Precise Onboard Real-Time Orbit Determination with a Low-Cost Single-Frequency GPS/BDS Receiver

remote sensing

Article Precise Onboard Real-Time Orbit Determination with a Low-Cost Single-Frequency GPS/BDS Receiver

Xuewen Gong 1,2 , Lei Guo 1,* , Fuhong Wang 1, Wanwei Zhang 1, Jizhang Sang 1, Maorong Ge 2 and Harald Schuh 2

1 School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China; [email protected] (X.G.); [email protected] (F.W.); [email protected] (W.Z.); [email protected] (J.S.) 2 Department of Geodesy, GeoForschungsZentrum (GFZ), 14473 Potsdam, Germany; [email protected] (M.G.); [email protected] (H.S.) * Correspondence: [email protected]; Tel.: +86-13387664923

Received: 12 May 2019; Accepted: 8 June 2019; Published: 11 June 2019

Abstract: The low-cost single-frequency GNSS receiver is one of the most economical and affordable tools for the onboard real-time navigation of numerous remote sensing small/micro satellites. We concentrate on the algorithm and experiments of onboard real-time orbit determination (RTOD) based on a single-frequency GPS/BDS receiver. Through various experiments of processing the real single-frequency GPS/BDS measurements from the Yaogan-30 (YG30) series and FengYun-3C (FY3C) satellites of China, some critical aspects of the onboard RTOD are investigated, such as the optimal force models setting, the effect of different measurements, and the impact of GPS/BDS fusion. The results demonstrate that a gravity model truncated to 55 55 order/degree for YG30 and 45 45 for FY3C × × and compensated with an optimal stochastic modeling of empirical accelerations, which minimize the onboard computational load and only result in a slight loss of orbit accuracy, is sufficient to obtain high-precision real-time orbit results. Under the optimal force models, the real-time orbit accuracy of 0.4–0.7 m for position and 0.4–0.7 mm/s for velocity is achievable with the carrier-phase-based solution, while an inferior real-time orbit accuracy of 0.8–1.6 m for position and 0.9–1.7 mm/s for velocity is achieved with the pseudo-range-based solution. Furthermore, although the GPS/BDS fusion only makes little change to the orbit accuracy, it increases the number of visible GNSS satellites significantly, and thus enhances the geometric distribution of GNSS satellites that help suppress the local orbit errors and improves the reliability and availability of the onboard RTOD, especially in some anomalous arcs where only a few GPS satellites are trackable.

Keywords: onboard RTOD; single-frequency GPS/BDS receiver; optimal force model; pseudo-range- and carrier-phase-based solutions; GPS/BDS fusion

1. Introduction Since the first GPS receiver flying onboard Landsat-4 in 1982 [1], GPS has become the most effective real-time navigation technique for space missions on low Earth orbit (LEO). With the dual-frequency GPS receivers, the onboard real-time navigation can obtain the 3-dimensional (3D) real-time orbit accuracy of 0.3–0.5 m level for many LEO satellites when using dual-frequency carrier-phase measurements and GPS broadcast ephemeris [2,3], while the post precise orbit accuracies are up to 2–5 cm if the precise GPS orbit and clock products are used [4,5]. However, expensive and energy-consuming dual-frequency GPS receivers are not always available onboard all LEO missions, especially for numerous low-cost remote sensing small/micro satellites [6]. For these low-cost small/micro satellites as well as those only requiring meter or sub-meter level orbit accuracy, the low-cost single-frequency GPS receiver is a

Remote Sens. 2019, 11, 1391; doi:10.3390/rs11111391 www.mdpi.com/journal/remotesensing Remote Sens. 2019, 11, 1391 2 of 23 more attractive real-time navigation tool. This has driven researches and experiments on the onboard real-time orbit determination (RTOD) with a low-cost single-frequency GPS receiver for years. Back in 1994, the US Air Force Phillips Laboratory’s Technology for Autonomous Operational Survivability (TAOS) satellite carried with a single-frequency 6-channel GPS Rockwell AST V receiver [7]. Due to the Anti-Spoofing (AS) operations of GPS, the Rockwell receiver only provided single-frequency pseudo-range and Doppler measurements. A fully autonomous spacecraft navigation system, namely Microcosm Autonomous Navigation System (MANS) developed by Microcosm, Inc., was scheduled for the onboard real-time navigation of TAOS satellite, which used the single-frequency GPS measurements as the main data source. Limited by the effect of Selective Availability (SA) at that time, the difference between the onboard real-time orbits and post precise orbits were about 58.1 m (1σ) in terms of 3D position [8]. In addition, another autonomous Orbit Control Kit (OCK) of Microcosm, Inc. also used the GPS receiver as a navigation source and it has been applied to the Surrey Satellite Technology Limited (SSTL) UoSAT-12 spacecraft in 1999 [9] and the TacSat-2 mission of US Air Force in 2006 [10]. In 1996, a GPS Attitude Determination Flyer (GADFLY) experiment was flying on the Small Satellite Technology Initiative Lewis (SSTI Lewis) spacecraft by the Goddard Space Flight Center (GSFC) Navigation, Guidance, and Control Branch, which aimed to demonstrate and validate the ability to provide precise time and to determinate spacecraft orbit and attitude using GPS [11]. A space-qualified 9-channel GPS L1 frequency, Coarse/Acquisition (C/A) code receiver was included in the GADFLY component, and the preflight performance assessments indicate that the onboard real-time navigation could provide a real-time position accuracy of better than 10 m and velocity accuracy of better than 0.01 m/s, with SA at typical levels [12]. The German small satellite mission BIRD, carried out by the German Aerospace Center (DLR) in 2001, was also equipped with a GPS Embedded Module (GEM)-S 5-channel L1 frequency, C/A- and P-code receiver [13]. Using the single-frequency GPS measurements, the onboard real-time navigation system could generate real-time orbits with the accuracy of 5.0 m for position and 6 mm/s for velocity [14]. The good performance of BIRD demonstrated that the low-cost single-frequency GPS receiver is also able to provide superior real-time orbits for other onboard Earth observation devices, which could be treated as a milestone for GPS-based onboard real-time navigation [15]. Soon afterwards, DLR developed a miniature Phoenix GPS receiver with integrated onboard real-time navigation system for LEO satellites, which aimed to achieve real-time orbit accuracy of 1.0–2.0 m using single-frequency GPS measurements [16]. The Phoenix receiver was applied to many small satellites, including the X-SAT mini-satellite developed by the Satellite Engineering Centre of the Nanyang Technological University at Singapore [17], the SunSat 2004 remote sensing micro-satellite built by Stellenbosch University of South Africa [18], and the PROBA-2 micro-satellite missions of DLR [19]. Montenbruck et al. analyzed the real-time navigation performance of Phoenix receiver onboard the PROBA-2 satellite. The results demonstrated that the real-time orbit accuracy of 1.1 m for the 3D position was achieved and the accuracy improvement to 0.7 m was feasible if using more refined filter setting on the ground [20]. The research and experiments in the past were mostly based on the single-frequency GPS measurements, because only the single-frequency GPS receiver was equipped for most small/micro satellites. Along with the development of other GNSS systems, such as Chinese BeiDou Navigation System (BDS), and the manufacture of multi-GNSS receiver, onboard RTOD using multi-GNSS measurements has been practicable. Therefore, we will focus on the onboard RTOD algorithm and experiments based on the low-cost single-frequency GPS/BDS receiver in this paper. The real space-borne single-frequency GPS/BDS data from the Yaogan-30 (YG30) series and FengYun-3C (FY3C) satellites will be processed in the experiments. The YG30 series satellites, launched in 2017, were a group of remote sensing satellites developed by the Innovation Academy for Microsatellites of Chinese Academy of Sciences. They consist of 12 small satellites distributed in orbit planes with an inclination of 35 degree, and six of them (SAT1–SAT6) will be tested in the experiments. Each YG30 satellite is equipped with a low-cost single-frequency GPS/BDS receiver that only provides single-frequency pseudo-range observations without carrier-phases output. The FY3C satellite, a Chinese meteorological satellite, Remote Sens. 2019, 11, 1391 3 of 23 was launched on 23 September 2013 and developed by the Meteorological Administration/National Satellite Meteorological Center (CMA/NSMC) of China. It was equipped with a GNSS Occultation Sounder (GNOS) instrument onboard which can provide dual-frequency GPS/BDS pseudo-range and carrier-phase observations [21], but only single-frequency GPS/BDS pseudo-range and carrier-phase measurements are employed in our experiments. Based on these real single-frequency GPS/BDS data, we will explore the optimal force models for the onboard RTOD, which should keep the balance between the real-time computational load and accuracy. The different solutions based on pseudo-range and carrier-phase measurements will be compared in detail and the real-time orbit accuracy will be analyzed in-depth. Furthermore, the effect of the GPS/BDS fusion on solutions will be investigated from several aspects including the geometry of GNSS satellites, the estimation of system bias, real-time orbit accuracy, filter convergence and reliability of the onboard RTOD. The onboard RTOD algorithm is implemented in the prototype software named SATPODS (Space-borne GNSS AuTonomous Precise Orbit Determination Software), which is use to perform all experiments. In order to assess the real-time orbit accuracy of the proposed onboard RTOD algorithm, the post precise orbits will be used as reference, which are generated by the PANDA (Positioning and Navigation Data Analyst) software [22]. In the following, the dynamical models of LEO satellites and the construction of state propagation equation will be introduced first. Secondly, GNSS measurements, the pseudo-range-based and carrier-phase-based solutions and the corresponding estimated parameters and observation equations will be elaborated in detail. Afterward, the parameter estimation by extend Kalman filter will be presented briefly. Then, a series of experiments will be shown, discussed and analyzed in detail to assess the effect of different force models, different measurement types and different GPS/BDS fusion on solutions. Finally, the conclusion is made.

