant3 of111t gravity, , , Q and ,...,Q theds , massP ,..., ofP Earth, is the geocentric radius(3) . r (k) (k) vector, r is the velocity of the LEO satellite, with the initial condition r (t0) = r (a, e, i, Ω, ω, T0; t0), kwhere= 0, 1the, whereas GM isa the, e, iproduct, Ω, ω, T0 ofrepresent the constant the of six gravity Keplerian and the elements mass of atEarth, the timer is tthe0. geocentricf1 is the perturbingradius vector, acceleration, r is andthe f isvelocity the total of acceleration.the LEO satellite,Q1, ... , Qdwithrepresent the theinitial force condition model parameters,()kk and() P , ...., P represent the pseudo-stochastic orbit parameters. There are different rr()taeiTt=Ω(),,,1 ,ωs , ; , k = 0,1 , whereas aei,,,Ω ,ω , T represent the six Keplerian kinds of000 pseudo-stochastic orbit parameters, such as instantaneous0 velocity changes, piecewise constantelements accelerations, at the time andt0 . piecewise f1 is the linear perturbing accelerations. acceleration, Usually, and instantaneous f is the velocity total acceleration. pulses and piecewise constant accelerations are adopted in the orbit improved and final orbit determination QQ1,..., d represent the force model parameters, and P1,...., Ps represent the pseudo-stochastic procedure, respectively. For the instantaneous velocity changes added at epoch t in the direction e(t), orbit parameters. There are different kinds of pseudo-stochastic orbit parameters,i such as the pseudo-stochastic parameter P = V in Equation (3) is written as P = V δ(t t ) e(t), and the instantaneous velocity changes, piecewisei i constant accelerations, and piecewisei i · linear− i · accelerations. δ(t) is Dirac’s delta function. Moreover, the piecewise constant accelerations are generally adopted as Usually, instantaneous velocity pulses and piecewise constant accelerations are adopted in the orbit improved and final orbit determination procedure, respectively. For the instantaneous velocity

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the empirical accelerations for orbit determination. Adding the change of acceleration ai in direction e(t) during the time ti 1 < t < ti, the pseudo-stochastic parameter Pi = ai in Equation (3) is written as − P = a e(t). i i · 4.2. POD Strategy of Luojia-1A Satellite The reduced dynamic orbit determination method is used for Luojia-1A satellite POD. In the data processing, both code and carrier-phase observations are used to form the ionosphere-free linear combination, and the observations are resampled into 5 s. The details of the data processing strategies for Luojia-1A satellite POD are presented in Table1

Table 1. The data processing strategies for Luojia-1A satellite POD.

Type of Parameter Settings Observations BDS B1 and B2 code and carrier-phase observations Sample rate 5 s Cut off angle 0º BeiDou antenna phase center PCV.atx [27] Relativistic effects IERS 2010 Ionospheric error Ionosphere-free linear combination Earth rotation ERP products Provided by Wuhan University Nutation IAU2000R06 Polar motion IERS2010XY BeiDou precise orbit and clock products WUM products Provided by Wuhan University Earth’s gravity field EGM2008 model with the degree of 120 Solid Earth tides IERS 2010 Ocean tides FES 2004 Planetary calendar DE405 Solar pressure ECOM Pseudo-stochastic parameters Radial, along-track, and cross-track directions

5. Luojia-1A Satellite POD Results and Discussion

5.1. The Procedure of Luojia-1A Satellite POD Several steps are required for the procedure of Luojia-1A satellite reduced dynamic orbit determination, which can be briefly described as follows. Firstly, obtaining the prior orbit of the satellite using the BDS pseudorange observations. The precise orbit and clocks products of BeiDou satellites, as well as the ERP products, are adopted to process the pseudorange positioning orbit result and then fitting the prior orbit using numerical integration. Secondly, data preprocessing for the zero difference phase observations, including cycle slip detection and outlier detection. After that, estimate the orbital parameters, dynamic parameters, and the pseudo-stochastic parameters for solving the improved satellite orbit, which is determined based on the prior orbit and the screened phase observations. The data preprocessing as well as the orbit improving procedure are iterated for the precise solution. Finally, the orbit generated from the last step is introduced as the prior orbit, and the screened phase observations are used to estimate the parameters; thus, the precise orbit result is obtained. With regards to the orbit precision evaluation method, the overlapping comparison method is used since the Luojia-1A satellite does not equip with the laser reflector array (LRA), and there is no precise reference orbit for comparison. The overlapping comparison strategy for the Luojia-1A satellite is illustrated in Figure7. The observations of the Luojia-1A satellite in 2018 DOY159 are divided into two observational arcs, one contains the begin epoch until 9:00:30, the other contains 2:14:52 until the last epoch, and then two observational arcs are determined respectively. Therefore, the difference of two solutions in the overlapped part is used to assess the Luojia-1A orbit precision in this study. Remote Sens. 2018, 10, x FOR PEER REVIEW 9 of 17 divided into two observational arcs, one contains the begin epoch until 9:00:30, the other contains 2:14:52 until the last epoch, and then two observational arcs are determined respectively. Therefore,

Remotethe difference Sens. 2020, of12 ,two 2063 solutions in the overlapped part is used to assess the Luojia-1A orbit precision9 of 17 in this study.