2. Algorithm The onboard RTOD algorithm is elaborated from three aspects including the dynamical models, GNSS measurements and parameter estimation. The extend Kalman ﬁlter is used to estimate the unknown parameters, thus the following description is mainly related to the construction of state propagation equations, the formation of observation equations, and the linearization of Kalman ﬁlter.

2.1. Dynamical Models For the onboard RTOD, besides the measurements from the GNSS receiver, the dynamics of LEO satellites has to be considered. The motion equation of a LEO satellite can be expressed as:

.. GM . r = ⊕ r + g r, r, e, p (1) − r3 ·

. .. where r, r, r are the position, velocity and acceleration vectors in a geocentric inertial coordinate frame, respectively, e = (eR, eT, eN) denotes the empirical accelerations in radial, tangential and normal directions, which account for those un-modeled forces, and p represents the dynamical parameters in force models and mainly consists of the atmospheric drag coefficient Cd and the solar radiation pressure coefficient Cr. The first part on the right side of Equation (1) is the central gravity of the Earth, and g( ) represents other accelerations that include the non-spherical gravity of the Earth, the · lunisolar gravitational perturbation, atmospheric drag, solar radiation pressure, ocean tide, and so on. However, concerning the computation workload of onboard space-borne processers and the required real-time orbit accuracy, including full perturbations in the onboard RTOD processing is not necessary, and some force models must be simplified or can be neglected. Strategies on the dynamical models setting will be given below. The impact of force models, especially the gravity model and the empirical accelerations, will be discussed in the experiments. Remote Sens. 2019, 11, 1391 4 of 23

All parameters in the dynamical models to be estimated, p, can be included in a state vector h . i X = r, r, e, p . Based on motion Equation (1), the diﬀerential equation of the state vector is obtained:

.  .   r   r  r  0           .   ..   3 .    d  r   r   GM /r r + g r, r, e, p   0  .   =  .  =  − ⊕ ·  +   X = F(X) + u (2) dt e   e     u  X      e/τ   e  ⇔    .   −    p p 0 up where a ﬁrst-order Gauss-Markov model and a random walk model are used to express the state propagation of the empirical accelerations e and the dynamical parameters p, τ is the correlation time, 2 2 both ue and up are the white noise, and σ is the power spectral density of ue. τ and σ are essential stochastic parameters to adjust the ﬁtness of force models to measurements. From Equation (2), the discrete state propagation equation can be derived out easily:

Xk = Φ(Xk 1) + Wk (3) − where X denotes the state vector at epoch t , W is considered as the white noise, and Φ( ) represents k k k · the implicit state transition function. In general, it is diﬃcult to derive out the explicit expression of Φ( ) because of the high non-linearity of the motion equation, but it can be computed by the numerical · integration. More details of the state propagation equation are well documented in the literature [23,24].

2.2. GNSS Measurements The single-frequency GNSS receiver onboard LEO satellites could produce the single-frequency pseudo-range and carrier-phase data. The LEO satellites ﬂy on the orbits above the troposphere, so the GNSS observations are not aﬀected by the tropospheric delays. Therefore, the single-frequency pseudo-range C1 and carrier-phase L1 measurements can be expressed as:

( s S C1 = T r r + cδR cδ + I + εP1 | · − s | − S (4) L = T r r + cδR cδ I + λ N + εL 1 | · − | − − 1 1 1 where rs is the position of GNSS satellite in the Earth-fixed system, T denotes the transformation matrix from the inertial system, in which the LEO satellite orbit is determined, to the Earth-fixed system, δR S and δ represent the clock offsets of the receiver and GNSS satellite, respectively. λ1 and N1 are the wavelength and ambiguity of the single-frequency carrier-phase (GPS L1, BDS B1), respectively, and

I denotes the ionospheric error. εC1 and εL1 represent the noises of these two measurements, which include the multipath error. For some LEO satellites, such as the YG30 series satellites, only pseudo-range data is available for the onboard RTOD. The solution only using single-frequency pseudo-range measurements is abbreviated as “pseudo-range-based solution”. In this solution, the state vector is expanded to h . i X = r, r, e, p, h , which includes the receiver clock parameters h. The clock parameters h involve the . clock offset δR, the clock rate δR and the bias η between GPS and BDS system if the GPS/BDS data is fused. The bias η mainly accounts for the difference in receiver hardware delays between GPS and BDS signals [25], as well as the difference in system times between GPS and BDS [26]. Theoretically, in order to realize the fusion of GPS and BDS, the bias η, which is of high stability, should be precisely calibrated [27]. However, the onboard RTOD processing is completed on the micro-processor onboard LEO satellites, so precisely calibrated parameter on the ground will not be available for the onboard GPS/BDS receiver. The alternative method to deal with the bias is to estimate it with a reasonable stochastic model in the Kalman filter. Due to the affection of the error of GNSS broadcast ephemeris, the residual ionospheric error, the measurement noises and other uncorrectable errors, the estimated value of the bias in the onboard RTOD will show certain fluctuations, and thus the bias η is modeled by a random walk process, instead by a constant. Meanwhile, the random walk process is also used to Remote Sens. 2019, 11, 1391 5 of 23