Figure 7. TheThe concept of overlap comparison for the Luojia-1A precise orbit. 5.2. Precise Orbit Determination with BeiDou Observations 5.2. Precise Orbit Determination with BeiDou Observations The precise orbit determination of LEO requires precise BeiDou orbit, satellite clocks, and earth The precise orbit determination of LEO requires precise BeiDou orbit, satellite clocks, and earth rotation parameters (ERP) products. One of the challenges for Beidou-based LEO POD is that the rotation parameters (ERP) products. One of the challenges for Beidou-based LEO POD is that the BeiDou orbit is generally not as good as the GPS orbit for many reasons. Presently, several analysis BeiDou orbit is generally not as good as the GPS orbit for many reasons. Presently, several analysis centers are providing multi-GNSS precise products with BeiDou constellation support. Four types of centers are providing multi-GNSS precise products with BeiDou constellation support. Four types of BeiDou precise products from different multi-GNSS experiment (MGEX) analysis centers are listed BeiDou precise products from different multi-GNSS experiment (MGEX) analysis centers are listed in Table2. However, a few issues arise in applying these products at the current stage, such as that in Table 2. However, a few issues arise in applying these products at the current stage, such as that the Technische Universität München (TUM) products currently does not involve the ERP products. the Technische Universität München (TUM) products currently does not involve the ERP products. The precise products provided by CODE do not involve the BDS-2 GEO satellites. The German Research The precise products provided by CODE do not involve the BDS-2 GEO satellites. The German Centre for Geosciences (GFZ) and Wuhan University provide the full BDS-2 satellites orbit and clock, Research Centre for Geosciences (GFZ) and Wuhan University provide the full BDS-2 satellites orbit but different phase center variation (PCV) models are applied for different ACs, so the products are not and clock, but different phase center variation (PCV) models are applied for different ACs, so the directly compatible. In this study, the Wuhan University Multi-GNSS (WUM) products are adopted to products are not directly compatible. In this study, the Wuhan University Multi-GNSS (WUM) determine the precise orbit of the Luojia-1A satellite. products are adopted to determine the precise orbit of the Luojia-1A satellite. Table 2. The information about BDS orbit and clock products. Table 2. The information about BDS orbit and clock products. Institution Indicate Constellations Orbit Clocks ERP Institution Indicate Constellations Orbit Clocks ERP CODECODE COM COM GPS GPS+ +GLO GLO+ +GAL GAL+ +BDS BDS+ QZS+ QZS 5 5 min min 30 30 s /s/55 min min 12 hh GFZ GBM GPS + GLO + GAL + BDS + QZS 15 min 30 s/5 min 24 h WHUGFZ WUM GBM GPS GPS+ +GLO GLO+ +GAL GAL+ +BDS BDS+ QZS+ QZS 15 15 min min 30 30 s /s/55 min min 24 hh TUMWHU TUM WUM GPS + GALGLO+ +BDS GAL+ +QZS BDS + QZS 5 15 min min 30 5 mins/5 min 24/ h TUM TUM GAL + BDS + QZS 5 min 5 min / The BDS precise orbit and clocks products are critical factors for the Luojia-1A satellite POD. As The BDS precise orbit and clocks products are critical factors for the Luojia-1A satellite POD. reported, the precise orbit products of BDS GEO satellites are not as good as the IGSO and MEO As reported, the precise orbit products of BDS GEO satellites are not as good as the IGSO and MEO satellites due to poor geometry relationship between the ground stations and the GEO satellites. satellites due to poor geometry relationship between the ground stations and the GEO satellites. Generally, the BeiDou GEO precise orbit only achieves a few decimeters accuracy, whereas the IGSO Generally, the BeiDou GEO precise orbit only achieves a few decimeters accuracy, whereas the IGSO and MEO satellites achieve centimeter-level accuracy. Consequently, adopting the GEO satellites and MEO satellites achieve centimeter-level accuracy. Consequently, adopting the GEO satellites might might decrease the LEO POD precision. WUM product employs a three-day arc for BDS orbit decrease the LEO POD precision. WUM product employs a three-day arc for BDS orbit determination, determination, and only the middle day is adopted as the final product. So the discontinuity is and only the middle day is adopted as the final product. So the discontinuity is inevitably present inevitably present at the day boundary (DBD) in orbit products [27], which can be used to roughly at the day boundary (DBD) in orbit products [27], which can be used to roughly evaluate the orbit evaluate the orbit precision [28]. To analyze the orbit precise of BeiDou satellites, the average DBDs precision [28]. To analyze the orbit precise of BeiDou satellites, the average DBDs of 2018 DOY 153 of 2018 DOY 153 to DOY 163 are calculated. Since the last epoch of WUM orbit products is 23h 45m, to DOY 163 are calculated. Since the last epoch of WUM orbit products is 23h 45m, the orbit is the orbit is extrapolated to the 24 h at midnight, then the difference is calculated at the day boundary extrapolated to the 24 h at midnight, then the difference is calculated at the day boundary with the next with the next day orbit product in radial, along-track, and cross-track directions. After that, the day orbit product in radial, along-track, and cross-track directions. After that, the average 3D DBDs of average 3D DBDs of BDS satellites orbit products are shown in Figure 8. According to the figure, the BDS satellites orbit products are shown in Figure8. According to the figure, the GEO satellites present GEO satellites present more significant DBDs than the IGSO and MEO satellites. The average DBDs more significant DBDs than the IGSO and MEO satellites. The average DBDs of GEO satellites vary of GEO satellites vary between 58 cm and 130 cm. As for IGSO satellites, the DBDs are less than 50 between 58 cm and 130 cm. As for IGSO satellites, the DBDs are less than 50 cm. The DBDs of MEO cm. The DBDs of MEO satellites are the smallest, which is generally less than 20 cm. The statistic satellites are the smallest, which is generally less than 20 cm. The statistic result of different types of the BeiDou satellites are listed in Table3, the average DBD of GEO satellites is 82.28 cm, which is more than three times of IGSO satellites. The precision of MEO satellites is best, which is 17 cm. Remote Sens. 2018, 10, x FOR PEER REVIEW 10 of 17