. describe the state propagation of clock offset δR and clock rate δR parameters. Therefore, the discrete formulation of the state equation of h can be derived out as:           δR   0 1 0  δR   u1   1 ∆t 0  d  .    .        =    +  u  = + =    δR   0 0 0  δR   2  hk Φh hk 1 uk, Φh  0 1 0  (5) dt       ⇒ · −   η 0 0 0 η u3 0 0 1 where u1, u2 and u3 are all the white noise, Φh is the state transition matrix, hk represents the clock parameters at the epoch tk, ∆t = tk tk 1 denotes the interval between two epochs, and uk is treated − − as the white noise. The estimation of the bias η will be discussed in-depth in the experiments. The position rS and clock offset δS of GNSS satellite are computed by using real-time broadcast ephemeris, which are marked as (r∗, δ∗). Then, the observation equation for the pseudo-range-based solution is formed as: C = T r r∗ + c (δR δ∗) + I + ε C = p(X) + ε (6) 1 | · − | · − C1 ⇒ 1 C1 where p( ) denotes the related function between the pseudo-range measurements C and the state · 1 vector X. It should be noted that, the ionospheric error I is usually computed by the Klobuchar model with an altitude dependent scale factor [28]. For some LEO satellites, both the single-frequency carrier-phase and pseudo-range data are available and the onboard space-borne processers have enough computational capacity to process carrier-phase data, these two types of measurements can be all used for the onboard RTOD. The solution using both carrier-phase and pseudo-range measurements is called “carrier-phase-based solution” in this paper. A study into the error sources for the onboard RTOD revealed that, the orbit and clock offset errors of the GNSS broadcast ephemeris in the line-of-sight (LOS) are the main factor that limits the real-time orbit accuracy, but they could be reflected in the carrier-phase observation equation, and the total LOS error and ambiguity can be estimated as a coupled parameter, named pseudo-ambiguity, and then the real-time orbit accuracy can be improved effectively due to the absorption of the LOS error by the pseudo-ambiguity. The detail of pseudo-ambiguity has been well descripted in the literature [3]. With the single-frequency pseudo-range and carrier-phase measurements (C1, L1), the combination, known as Group and Phase Ionospheric Correction (GRAPHIC) LH, can be treated as an ionosphere-free phase measurement but with a higher noise, and then the observation equation of L is: H C1 + L1 λ1N1 LH = = T r r∗ + c (δR δ∗) + dρ cdδ + + εL (7) 2 | · − | · − − 2 H where dρ is the range error in the line-of-sight (LOS) caused by the orbit error of the GNSS broadcast ephemeris, dδ is the clock offset error of GNSS satellite, dρ cdδ is the total LOS error caused by the − GNSS broadcast ephemeris, and thus the pseudo-ambiguity is defined as: B = ρ cdδ + λ N /2. − 1 1 Obviously, for the carrier-phase-based solution, the state vector should be expanded to h . i X = r, r, e, p, h, b , where b = (B1, B2, ... , Bn) represents the n pseudo-ambiguity parameters where n is the number of tracked GNSS satellites. With the standalone receiver and GNSS broadcast ephemeris, the true ambiguity is affected by several errors, including the predominant LOS error caused by broadcast ephemeris, the un-calibrated phase delay, the measurement noises, and so on, so it cannot be estimated precisely. Actually, it is not necessary to fix the ambiguity, because the introduction of the pseudo-ambiguity, with a reasonable stochastic model, is to absorb the LOS errors and then improve the real-time orbit accuracy [3]. On the basis of the characteristic of the LOS error, a random walk process is used to express the state propagation of the pseudo-ambiguity parameters. So, the differential equation and the discrete state propagation of the pseudo-ambiguity parameters can be derived out easily: . b = ub bk = bk 1 + µk (8) ⇒ − Remote Sens. 2019, 11, 1391 6 of 23

where, ub is a white noise, bk represents the pseudo-ambiguity parameters at the epoch tk. µk is considered as white noise, too. The observation equation is formed as:

LH = T r r∗ + c (δR δ∗) + B + εL LH = l(X) + εL (9) | · − | · − H ⇔ H where l( ) denotes the related function between the carrier-phase measurements LH and the state · vector X.

2.3. Parameter Estimation An extended Kalman filter is used to estimate all unknown parameters in the dynamical model h . i and observation equation. The filter state is either X = r, r, e, p, h for the pseudo-range-based h . i solution or X = r, r, e, p, h for the carrier-phase-based solution. The dynamical equation of Kalman filter can be set up directly by combining the state propagation equations of these to-be-estimated parameters, such as Equations (3), (5), and (8). The observation equation of Kalman filter is either pseudo-range Equation (6) or carrier-phase Equation (9). Obviously, for the pseudo-range-based and carrier-phase-based solutions, the filter equations are different, due to the different used measurements and to-be-estimated parameters. In any case, the unified formula of Kalman filter and its linearization can be derived out:  ∂Xk (  Xk = Φ X∗ + Xk 1 X∗ + wk x = x + w  k 1 ∂Xk 1 k 1 k φk,k 1 k 1 k − − − − − − − (10)  ∂Yk  Yk = H X∗ + Xk X∗ + vk ⇒ yk = hkxk + vk k ∂Xk − k where, Y is the measurement vector, Φ( ) and H( ) represent the dynamical and observation functions, k · · respectively, and wk is the process noise, which is usually treated as the white noise but with different variance for different to-be-estimated parameters. The linearization is based on the nominal filter ˆ state Xk∗ 1 at the epoch tk 1, which usually adopts the estimated state Xk 1 to reduce the linearization − − − error. X = Φ X is the one-step predicted state from epoch t t . φ = ∂X /∂X is the k∗ k∗ 1 k 1 k k,k 1 k k 1 − − → − − state transition matrix. Both X∗ and φk,k 1 are computed by numerical integration. xk = Xk X∗ is k − − k the estimated correction of the filter state Xk relative to the predicted state X∗. H X∗ is the computed k k measurements and y = Y H X∗ is the prior residuals equal to the observed measurements minus k k − k computed measurements. hk is the design matrix, also known as observation matrix and vk is the observation error. The detailed proceeding of the extended Kalman filter is documented in the literatures [23,24].

3. Experiments The above onboard RTOD algorithm is implemented in the prototype software named SATPODS. Thanks to the YG30 series and FY3C satellites, the real single-frequency GPS/BDS data from the space-borne GNSS receivers is available for the experiments. Following the introduction about the employed datasets and comparison between the onboard RTOD and post POD strategies, the reference orbits used for real-time orbit accuracy assessment will be analyzed in detail. Then various experiments will be conducted to assess the effects of different force models, different measurement types and different GPS/BDS fusion on the onboard RTOD solutions.

3.1. Datasets and Orbit Determination Strategies The real single-frequency GPS/BDS data, which covers two sessions, from 2017/340 to 2017/345 and from 2015/69 and 2015/75, respectively, for the YG30/SAT1–SAT6 satellites and FY3C satellite, are processed in the experiments. It should be noted that the GNSS data of SAT5 and SAT6 satellites on day 2017/345 is not complete, and only the data at the first few hours are available. The GNSS receivers onboard the YG30/SAT1–SAT6 satellites only provide single-frequency pseudo-range measurements, Remote Sens. 2019, 11, 1391 7 of 23 so only the single-frequency GPS/BDS data from the FY3C satellite, including both carrier-phase and pseudo-range measurements, is tested, even if the data is from the dual-frequency GNSS receiver. The SATPODS software is capable of processing both single-frequency pseudo-range and carrier-phase measurements and simulating the onboard operational scenarios as realistically as possible. In order to assess the real-time orbit accuracies of SATPODS, the orbit results generated by the precise orbit determination (POD) software PANDA are used as reference. The algorithm of the onboard RTOD is different from that of the post POD. The strategies employed by the two algorithms are shown in Table1. The di fference between the onboard RTOD and post POD exists in many aspects. Firstly, only the GPS/BDS broadcast ephemeris is available, and the pseudo-ambiguities will be estimated instead of the true constant ambiguities for the onboard RTOD. Then, the dynamical models are set as complete and precise as possible to get high-precision orbit results for the post POD. In contrast, the dynamical models of the onboard RTOD are simplified to the maximum extent in order to reduce the computational load, at the same time without notable loss of orbit accuracy. Similarly, compared to complex precession and nutation models and precise Earth orientation parameter (EOP) products used in the post POD, only simplified models and rapid predicted EOP products are applied for the transformation of coordinate systems for the onboard RTOD. Of course, the most striking difference is that the onboard RTOD always uses the EKF filter, and processes data in real-time mode, while the post POD uses the least square estimator in batch mode.

Table 1. Comparison of onboard real-time orbit determination (RTOD) and post POD strategies.

Model Onboard RTOD (SATPODS) Post POD (PANDA) Measurement model Single-frequency GPS/BeiDou Navigation System (BDS) phase Dual-frequency GPS/BDS phase and pseudo-range measurements and pseudo-range measurements (LH + C1) for the FY3C satellite; for the FY3C satellite; and GNSS data and single-frequency single-frequency pseudo-range pseudo-range data (C1) for the data (C1) for the YG30/SAT1–SAT6 YG30/SAT1–SAT6 satellites satellites (interval 10 s) (interval 30 s) IGS MGEX final precise orbit and GNSS orbit and clock Broadcast ephemeris clock products Epoch-wise receiver clock offset A receiver clock offset and a bias Receiver clock and a bias between GPS and BDS between GPS and BDS system system Pseudo-Ambiguity with a random Real constant value for each Ambiguity walk process ambiguity pass Dynamical model Truncated EGM 2008 with low EIGEN-6S, adopt 120 120, Earth gravity field order and degree, neglect the × include the time-varying part time-varying part Moon and Sun only, low precision Moon, Sun and other planets, JPL N-body gravitation analytic method (position) DE405 (position) Remote Sens. 2019, 11, 1391 8 of 23

Table 1. Cont.

Model Onboard RTOD (SATPODS) Post POD (PANDA)

Solid earth tide Simplified model, k20 solid only IERS Conventions 2010 Earth pole tide Neglected IERS Conventions 2010 Ocean tide Neglected FES2004 Ocean pole tide Neglected IERS Conventions 2010 Relativistic effects Neglected IERS Conventions 2010 Modified Harris-Priester model DTM94 model (density), macro (density), fixed effective area, drag Atmosphere drag model, drag coefficients every 360 coefficient with a random walk min process Cannonball model, fixed effective Macro model, radiation pressure Solar radiation area, radiation pressure coefficient coefficients every 360 min with a random walk process Macro model, radiation pressure Earth radiation Neglected coefficients every 360 min three empirical accelerations in One-cycle-per-orbit-revolution Empirical acceleration radial, along-track and cross with (1CPR) empirical accelerations in a first-order Gauss-Markov model radial, along-track and cross Reference frame Coordinate system WGS84/CGCS2000 ITRF 2008/ITRF 2014 IAU1976/IAU 1980 simplified Precession/nutation IAU 2006/IAU 2000R06 model model Rapid predicted EOP in IERS Earth rotation parameter IERS final EOP products Bulletin A Estimation Estimator EKF filter Least square Mode Sequential processing in real-time Batch processing

3.2. Reference Orbits It is necessary to assess the post orbit accuracy firstly before using the post POD orbits of PANDA as reference. For the FY3C satellite with dual-frequency GPS/BDS receiver, the post POD orbits are generated by processing dual-frequency carrier-phase and pseudo-range measurements. Figure1 shows the root mean square (RMS) statistics of the radial (R), along-track (A), cross (C), 1-dimensional (1D) and 3-dimensional (3D) orbit differences in the middle 5h-overlap arc between two 30 h POD arcs, and the daily RMS statistics of the difference between the post POD orbits and the external precise orbit products released by CMA/NSMC [5]. As can be observed from sub-graph (a), the 1D and 3D overlap differences are about 0.5–2.0 cm and 1.5–3.0 cm, respectively. Sub-graph (b) illustrates the 1D and 3D orbit differences from the external precise orbit products are about 1.0–2.0 cm and 2.5–3.5 cm, respectively, except for a poor performance on day 2015/74 due to less tracked GPS satellites over the arc of 16–24 h. Obviously, the post orbit accuracy of FY3C satellite is at the centimeter level (2–4 cm), so the post POD orbits can be treated as reference to assess the real-time orbit accuracy of SATPODS. Remote Sens. 2019, 11, 1391 9 of 23 Remote Sens. 2019, 11, x FOR PEER REVIEW 9 of 23

] 3 9 R A C 1D 3D R A C 1D 3D 2 6

1 3

0 0 70/71 71/72 72/73 73/74 69 70 71 72 73 74 75 Overlap Orbit Difference[cm Overlap Time [d/d] [cm] External Difference Orbit DOY [d]

(a) (b)

FigureFigure 1.1. TheThe postpost orbitorbit accuracyaccuracy assessmentassessment ofof thethe FY3CFY3C satellitesatellite usingusing dual-frequencydual-frequency carrier-phasecarrier-phase andand pseudo-rangepseudo-range measurements measurements in in 2015 2015/69/69–75:−75: (a )(a The) The RMS RMS statistics statistics of theof the radial radial (R), along-track(R), along-track (A), cross(A), cross (C), 1-dimensional(C), 1-dimensional (1D) (1D) and 3-dimensionaland 3-dimensional (3D) (3D) orbit orbit diﬀerence difference in the in 5h-overlap the 5h-overlap arcs; arcs; (b) The (b) dailyThe daily RMS RMS statistics statistics of the of R the/A/ CR/A/C/1D/3D/1D/3D orbit diorbitﬀerence difference from externalfrom external precise precise orbit products.orbit products.

ForFor the YG30/SAT1 YG30/SAT1–SAT6−SAT6 satellites, satellites, it itis isdifficult difficult to toobtain obtain high-precision high-precision post post POD POD orbits orbits with withaccuracies accuracies at the at thecentimeter centimeter level, level, because because the the space-borne space-borne GPS/BDS GPS/BDS receivers receivers only provideprovide single-frequencysingle-frequency pseudo-rangepseudo-range data.data. It seemsseems impossibleimpossible toto assessassess thethe absoluteabsolute accuracyaccuracy ofof thethe postpost PODPOD orbits,orbits, duedue toto lacklack ofof externalexternal preciseprecise orbitorbit products.products. TheThe RMSRMS statisticsstatistics ofof thethe overlapoverlap orbitorbit didifferencesfferences ofof thethe YG30 YG30/SAT1/SAT1–SAT6−SAT6 satellites satellites are are all all shown shown in in sub-graph sub-graph (a) (a) and and (b) (b) of of Figure Figure2. As2. As a comparison,a comparison, the the RMSs RMSs of overlap of overlap and external and external orbit di orfferencesbit differences of the FY3C of the satellite FY3C are satellite also illustrated are also inillustrated sub-graph in (c)sub-graph and (d), where(c) and the (d), precise where orbits the areprecise generated orbits byare using generated only single-frequency by using only pseudo-rangesingle-frequency measurements. pseudo-range As measurements. can be observed As from can sub-graphs be observed (c) from and (d),sub-graphs the RMSs (c) of and 1D and(d), 3Dthe overlapRMSs of orbit 1D and differences 3D overlap of the orbit FY3C differ satelliteences are of aboutthe FY3C 4–7 cmsatellite and 7–11are about cm, respectively, 4−7 cm and and 7−11 those cm, ofrespectively, 1D and 3D and external those orbit of 1D di andfferences 3D external 5–8 cm orbit and 12–14differences cm, respectively. 5−8 cm and Sub-graphs12−14 cm, respectively. (a) and (b) illustrateSub-graphs the (a) RMSs and of (b) overlap illustrate orbit the di RMSsfferences of overlap of SAT1–SAT6 orbit differences satellites areof allSAT1-SAT6 of similar satellites magnitude, are andall of the similar 1D and 3Dmagnitude, differences and are aboutthe 1D 7–11 and cm 3D and differences 10–20 cm, respectively.are about 7 It−11 may cm be possibleand 10− to20 givecm, anrespectively. external accuracy It may be indicator possible for to the give YG30 an external series satellites accuracy by indicator applying for an the RMS YG30 ratio series parameter satellites of FY3Cby applying satellite. an The RMS FY3C ratio RMS parameter ratio is of defined FY3C assatellit the ratioe. The of FY3C the RMS RMS of ratio the external is defined orbit as ditheff erencesratio of overthe RMS the RMSof the of external the overlap orbit orbit differences differences. over the If the RMS YG30 of the series overlap satellites orbit are differences. assumed toIf the have YG30 the sameseries RMS satellites ratio are of theassumed FY3C satellite,to have the then same the RMS RMSs ratio of the of “external”the FY3C satellite, orbit diff thenerences the ofRMSs the YG30of the series“external” satellites orbit are differences inferred from of the the YG30 RMSs series of the overlapsatellites orbit are diinferredfferences, from even the if RMSs the external of theorbits overlap of STA1–SAT6orbit differences, satellites even don’t if the exist. external orbits of STA1−SAT6 satellites don’t exist.

R A C 1D 3D R A C 1D 3D 20 SAT1 20 SAT4 10 10 0 0 20 SAT2 20 SAT5 10 10 0 0 20 SAT3 20 SAT6 Overlap Orbit Difference [cm] 10 Overlap Orbit Difference [cm] 10

0 0 340/341 341/342 342/343 343/344 344/345 340/341 341/342 342/343 343/344 Time [d/d] Time [d/d]

(a) (b)

Remote Sens. 2019, 11, x FOR PEER REVIEW 9 of 23

] 3 9 R A C 1D 3D R A C 1D 3D 2 6

1 3

0 0 70/71 71/72 72/73 73/74 69 70 71 72 73 74 75 Overlap Orbit Difference[cm Overlap Time [d/d] [cm] External Difference Orbit DOY [d]

(a) (b)

Figure 1. The post orbit accuracy assessment of the FY3C satellite using dual-frequency carrier-phase and pseudo-range measurements in 2015/69−75: (a) The RMS statistics of the radial (R), along-track (A), cross (C), 1-dimensional (1D) and 3-dimensional (3D) orbit difference in the 5h-overlap arcs; (b) The daily RMS statistics of the R/A/C/1D/3D orbit difference from external precise orbit products.