result of different types of the BeiDou satellites are listed in Table 3, the average DBD of GEO satellites is 82.28 cm, which is more than three times of IGSO satellites. The precision of MEO satellites is best, Remotewhich Sens. is 202017 cm., 12, 2063 10 of 17

FigureFigure 8. 8.The The average average day day boundary boundary of of BDS BDS orbit orbit products. products. Table 3. The average 3D day boundary of different orbital type of BeiDou satellites (Unit: cm). Table 3. The average 3D day boundary of different orbital type of BeiDou satellites (Unit: cm). GEO IGSO MEO GEO IGSO MEO 3D DBD3D DBD 82.28 82.28 26.74 26.74 17.00 17.00 According to the analysis of BeiDou satellites DBDs, the precision of GEO satellites orbit is not as goodAccording as that to of the IGSO analysis and ofMEO BeiDou satellites. satellites Therefore, DBDs, theadopting precision the ofGEO GEO satellite satellites observations orbit is not as assame good as as IGSO that ofand IGSO MEO and satellite MEO satellites.observations Therefore, might decrease adopting the the LEO GEO POD satellite precision. observations Li et al. [22] as sameimprove as IGSO the andBDS-based MEO satellite precise observationsorbit determination mightdecrease result of theFY-3C LEO by POD excluding precision. the LiGEO et al.satellite [22] improveobservations. the BDS-based However, precise the GEO orbit satellite determination observations result of ofthe FY-3C Luojia-1A by excluding satellite thecontribute GEO satellite to more observations.than 40% observations. However, the Excluding GEO satellite the observationsGEO satellite of observations the Luojia-1A will satellite dramatically contribute reduce to more the thannumber 40% observations.of effective observations Excluding the and GEO lead satellite to a negative observations effect on will the dramatically reliability of reduce the POD the results. number of effectiveTo determine observations the precise and lead orbit to aof negative the Luojia-1A effect sate on thellite, reliability the satellite of the state POD vector results. can be expressed as followsTo determine [29]: the precise orbit of the Luojia-1A satellite, the satellite state vector can be expressed as follows [29]: T xx(t())ttvtpq==(r ((rt(),(),), v(t), p, q) ,T ) (4)(4) ( ) ( ) where r tr and()t v t arev () thet position and the velocity of the LEO satellite at epoch t. p representst thep forcewhere model parameter, and e.g., are the the scaling position factors and of the the velocity perturbing of the accelerations LEO satellite acting at onepoch the LEO. satellite.representsq is the pseudo-stochasticforce model paramete orbitr, parameter, e,g., the scaling which factors compensates of the forperturbing the inaccuracy accelerations force model. acting q Theon observationsthe LEO satellite. equation can is bethe briefly pseudo-stochastic described as orbit parameter, which compensates for the inaccuracy force model. The observations equation can be briefly described as z = h(x0) + ε (5) =+()ε zhx0 (5) where z is the n-dimensional observation vector and x0 is the initial state vector. The recursive least-square estimator is adopted in LEO POD and the weightedx least-squares can be written as: where z is the n-dimensional observation vector and 0 is the initial state vector. The recursive least-square estimator is adopted in LEOlsq POD and the1 weighted least-squares can be written as: ∆x = HTWH − HTW∆z (6) 0 − lsq 1 Δ =ΔTT (6) where the H matrix is the partial derivativesx0 ()()H of theWH observations H W zconcerning the state vector at the re f reference epoch t , W is the weighting matrix, and ∆z = z h x . Generally, the weight of different 0 − 0 satelliteswhere the observations H matrix is is usually the partial set as derivatives the same: of the observations concerning the state vector at the Δ−zz= hx()ref t0 2 0 reference epoch , W is the weighting matrix, σand0 . Generally, the weight of P = (7) 2 different satellites observations is usually set as theσi same: 2 where the σ is a priori unit weight factor, which is 0.001mσ in this study. However, as for the BeiDou 0 P = 0 system, the precision of GEO satellites orbit is not as goodσ 2 as that of IGSO and MEO satellites. To adopt(7) i the GEO satellite observations to improve the reliability of the Luojia-1A precise orbit and reduce the effect from the orbit error of GEO satellites, the strategy of estimation that reduces the weight of GEO satellites observations are proposed in this study. To fully discover the effect of the reduced weight strategy, six different weighting strategies are designed and listed in Table4. In this study, the IGSO Remote Sens. 2020, 12, 2063 11 of 17 and MEO satellites are treated as the equal-weighted, and the weight of GEO satellite observations is relative to the IGSO and MEO satellite observations; all the IGSO and MEO satellite observations are set unit weight.