For the YG30/SAT1−SAT6 satellites, it is difficult to obtain high-precision post POD orbits with accuracies at the centimeter level, because the space-borne GPS/BDS receivers only provide single-frequency pseudo-range data. It seems impossible to assess the absolute accuracy of the post POD orbits, due to lack of external precise orbit products. The RMS statistics of the overlap orbit differences of the YG30/SAT1−SAT6 satellites are all shown in sub-graph (a) and (b) of Figure 2. As a comparison, the RMSs of overlap and external orbit differences of the FY3C satellite are also illustrated in sub-graph (c) and (d), where the precise orbits are generated by using only single-frequency pseudo-range measurements. As can be observed from sub-graphs (c) and (d), the RMSs of 1D and 3D overlap orbit differences of the FY3C satellite are about 4−7 cm and 7−11 cm, respectively, and those of 1D and 3D external orbit differences 5−8 cm and 12−14 cm, respectively. Sub-graphs (a) and (b) illustrate the RMSs of overlap orbit differences of SAT1-SAT6 satellites are all of similar magnitude, and the 1D and 3D differences are about 7−11 cm and 10−20 cm, respectively. It may be possible to give an external accuracy indicator for the YG30 series satellites by applying an RMS ratio parameter of FY3C satellite. The FY3C RMS ratio is defined as the ratio of the RMS of the external orbit differences over the RMS of the overlap orbit differences. If the YG30 series satellites are assumed to have the same RMS ratio of the FY3C satellite, then the RMSs of the Remote“external” Sens. 2019 orbit, 11 ,differences 1391 of the YG30 series satellites are inferred from the RMSs of the overlap10 of 23 orbit differences, even if the external orbits of STA1−SAT6 satellites don’t exist.

R A C 1D 3D R A C 1D 3D 20 SAT1 20 SAT4 10 10 0 0 20 SAT2 20 SAT5 10 10 0 0 20 SAT3 20 SAT6 Overlap Orbit Difference [cm] 10 Overlap Orbit Difference [cm] 10

0 0 340/341 341/342 342/343 343/344 344/345 340/341 341/342 342/343 343/344 Time [d/d] Time [d/d]

Remote Sens. 2019, 11, x FOR PEER REVIEW(a) (b) 10 of 23

R A C 1D 3D R A C 1D 3D 15 15 FY3C FY3C

10 10

5 5

0 0 70/71 71/72 72/73 73/74 70 71 72 73 Overlap Orbit Difference [cm] Orbit Difference Overlap Time [d/d] Time [d] External Orbit Difference [cm]

Figure 2.2. The post orbit accuracy assessment of the YG30/SAT1 YG30/SAT1–SAT6−SAT6 and FY3CFY3C satellitessatellites usingusing single-frequency pseudo-rangepseudo-range measurements measurements (C1): (C1): (a) The(a) RMSThe statisticsRMS statistics of the radial of the (R), along-trackradial (R), (A),along-track cross (C), (A), 1-dimensional cross (C), 1-dimensional (1D) and 3-dimensional (1D) and 3-dimensional (3D) overlap (3D) orbit overlap difference orbit of difference the SAT1–SAT3 of the satellites;SAT1−SAT3 (b) Thesatellites; RMS statistics(b) The ofRMS the Rstatistics/A/C/1D /3Dof the overlap R/A/C/1D/3D orbit difference overlap of the orbit SAT4–SAT6 difference satellites; of the (SAT4c) The−SAT6 RMS satellites; statistics of(c) the The R /RMSA/C/ 1Dstatistics/3D overlap of the orbitR/A/C/1D/3D difference overlap of the FY3Corbit difference satellite; ( dof) Thethe FY3C RMS statisticssatellite; of(d the) The R/ ARMS/C/1D statistics/3D orbit of di fftheerences R/A/C/1D/3D of the FY3C orbit satellite differences from external of the precise FY3C orbitsatellite products. from external precise orbit products. Table2 summarizes the overall statistics of the overlap and external orbit di fference of the FY3C satellites.Table 2 The summarizes RMS of the the 3D overall overlap statistics orbit of di fftheerences overlap is 8.9and cm, external while orbit that difference of the corresponding of the FY3C externalsatellites. orbit The diRMSfferences of the is 3D 13.2 over cm.lap It illustratesorbit differences that the is true 8.9 orbitcm, while external that accuracy of the corresponding is worse than theexternal internal orbit overlap differences accuracy. is 13.2 Moreover, cm. It illustrates it is reasonable that the to true believe orbit that, external the post accuracy orbit accuracy is worse of than the FY3Cthe internal satellite overlap is at about accuracy. 10–15 Moreover, cm level if it only is reasonable using single-frequency to believe that, pseudo-range the post orbit measurements. accuracy of Thethe ratiosFY3C ofsatellite the external is atover about the overlap10−15 cm statistics level valuesif only are using respectively single-frequency 1.35, 1.59 andpseudo-range 1.33 in the Rmeasurements./A/C directions, The and ratios the of 1D the and external 3D ratios over are the both overlap 1.49. statistics Applying values these are ratios, respectively the RMSs 1.35, of 1.59 the “external”and 1.33 in orbit the R/A/C differences directions, of the YG30and the/SAT1–SAT6 1D and 3D satellites ratios are in Rboth/A/C 1.49./1D/ 3DApplying directions these are ratios, obtained, the whichRMSs areof the listed “external” in Table2 ,orbi too.t Thedifferences inferred of “external” the YG30/SAT1 orbit accuracies−SAT6 satellites are about in 10–15R/A/C/1D/3D cm and 15–26directions cm inare terms obtained, of 1D which and 3D are orbit listed di infferences, Table 2, respectively.too. The inferred It may “external” be concluded orbit accuracies that the true are postabout orbit 10−15 accuracies cm and of15− the26 YG30cm in/ SAT1–SAT6terms of 1D satellites and 3D areorbit at differences, about 20–25 respectively. cm level if onlyIt may using be single-frequencyconcluded that the pseudo-range true post orbit measurements. accuracies of Compared the YG30/SAT1 to the FY3C−SAT6 satellite, satellites the are inferior at about post 20 POD−25 orbitscm level are achievedif only using for the single-frequency YG30/SAT1–SAT6 pseudo-range satellites. The measurements. reason of the poorCompared POD performance to the FY3C is probablysatellite, the that inferior the pseudo-range post POD measurementorbits are achieved noises for of thethe single-frequencyYG30/SAT1−SAT6 GPS satellites./BDS receiver The reason onboard of thethe YG30poor/ SAT1–SAT6POD performance satellites areis probably higher than that that the of thepseudo-range dual-frequency measurement GPS/BDS receiver noises onboard of the thesingle-frequency FY3C satellite. GPS/BDS Figure3 receiverpresents onboard the RMS the statistics YG30/SAT1 of single-frequency−SAT6 satellites pseudo-range are higher than residuals that of afterthe dual-frequency POD processing. GPS/BDS The pseudo-range receiver onboard residuals th ofe theFY3C FY3C satellite. satellites Figure are at about3 presents 0.3–0.7 the m level,RMS whilestatistics those of ofsingle-frequency the YG30/SAT1–SAT6 pseudo-range satellites areresiduals about 0.8–1.8after POD m. processing. The pseudo-range residuals of the FY3C satellites are at about 0.3−0.7 m level, while those of the YG30/SAT1−SAT6 satellites are about 0.8−1.8 m.

Table 2. The overall statistics of overlap and external orbit difference and the inferred orbit accuracy of the FY3C and YG30/SAT1−SAT6 satellites.