Table 4. The weight setting for GEO satellite observations.

Scheme ID A B C D E F Weighting Strategy 1 1/25 1/49 1/100 1/900 1/2500

The precise orbit determination results of the Luojia-1A satellite with six diﬀerent weighting strategies are listed in Table5, and the equal weight of all satellites strategy is used as the reference; the LEO POD precision can be improved by re-weighting the GEO satellite observations. According to the results, the strategy that reduces the weight of GEO satellites can improve the BDS based Luojia-1A satellite POD signiﬁcantly. As the weight reduces to 1/49, the improvement of precision can reach 44.21%. Moreover, as the weight of GEO satellites is reduced, the precision of Luojia-1A orbit is also improved gradually. Luojia-1A satellite orbit precision can be improved to 2.316 cm in the 3D direction by reducing the GEO satellite weight down to 1/900, in which the RMS of radial direction is better than 1 cm and the improvement is 63.43% refers to the equal-weight strategy. The POD precision will not be continuously improved as the GEO satellite weight reduces to 1/2500. Therefore, 1/900 is considered as the optimal GEO weighting strategy for the Luojia-1A satellite POD.

Table 5. The precision of Luojia-1A precise orbit results with diﬀerent GEO weighted strategies (Unit: cm).

Weight Radial Along-Track Cross-Track 3D Improvement 1 2.02 3.32 5.00 6.333 / 1/25 1.50 2.29 4.37 5.157 18.57% 1/49 1.28 2.20 2.45 3.533 44.21% 1/100 1.11 2.00 1.98 3.025 52.23% 1/900 0.94 1.75 1.19 2.316 63.43% 1/2500 0.94 1.73 1.23 2.322 63.43%

The overlapping orbit comparison result of the Luojia-1A satellite precise orbit is shown in Figure9, which represents the LEO orbit computed from the different GEO weighting strategies. The panel (a) presents the overlapped orbit residuals for the all satellite equally weighted cases. The figure shows that orbit presents systematical biases for the whole arc and significant fluctuation presence, especially at the end of the arc. As the weight of the GEO satellites is reduced to 1/25, the fluctuation of orbit error series is dramatically mitigated, but the constant bias on the cross-track direction is still present. Further reducing the weight of the GEO satellites can reduce the constant bias in the cross-track direction. A weight of 1/900 for the GEO satellites is considered as the optimal choice for the Luojia-1A satellite since the orbit error is noticeably reduced and without significant fluctuation. The analysis also indicates that the GEO satellite orbit error may introduce fluctuations and constant bias in the LEO orbit determination. Remote Sens. 2020, 12, 2063 12 of 17 Remote Sens. 2018, 10, x FOR PEER REVIEW 12 of 17 Remote Sens. 2018, 10, x FOR PEER REVIEW 12 of 17