Overlap Orbit Difference External Orbit Difference (cm) 3D Orbit SAT (cm) Accuracy (cm) R A C 1D 3D R A C 1D 3D FY3C 2.4 6.7 5.3 5.1 8.9 3.3 10.7 7.0 7.6 13.2 10−15 Ratio 1.35 1.59 1.33 1.49 1.49 SAT1 2.9 8.5 9.7 7.7 13.3 ~3.9 ~13.6 ~12.9 ~11.4 ~19.7 SAT2 5.8 14.5 7.6 10.0 17.4 ~7.8 ~23.2 ~10.0 ~14.9 ~25.9 SAT3 3.6 9.4 9.6 8.0 13.9 ~4.9 ~14.9 ~12.7 ~11.9 ~20.6 ~20−25 SAT4 4.0 12.0 8.8 8.9 15.5 ~5.4 ~19.2 ~11.7 ~13.3 ~23.0 SAT5 3.6 9.8 6.5 7.1 12.3 ~4.8 ~15.6 ~8.7 ~10.6 ~18.3 SAT6 4.1 12.2 10.4 9.6 16.6 ~5.6 ~19.5 ~13.8 ~14.2 ~24.6

Remote Sens. 2019, 11, 1391 11 of 23

Table 2. The overall statistics of overlap and external orbit diﬀerence and the inferred orbit accuracy of the FY3C and YG30/SAT1–SAT6 satellites.

Overlap Orbit Diﬀerence (cm) External Orbit Diﬀerence (cm) 3D Orbit SAT Accuracy (cm) R A C 1D 3D R A C 1D 3D FY3C 2.4 6.7 5.3 5.1 8.9 3.3 10.7 7.0 7.6 13.2 10–15 Ratio 1.35 1.59 1.33 1.49 1.49 SAT1 2.9 8.5 9.7 7.7 13.3 ~3.9 ~13.6 ~12.9 ~11.4 ~19.7 SAT2 5.8 14.5 7.6 10.0 17.4 ~7.8 ~23.2 ~10.0 ~14.9 ~25.9 SAT3 3.6 9.4 9.6 8.0 13.9 ~4.9 ~14.9 ~12.7 ~11.9 ~20.6 ~20–25 SAT4 4.0 12.0 8.8 8.9 15.5 ~5.4 ~19.2 ~11.7 ~13.3 ~23.0 SAT5 3.6 9.8 6.5 7.1 12.3 ~4.8 ~15.6 ~8.7 ~10.6 ~18.3 SAT6 4.1 12.2 10.4 9.6 16.6 ~5.6 ~19.5 ~13.8 ~14.2 ~24.6 Remote Sens. 2019, 11, x FOR PEER REVIEW 11 of 23

1.5 FY3C 1.0 0.5 C1 Residual [m] C1 Residual 0.0 SAT1-SAT6 1.5 1.0 0.5

C1 Residual C1 Residual [m] 0.0 C02 C04 C06 C08 C10 C12 C14 G02 G04 G06 G08 G10 G12 G14 G16 G18 G20 G22 G24 G26 G28 G30 G32 Satellite FigureFigure 3.3. TheThe RMS RMS statistics statistics of single-frequency ps pseudo-rangeeudo-range residuals for thethe FY3CFY3C andand YG30YG30/SAT1/SAT1–SAT6−SAT6 satellites. satellites.

InIn summary,summary, thethe referencereference orbitsorbits ofof thethe FY3CFY3C satellitesatellite areare generatedgenerated byby usingusing dual-frequencydual-frequency carrier-phasecarrier-phase and and pseudo-range pseudo-range measurements, measurements, and and the orbitthe accuracyorbit accuracy is at 2–4 is cmat level.2−4 cm The level. reference The orbitsreference of the orbits YG30 of/SAT1–SAT6 the YG30/SAT1 satellites−SAT6 are satellites generated are by generated processing by single-frequency processing single-frequency pseudo-range measurementspseudo-range measurements and the orbit accuracies and the orbit are ataccuracies 20–25 cm are level. at 20−25 cm level.

3.3. Force Model Trade-off 3.3. Force Model Trade-off Among all perturbations on the LEO satellite orbits, the gravity of the Earth is absolutely a Among all perturbations on the LEO satellite orbits, the gravity of the Earth is absolutely a dominant factor for the overall computational load and real-time orbit accuracy. An Earth gravity dominant factor for the overall computational load and real-time orbit accuracy. An Earth gravity model of high order/degree is required to obtain high-precision real-time orbit results, but this would model of high order/degree is required to obtain high-precision real-time orbit results, but this lead to unacceptable onboard computing requirements. Therefore, it is necessary to truncate the would lead to unacceptable onboard computing requirements. Therefore, it is necessary to truncate Earth gravity model that reduces the onboard computational load and prevents notable loss of orbit the Earth gravity model that reduces the onboard computational load and prevents notable loss of accuracy at the same time. In addition, the empirical accelerations are estimated with a first-order orbit accuracy at the same time. In addition, the empirical accelerations are estimated with a Gauss-Markov model to compensate the inaccuracy of the force models. The correlated time and the first-order Gauss-Markov model to compensate the inaccuracy of the force models. The correlatedτ 2 power spectral density σ of the process noise2 in the first-order Gauss-Markov model are essential time τ and the power spectral density σ of the process noise in the first-order Gauss-Markov stochastic parameters to make the force models fit with the measurements. Therefore, several tests model are essential stochastic parameters to make the force models fit with the measurements. with different sizes of gravity models and different stochastic parameters τ, σ2 are conducted to Therefore, several tests with different sizes of gravity models and different stochastic parameters assess the impact of force models with different qualities on orbit determination accuracy, and to ()τσ, 2 are conducted to assess the impact of force models with different qualities on orbit identify a reasonable force model for the FY3C and YG30/SAT1–SAT6 satellites. The EGM2008 gravity modeldetermination is used inaccuracy, the tests. and The orderto identify/degree ofa thereasonable model variesforce frommodel 5 5for to 100the FY3C100 inand an × × increaseYG30/SAT1 of 5−SAT6 orders satellites. and degrees, The andEGM2008 thus theregravity are model 18 truncated is used models in the tests. tested. The At order/degree the same time, of 4the sets model of stochastic varies from parameters 5 × 5 to 100τ, ×σ 2100are in employedan increase for of 5 each orders truncated and degrees, gravity and model, thus there namely are (1)18 2 14 2 5 2 16 2 5 ()τσ, 2 18 2 5 τtruncated= 1 s, σ models= 1 10 tested.− m At/s the; (2) sameτ = ti30me,s, σ4 sets= 1 of stochastic10− m / sparameters; (3) τ = 60 s, σ = are1 employed10− m / fors ; ×2 20 2 5 21425× − 2× − 1625 (4) τ = 900 s, σ = 1 10− m /s , whichτσ==× are1,s abbreviated 1 10 respectivelyms τσ as==×30 “M1”,s , “M2”, 1 10 “M3”ms and each truncated gravity× model, namely (1) ; (2) ; (3) − − τσ==×60s ,21825 1 10 ms; (4) τσ==×900s ,2 1 10 2025ms, which are abbreviated respectively as “M1”, “M2”, “M3” and “M4”. In these four sets of parameters, the one resulting in the best orbit accuracy is marked as “M” for each truncated gravity model. It should be noted that, both GPS and BDS measurements are used in the tests. Figure 4 shows the real-time orbit accuracies in terms of the 3D position for the FY3C satellite under the force models of different settings, and sub-graphs (a) and (b) are corresponding to the carrier-phase-based and pseudo-range-based solutions, respectively. In the bottom panel of sub-graph (a), each curve represents a set of stochastic parameters ()τσ, 2 and 18 truncated gravity models, and 4 curves correspond to M1, M2, M3 and M4. As can be seen, if the size of the gravity model is 15 × 15 or 20 × 20, the optimal stochastic parameters resulting in the best real-time orbit accuracy are M2. When the size is more than 35 × 35, the optimal setting is M3. The real-time orbit accuracies under different sizes of gravity models with the corresponding optimal stochastic parameters setting are listed in the top panel. As can be observed clearly, when the size increases from 5 × 5 to 45 × 45, the real-time orbit accuracy is improved from 2.730 m to 0.527 m monotonically. If the size increases from 45 × 45 to 100 × 100, the real-time orbit accuracy varies

Remote Sens. 2019, 11, 1391 12 of 23

“M4”. In these four sets of parameters, the one resulting in the best orbit accuracy is marked as “M” for each truncated gravity model. It should be noted that, both GPS and BDS measurements are used in the tests. Figure4 shows the real-time orbit accuracies in terms of the 3D position for the FY3C satellite under the force models of different settings, and sub-graphs (a) and (b) are corresponding to the carrier-phase-based and pseudo-range-based solutions, respectively. In the bottom panel of sub-graph (a), each curve represents a set of stochastic parameters τ, σ2 and 18 truncated gravity models, and 4 curves correspond to M1, M2, M3 and M4. As can be seen, if the size of the gravity model is 15 15 or 20 20, the optimal stochastic parameters resulting in the best real-time orbit accuracy are × × M2. When the size is more than 35 35, the optimal setting is M3. The real-time orbit accuracies × under different sizes of gravity models with the corresponding optimal stochastic parameters setting are listed in the top panel. As can be observed clearly, when the size increases from 5 5 to 45 45, × × the real-time orbit accuracy is improved from 2.730 m to 0.527 m monotonically. If the size increases Remote Sens. 2019, 11, x FOR PEER REVIEW 12 of 23 from 45 45 to 100 100, the real-time orbit accuracy varies from 0.527 m to 0.513 m, where the × × accuracy is only marginally improved by 1.4 cm. Obviously, it is not worthy to achieve this slight from 0.527 m to 0.513 m, where the accuracy is only marginally improved by 1.4 cm. Obviously, it is accuracy improvement of less than 2.0 cm with the greatly increased onboard computational load not worthy to achieve this slight accuracy improvement of less than 2.0 cm with the greatly caused by the large size of the gravity model. Therefore, 45 45 is the minimal order and degree, increased onboard computational load caused by the large size× of the gravity model. Therefore, 45 × namely the most time-saving size, to get a reasonable orbit result without sacrificing the real-time orbit 45 is the minimal order and degree, namely the most time-saving size, to get a reasonable orbit accuracy notably ( 2 cm). Sub-graph (b) is similar to (a), where 45 45 is the optimal size and M3 result without sacrificing≤ the real-time orbit accuracy notably (≤ 2 ×cm). Sub-graph (b) is similar to is the optimal stochastic parameters setting, too. As a whole, the gravity model of 45 45 together (a), where 45 × 45 is the optimal size and M3 is the optimal stochastic parameters setting,× too. As a with the stochastic parameters of M3 are the optimal force model setting for the FY3C satellite, which whole, the gravity model of 45 × 45 together with the stochastic parameters of M3 are the optimal results in the superior real-time orbit accuracies of 0.527 m and 0.847 m for the carrier-phase-based and force model setting for the FY3C satellite, which results in the superior real-time orbit accuracies of pseudo-range-based solutions, respectively. 0.527 m and 0.847 m for the carrier-phase-based and pseudo-range-based solutions, respectively.

4 4 M M 3 3

2 2 ≤ ≤  ±2cm 1   ±2cm 1  Orbit Accuracy [m] 0 Orbit Accuracy [m] 0 6 M1 M2 M3 M4 6 M1 M2 M3 M4

4 4

2 2 Orbit Accuracy [m] Accuracy Orbit 0 [m] Accuracy Orbit 0 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100 Gravity Order and Degree Gravity Order and Degree

(a) (b)

FigureFigure 4.4. ComparisonComparison ofof real-timereal-time orbitorbit accuraciesaccuracies ofof thethe FY3CFY3C satellitesatellite underunder didifferentfferent orderorder/degree/degree of truncated gravity models (15×15~100×100) and different stochastic parameters setting (M1: of truncated gravity models (15 15~100 100) and different stochastic parameters setting (M1: 21425− × × 2− 1625 21825− ττσ===×11,ss, σ2 = 11 1010 14msm2/s;5 ;M2: M2: ττσ===×3030ss, ,σ2 = 1 1010 16msm2/s5;; M3:M3: ττσ===×6060s,s ,σ2 = 1 110 18 msm2/s5;; × − × − × − 22− 202520 2 5 M4: ττσ===×900900ss, ,σ = 11 1010− msm /s ): (a) The position accuracy of the carrier-phase-based solution; M4: × ): (a) The position accuracy of the carrier-phase-based solution; (b) The position accuracy of the pseudo-range-based solution. (b) The position accuracy of the pseudo-range-based solution. The real-time orbit accuracies of the YG30/SAT1–SAT6 satellites under different sizes of the gravity The real-time orbit accuracies of the YG30/SAT1−SAT6 satellites under different sizes of the model are shown in Figure5. For each size, only the orbit accuracy corresponding to the optimal gravity model are shown in Figure 5. For each size, only the orbit accuracy corresponding to the stochastic parameters setting M is presented here. Similarly, for the YG30/SAT1–SAT6 satellites, the optimal stochastic parameters setting M is presented here. Similarly, for the YG30/SAT1–SAT6 increase of the size after 55 55 only improves the orbit accuracy by less than 2.0 cm, so the most satellites, the increase of the× size after 55 × 55 only improves the orbit accuracy by less than 2.0 cm, reasonable size of the gravity model is 55 55. Moreover, with the size of 55 55, the optimal stochastic so the most reasonable size of the gravity× model is 55 × 55. Moreover, with× the size of 55 × 55, the parameters setting is all M3. With the optimal force models, the real-time 3D orbit accuracies of 1.143 m, optimal stochastic parameters setting is all M3. With the optimal force models, the real-time 3D orbit accuracies of 1.143 m, 1.140 m, 1.144 m, 1.171 m, 1.1200 m and 1.197 m are achieved for the SAT1−SAT6 satellites, respectively. Compared to the FY3C satellite, a slightly bigger size of the gravity model is required for the YG30 series satellites. The reason for the difference is that the YG30 satellites are at a lower altitude of about 600 km compared to the FY3C at 836 km, which would require higher order/degree of the gravity model to account for the stronger gravity perturbation. In the following experiments, the gravity model is truncated to 45 × 45 and 55 × 55 for the FY3C and YG30 series satellites, respectively, and the stochastic parameters of empirical τσ==×60s ,21825 1 10− ms accelerations are all set as M3: .

Remote Sens. 2019, 11, 1391 13 of 23

1.140 m, 1.144 m, 1.171 m, 1.1200 m and 1.197 m are achieved for the SAT1–SAT6 satellites, respectively. Compared to the FY3C satellite, a slightly bigger size of the gravity model is required for the YG30 series satellites. The reason for the diﬀerence is that the YG30 satellites are at a lower altitude of about 600 km compared to the FY3C at 836 km, which would require higher order/degree of the gravity model to account for the stronger gravity perturbation. In the following experiments, the gravity modelRemote is Sens. truncated 2019, 11 to, x FOR 45 PEER45 and REVIEW 55 55 for the FY3C and YG30 series satellites, respectively, and 13 the of 23 × × 2 18 2 5 stochastic parameters of empirical accelerations are all set as M3: τ = 60 s, σ = 1 10− m /s . ×

7 7 6 SAT1 SAT2 SAT3 6 SAT4 SAT5 SAT6 5 5 4 4 3 size≥55×55;M=M3 3 size≥55×55;M=M3 ≤ ≤ 2  ±2cm 2  ±2cm 1 1 Orbit Accuracy [m] Accuracy Orbit 0 [m] Accuracy Orbit 0 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100 Gravity Order and Degree Gravity Order and Degree

(a) (b)

FigureFigure 5. Comparison5. Comparison of real-timeof real-time orbit orbit accuracies accuracies of theof the YG30 YG30/SAT1-SAT6/SAT1–SAT6 satellites satellites under under diﬀ differenterent forceforce models: models: (a) ( Thea) The position position accuracy accuracy of theof the SAT1–SAT3 SAT1-SAT3 satellites; satellites; (b) ( Theb) The position position accuracy accuracy of theof the SAT4–SAT6SAT4-SAT6 satellites. satellites.