FigureFigure 9.9. Orbit error error of of Luojia-1A Luojia-1A in different in different scenarios. scenarios. Figure 9. Orbit error of Luojia-1A in different scenarios. TheThe ionosphere-free ionosphere-free carriercarrier phase residuals residuals of of Luojia-1A Luojia-1A satellite satellite orbit orbit determination determination under under The ionosphere-free carrier phase residuals of Luojia-1A satellite orbit determination under differentdifferent weight weight strategy strategy is is also also analyzed,analyzed, and and the the re resultssults are are presented presented in Figure in Figure 10. In 10 this. In analysis, this analysis, thedifferent C12 satellite weight is strategyexcluded is inalso the analyzed, orbit determ and theine reproceduresults are presenteddue to the in absence Figure 10. of Inprecise this analysis, orbit the C12 satellite is excluded in the orbit determine procedure due to the absence of precise orbit products.the C12 Thesatellite figure is indicatesexcluded thatin the for orbit the determequal weightine procedure strategy, due the to residuals the absence of GEO of precise satellites orbit products.achieveproducts. The a minimum figureThe figure indicates while indicates the that IGSO forthat and the for equalMEO the equal satellite weight weights strategy,yield strategy, larger the residuals. residualsthe residuals By of GEOadopting of GEO satellites thesatellites re- achieve a minimumweightingachieve while astrategy, minimum the the IGSO whileresiduals and the MEO IGSOof GEO satellitesand satellites MEO yield satellite are slightly largers yield residuals.increased, larger residuals. which By adopting is By reasonable adopting the re-weighting since the re- strategy,theweighting GEO the satellite residuals strategy, orbit oferrorsthe GEO residuals are satellites assimilated of GEO are intosatellites slightly the residuals. are increased, slightly Meanwhile, increased, which the is which reasonableresiduals is reasonable of IGSO since and thesince GEO satelliteMEOthe orbitGEOsatellites satellite errors are are orbitdecreased assimilated errors toare less assimilated into than the 10 residuals. intomm. the Different residuals. Meanwhile, weighting Meanwhile, the strategies residuals the residuals did of not IGSO of IGSOlead and to and MEO satellitesdramaticMEO are satellites variation decreased are of thedecreased to carrier less than phaseto 10less residuals. mm. than Di10ff mm.erent Different weighting weighting strategies strategies did not did lead not tolead dramatic to dramatic variation of the carrier phase residuals. variation of the carrier phase residuals.

Figure 10. The RMS of diﬀerent satellite residuals, (a) C01–C05, (b) C06–C14.

The residuals of observations reﬂect the consistency of the reduce-dynamic model and the observation model, which can approach noise level if only the reduce-dynamic model can represent the Remote Sens. 2018, 10, x FOR PEER REVIEW 13 of 17

Figure 10. The RMS of different satellite residuals.

RemoteThe Sens. residuals2020, 12, 2063 of observations reflect the consistency of the reduce-dynamic model and13 ofthe 17 observation model, which can approach noise level if only the reduce-dynamic model can represent the motion of the LEO satellites appropriately. According to experiments, the precise orbit of the Luojia-1Amotion of thesatellite LEO satellitescan obtain appropriately. the best result According with the to experiments,condition that the the precise weight orbit of of GEO the Luojia-1A satellite observationssatellite can obtainis set 1/900. the best result with the condition that the weight of GEO satellite observations is set 1The/900. left panel of Figure 11 reflects the relationship between the carrier phase residuals and the elevationThe angle left panel of the of optimal Figure 11 results reﬂects of the the Luojia-1 relationshipA satellite, between which the carrierindicates phase that residualsthe residuals and are the elevation-dependent.elevation angle of the For optimal the high results elevation of the Luojia-1Aangle case, satellite, the carrier which phase indicates residuals that are the generally residuals less are thanelevation-dependent. 3 cm, while the carrier For the phase high residuals elevation with angle lower case, elevation the carrier angle phase are residuals generally are smaller generally than less 9 cm.than The 3 cm, figure while also the concludes carrier phase that residuals the residuals with lower are fairly elevation small angle in most are generally cases. The smaller carrier than phase 9 cm. residualsThe ﬁgure are also smaller concludes than that 2 cm the in residuals most cases; are fairly hence, small the in overall most cases. RMS The of carrierresiduals phase is 1.49 residuals cm. Accordingare smaller to than the 2right cm inpanel, most it cases; is apparent hence, thethat overall over 70% RMS residuals of residuals are smaller is 1.49 cm.than According 1 cm and toover the 93.01%right panel, residuals it is apparentare smaller that than over 3 cm. 70% residuals are smaller than 1 cm and over 93.01% residuals are smaller than 3 cm.

FigureFigure 11. 11. TheThe statistic statistic results results of of precise precise orbit orbit determination determination (POD) (POD) residuals. residuals. (a (a) )The The relationship relationship b betweenbetween POD residualsresiduals and and the the elevation elevation angle. angle. ( ) Frequency (b) Frequency distribution distribution histogram histogram of POD residuals.of POD 5.3.residuals. Tuning the Pseudo-Stochastics Parameters

5.3. TuningThe Luojia-1A the Pseudo is-Stochastics a nanosatellite Parameters which only weighs 19.8 kg. Such a nanosatellite has different dynamic characteristics compared to the large satellites. In the reduced dynamic orbit determination, the pseudo-stochasticThe Luojia-1A is a parameters nanosatellite are which introduced only weighs into the 19.8 orbit kg. determination Such a nanosatellite systems has to handle different the dynamicimpact of characteristics unmodeled forces. compared The pseudo-stochasticto the large satellites. parameter In the redu canced assimilate dynamic the orbit impact determination, of nuisance theforces pseudo-stochastic and prevent the parameters orbit solution are divergence.introduced into the orbit determination systems to handle the impactUsually, of unmodeled different forces. types The of pseudo-stochasticpseudo-stochastic parameter parameters can are assimilate introduced the for impact the preciseof nuisance orbit forcesdetermination, and prevent generally the orbit including solution divergence. the instantaneous velocity pulse and the piecewise constant accelerationUsually, [30different]. The empirical types of settingpseudo-stochastic of the pseudo-stochastic parameters parametersare introduced is adopted for the in precise the previous orbit determination,POD experiments, generally which setincluding up the velocitythe instantaneous pulses during velocity the orbit pulse improved and the procedure piecewise and constant estimate accelerationthe empirical [30]. accelerations The empirical in the setting final of POD the pseudo-stochastic step. The instantaneous parameters velocity is adopted pulses arein the added previous every POD15 min experiments, interval with which the priorset up sigma the 5.0velocity10 6 pum/lsess. The during piecewise the orbit constant improved accelerations procedure with and the · − estimatespacing ofthe 6 minempirical are used accelerations to improve in the the orbit final result POD at thestep. last The orbit instantaneous determination velocity procedure, pulses and are the ⋅ −6 addedprior sigmaevery is15 5 min10 interval9 m/s2. with These the pseudo-stochastic prior sigma 5.0 parameters 10 ms. The are piecewise added in constant the radial, accelerations along-track, × − withand the out-of-plane spacing of directions. 6 min are used to improve the orbit result at the last orbit determination procedure, and theIn prior this section, sigma theis 5 e×ff 10ect−9 ofm/s pseudo-stochastics2. These pseudo-stochastic parameters parameters on the Luojia-1A are added precise in the orbit radial, is analyzed along- track,by setting and out-of-plane different spacing directions. for both the instantaneous velocity pulses and the empirical accelerations. TheIn basic this precise section, orbit the determination effect of pseudo-stochasti strategy is listedcs parameters in Table1 and on thethe weightLuojia-1A of the precise GEO satelliteorbit is analyzedobservations by setting is set 1 different/900. Five spacing different for scenarios both the of instantaneous the spacing of velocity pseudo-stochastic pulses and parametersthe empirical are accelerations.analyzed in thisThe section,basic precise including orbit 30determination min, 15 min, strategy 10 min, is 6listed min, in and Table without 1 and pseudo-stochasticthe weight of the parameters. The five different spacing strategies are adopted for both the instantaneous velocity pulses and the empirical accelerations. Therefore, there are 25 different mixed strategies in all. The 3D precision of the Luojia-1A precise orbit with different settings for the pseudo-stochastic parameters is shown in Remote Sens. 2018, 10, x FOR PEER REVIEW 14 of 17

GEO satellite observations is set 1/900. Five different scenarios of the spacing of pseudo-stochastic parameters are analyzed in this section, including 30 min, 15 min, 10 min, 6 min, and without pseudo- stochastic parameters. The five different spacing strategies are adopted for both the instantaneous velocity pulses and the empirical accelerations. Therefore, there are 25 different mixed strategies in

Remoteall. The Sens. 3D2020 precision, 12, 2063 of the Luojia-1A precise orbit with different settings for the pseudo-stochastic14 of 17 parameters is shown in Figure 12. The bottom axis reflects the spacing for the empirical accelerations in the final POD step, whereas the bar in different colors represents different spacing for the Figureinstantaneous 12. The bottom velocity axis pulses reflects in the the orbit-improved spacing for the procedure. empirical accelerations The figure indicates in the final that POD the impact step, whereasof the theempirical bar in diaccelerationsfferent colors is represents greater than diff erentthat of spacing the instantaneous for the instantaneous velocity velocitypulse. Once pulses the inempirical the orbit-improved accelerations procedure. are absent The in figure the final indicates POD that step, the the impact precision of the of empirical the Luojia-1A accelerations orbit is isdramatically greater than thatdecreased. of the instantaneousThe precision is velocity improved pulse. significantly Once the empiricaleven with accelerations the sparse parameters are absent of inaccelerations. the final POD According step, the precisionto the results, of the the Luojia-1A shorter spacing orbit is adopted dramatically for the decreased. piece constant The acceleration, precision isthe improved more accurate significantly result even is withobtained. the sparse However, parameters there ofare accelerations. no pronounced According advantages to the from results, the theinstantaneous shorter spacing velocity adopted pulse for for the precise piece constant orbit determination acceleration, of the the more Luojia-1A accurate satellite. result is The obtained. optimal However,result of there the Luojia-1A are no pronounced satellite accessed advantages by fromadopting the instantaneous the strategy velocitywith 6 min pulse interval for precise piecewise- orbit determinationconstant accelerations of the Luojia-1A parameters satellite. and Thewithout optimal instantaneous result of the velocity Luojia-1A pulse, satellite which accessed achieves by the adoptingprecision the of strategynearly 2 withcm in 6 3D min direction. interval piecewise-constantThe empirical strategy accelerations adopted in parameters the previous and experiments without instantaneouscan achieve velocitythe near-optimal pulse, which result. achieves the precision of nearly 2 cm in 3D direction. The empirical strategy adopted in the previous experiments can achieve the near-optimal result.

FigureFigure 12. 12.The The 3D 3D precision precision of of Luojia-1A Luojia-1A with with di differentﬀerent settings settings for for the the pseudo-stochastic pseudo-stochastic parameters. parameters.

AsAs shown shown in Figurein Figure 12, the 12, piece the constant piece accelerationconstant acceleration is crucial for is the crucial precise for orbit the determination precise orbit ofdetermination the Luojia-1A satellite.of the Luojia-1A Therefore, satel thelite. results Therefore, of the strategiesthe results with of the diﬀ strategieserent spacing with of different the piecewise spacing constantof the accelerationspiecewise constant but without accelerations the instantaneous but without velocity the instantaneous pulse are further velo analyzed.city pulse The are statistic further resultsanalyzed. of Luojia-1A The statistic satellite results orbit of inLuojia-1A the radial, satellite along-track, orbit in and the cross-trackradial, along-track, directions and are cross-track listed in Tabledirections6, and the are detail listed of in the Table orbit 6, errorand the is shown detail inof Figurethe orbit 13 .error Panel is (a) shown shows in the Figure error 13. of thePanel Luojia-1A (a) shows satellitethe error orbit of without the Luojia-1A adding satellite any pseudo-stochastic orbit without parameters.adding any Thepseudo-stochastic ﬂuctuation of theparameters. along-track The errorfluctuation is pronounced, of the along-track in which theerror error is pronounced, of the start in epoch which is the larger error than of the 30 start cm. epoch In the is cross-track larger than direction,30 cm. In there the cross-track is nearly 10 direction, cm constant there error. is nearly As shown 10 cm in constant the other error. three As panels, shown the in errors the other of three three directionspanels, the are errors reduced of bythree adding directions the pseudo-stochastic are reduced by parameters.adding the Thepseudo-s shortertochastic spacing parameters. that the piece The constantshorter parameters spacing that are the set, piece the more constant precise parameters the result are of Luojia-1Aset, the more satellite precise orbit the achieved, result of especially Luojia-1A insatellite the beginning orbit achieved, of overlap especially comparison. in the beginning of overlap comparison.

TableTable 6. 6.The The RMS RMS precision precision of of Luojia-1A Luojia-1A with with di ﬀdifferenterent piecewise piecewise constant constant accelerations accelerations intervals intervals (Unit:(Unit: cm). cm).

AccelerationsAccelerations Interval Interval Radial Radial Along-Track Along-track Cross-Track Cross-track 3D 3D w/o w/o2.43 2.43 11.1711.17 12.8312.83 17.184 17.184 30 min 30 min 1.30 1.30 3.66 3.66 0.62 0.62 3.933 3.933 15 min 0.84 2.37 0.77 2.630 15 min 0.84 2.37 0.77 2.630 10 min 0.74 1.99 0.73 2.245 6 min 0.99 1.66 0.52 2.002

Remote Sens. 2018, 10, x FOR PEER REVIEW 15 of 17

10 min 0.74 1.99 0.73 2.245 Remote Sens. 2020, 12, 2063 15 of 17 6 min 0.99 1.66 0.52 2.002

FigureFigure 13.13.The The error error of Luojia-1Aof Luojia-1A satellite satellite orbit withorbit di ffwitherent different piecewise piecewise constant accelerationconstant acceleration parameter strategies.parameter (strategies.a) Without (a piecewise) Without constantpiecewise accelerations. constant accelerations. (b) 30 min (b interval.) 30 min (interval.c) 15 min (c) interval. 15 min (interval.d) 6 min ( interval.d) 6 min interval. 6. Conclusions 6. Conclusions This study focuses on the precise orbit determination of the Luojia-1A low earth orbit satellite This study focuses on the precise orbit determination of the Luojia-1A low earth orbit satellite with onboard BDS observations. The onboard BDS observations are analyzed firstly and demonstrated with onboard BDS observations. The onboard BDS observations are analyzed firstly and that the visibility and continuity of the Luojia-1A are good enough for precise orbit determination. demonstrated that the visibility and continuity of the Luojia-1A are good enough for precise orbit Moreover, the multipath error of onboard BDS code observations is modeled, which shows evident determination. Moreover, the multipath error of onboard BDS code observations is modeled, which elevation-dependent code bias for the multipath. An elevation-dependent code bias model using LEO shows evident elevation-dependent code bias for the multipath. An elevation-dependent code bias data is established, which is capable of modeling the elevation-dependent code bias for the Beidou model using LEO data is established, which is capable of modeling the elevation-dependent code GEO satellites. The daily boundary of the BeiDou orbit product is calculated and demonstrates that the bias for the Beidou GEO satellites. The daily boundary of the BeiDou orbit product is calculated and precision of the GEO satellite orbit is not as good as IGSO and MEO satellites. To overcome the poor demonstrates that the precision of the GEO satellite orbit is not as good as IGSO and MEO satellites. precision of the GEO satellite orbit, the strategy that reduces the weight of GEO satellite observations To overcome the poor precision of the GEO satellite orbit, the strategy that reduces the weight of is adopted. The precise orbit result is obtained by adopting the reduced weight strategy, which can GEO satellite observations is adopted. The precise orbit result is obtained by adopting the reduced achieve 2.3 cm precision by reducing the weight of GEO satellites down to 1/900, whereas the normal weight strategy, which can achieve 2.3 cm precision by reducing the weight of GEO satellites down equal weight strategy only achieves the 3D precision of 6.3 cm. Finally, the effect of pseudo-stochastic to 1/900, whereas the normal equal weight strategy only achieves the 3D precision of 6.3 cm. Finally, parameters is explored, and the optimal 3-dimensional orbit precision of the Luojia-1A can achieve the effect of pseudo-stochastic parameters is explored, and the optimal 3-dimensional orbit precision 2 cm. The strategy that reduces the weight of GEO satellite observations is effective for the precise of the Luojia-1A can achieve 2 cm. The strategy that reduces the weight of GEO satellite observations orbit determination of the Luojia-1A Satellite. is effective for the precise orbit determination of the Luojia-1A Satellite.

AuthorAuthor Contributions:Contributions: L.W.L.W. createdcreated thethe ideaidea andand contributedcontributed toto writingwritingthe the manuscript.manuscript. B.X.B.X. wrotewrote thethe firstfirst draftdraft and carried out the experiment. W.F., R.C., and T.L. helped to carry out the data analysis. Y.H. and H.Z. helped to and carried out the experiment. W.F., R.C., and T.L. helped to carry out the data analysis. Y.H. and H.Z. helped carry out the experiments. All authors have read and agreed to the published version of the manuscript. to carry out the experiments. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the National Natural Science Foundation of China (No. 41704002, 91638203Funding: and This 41904038) research was and supported the China Postdoctoralby the National Science Natural Foundation Science Foundation (No. 2017M620337 of China and (No. 2019M662713). 41704002 and This41904038) work isand also the partially China Postdoctoral funded by the Science Fundamental Foundation Research (No. Funds2017M for620337 the Centraland 2019M662713). Universities. This work is Conflictsalso partially of Interest: funded Theby the authors Fundamental declare no Re conflictsearch Funds of interest. for the Central Universities. Conflicts of Interest: The authors declare no conflict of interest.

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