3.4.3.4. Eﬀ Effectect of Measurementsof Measurements Type Type AsAs described described above, above, for for the the single-frequency single-frequency GPS GPS/BDS/BDS receiver, receiver, two two solutions solutions with with di ﬀdifferenterent typestypes of of measurements measurements can can be be obtained: obtained: (1) (1) the the pseudo-range-based pseudo-range-based solution, solution, namely namely only only using using the the single-frequencysingle-frequency GPS /GPS/BDSBDS pseudo-range pseudo-range measurement measurement (C1), which (C1), is abbreviated which asis “GPSabbreviated+BDS_C1”; as (2)“GPS+BDS_C1”; the carrier-phase-based (2) the solution,carrier-phase-based namely using solution, the GRAPHIC namely combinationusing the GRAPHIC (LH) and combination pseudo-range (LH) measurementand pseudo-range (C1), which measurement is shortened (C1), as “GPSwhich+ BDS_LHis shortened+C1”. as The “GPS+BDS_LH+C1”. daily position and velocityThe daily accuracyposition statistics and velocity of the accuracy GPS+BDS_C1 statistics and of GPSthe GPS+BDS_C1+BDS_LH+C1 and solutions GPS+BDS_LH+C1 for the FY3C solutions satellite for on the dayFY3C 2015 satellite/69–75 are on shown day 2015/69 in Figure−756 .are The shown R /A/ Cin/3D Figure position 6. The accuracies R/A/C/3D of theposition pseudo-range-based accuracies of the solutionpseudo-range-based are at 0.4–0.6 m,solution 0.5–0.8 are m, at 0.1–0.3 0.4−0.6 m m, and 0.5− 0.7–1.00.8 m, 0.1 m level,−0.3 m respectively, and 0.7−1.0 m while level, those respectively, of the carrier-phase-basedwhile those of the solution carrier-phase-based are improved dramaticallysolution are toimproved 0.1–0.3 m, dramatically 0.4–0.6 m, 0.1–0.3 to 0.1 m−0.3 and m, 0.4–0.7 0.4−0.6 m m, level.0.1−0.3 Similarly, m and the R0.4/A−/0.7C/3D m velocity level. accuraciesSimilarly, ofthe the pseudo-range-basedR/A/C/3D velocity solutionaccuracies are aboutof the 0.5–0.7pseudo-range-based mm/s, 0.4–0.5 mm solution/s, 0.1–0.4 are mm about/s and 0.5− 0.7–0.90.7 mm/s, mm /0.4s, respectively,−0.5 mm/s, 0.1 which−0.4 mm/s are inferior and 0.7 to− those0.9 mm/s, of therespectively, carrier-phase-based which are solution, inferior namely to those 0.4–0.5 of the mm carrier-phase-based/s, 0.2–0.3 mm/s, 0.1–0.3 solution, mm /namelys and 0.4–0.6 0.4−0.5 mm mm/s,/s. The0.2 results−0.3 mm/s, demonstrate 0.1−0.3 thatmm/s the and real-time 0.4−0.6 orbit mm/s. accuracy The results of carrier-phase-based demonstrate that solutions the real-time is notably orbit betteraccuracy than thatof ofcarrier-phase-based pseudo-range-based solutions solutions. is notably better than that of pseudo-range-based solutions.

GPS+BDS_C1 GPS+BDS_LH+C1 GPS+BDS_C1 GPS+BDS_LH+C1 0.6 0.4 0.6 0.3 R [m] 0.2 R [mm/s] 0.0 0.0 0.6 0.4 0.3 0.2 A [m] 0.0 A [mm/s] 0.0 0.3 0.3 0.2 0.2

C [m] 0.1 0.1 0.0 C [mm/s] 0.0 1.0 1.0

0.5 0.5 3D [m] 3D

0.0 3D [mm/s] 0.0 69 70 71 72 73 74 75 69 70 71 72 73 74 75 DOY [d] DOY [d]

(a) (b)

Figure 6. Comparison of daily real-time orbit accuracies of the pseudo-range-based and carrier-phase-based solutions for the FY3C satellite: (a) The radial (R), along-track (A), cross (C), 1-dimensional (1D) and 3-dimensional (3D) position accuracy; (b) The R/A/C/3D velocity accuracy.

Remote Sens. 2019, 11, x FOR PEER REVIEW 13 of 23

(a) (b)

Figure 5. Comparison of real-time orbit accuracies of the YG30/SAT1-SAT6 satellites under different force models: (a) The position accuracy of the SAT1-SAT3 satellites; (b) The position accuracy of the SAT4-SAT6 satellites.

3.4. Effect of Measurements Type As described above, for the single-frequency GPS/BDS receiver, two solutions with different types of measurements can be obtained: (1) the pseudo-range-based solution, namely only using the single-frequency GPS/BDS pseudo-range measurement (C1), which is abbreviated as “GPS+BDS_C1”; (2) the carrier-phase-based solution, namely using the GRAPHIC combination (LH) and pseudo-range measurement (C1), which is shortened as “GPS+BDS_LH+C1”. The daily position and velocity accuracy statistics of the GPS+BDS_C1 and GPS+BDS_LH+C1 solutions for the FY3C satellite on day 2015/69−75 are shown in Figure 6. The R/A/C/3D position accuracies of the pseudo-range-based solution are at 0.4−0.6 m, 0.5−0.8 m, 0.1−0.3 m and 0.7−1.0 m level, respectively, while those of the carrier-phase-based solution are improved dramatically to 0.1−0.3 m, 0.4−0.6 m, 0.1−0.3 m and 0.4−0.7 m level. Similarly, the R/A/C/3D velocity accuracies of the pseudo-range-based solution are about 0.5−0.7 mm/s, 0.4−0.5 mm/s, 0.1−0.4 mm/s and 0.7−0.9 mm/s, respectively, which are inferior to those of the carrier-phase-based solution, namely 0.4−0.5 mm/s, 0.2−0.3 mm/s, 0.1−0.3 mm/s and 0.4−0.6 mm/s. The results demonstrate that the real-time orbit Remoteaccuracy Sens. 2019 of, 11 carrier-phase-based, 1391 solutions is notably better than that of pseudo-range-based14 of 23 solutions.

GPS+BDS_C1 GPS+BDS_LH+C1 GPS+BDS_C1 GPS+BDS_LH+C1 0.6 0.4 0.6 0.3 R [m] 0.2 R [mm/s] 0.0 0.0 0.6 0.4 0.3 0.2 A [m] 0.0 A [mm/s] 0.0 0.3 0.3 0.2 0.2

C [m] 0.1 0.1 0.0 C [mm/s] 0.0 1.0 1.0

0.5 0.5 3D [m] 3D

0.0 3D [mm/s] 0.0 69 70 71 72 73 74 75 69 70 71 72 73 74 75 DOY [d] DOY [d]

(a) (b)

FigureFigure 6. 6. ComparisonComparison ofof dailydaily real-time real-time orbit orbit accuracies accuracies of of the the pseudo-range-based pseudo-range-based and and carrier-phase-basedcarrier-phase-based solutions solutions for for the the FY3C FY3C satellite: satellite: (a )(a The) The radial radial (R), (R), along-track along-track (A), (A), cross cross (C), (C), 1-dimensional1-dimensional (1D) (1D) and and 3-dimensional 3-dimensional (3D) (3D) position position accuracy; accuracy; (b) ( Theb) The R/A R/A/C/3D/C/3D velocity velocity accuracy. accuracy.

The R/A/C/3D position and velocity errors of the pseudo-range-based and carrier-phase-based solutions on day 2015/71 are illustrated in Figure7, and also their arithmetic mean and standard deviation statistics. As can be seen firstly, the variation magnitudes of the R/A/C/3D position and velocity errors of the carrier-phase-based solution are all notably smaller compared to those of the pseudo-range-based solution. Comparing the R/A/C error curves of these two solutions further, it can be observed that the R/A position and velocity errors of the pseudo-range-based solution all contains a notable overall deviation from the zero-axis, which indicates the real-time orbit results are affected by the systematic errors significantly. In the pseudo-range-based solution, the systematic errors include the residual ionospheric delay, broadcast ephemeris errors, group delay variation, phase center variation, and so on. According to the statistics, the arithmetic mean of the R/A position and velocity errors of the pseudo-range-based solution are 0.435 m, 0.368 m, 0.390 mm/s and 0.316 mm/s, respectively, while − − those of the carrier-phase-based solution are reduced to 0.071 m, 0.064 m, 0.041 mm/s and 0.073 mm/s − − dramatically, which means there is no obvious systematic error residua in the carrier-phase-based solution. The reason for the notably reduced systematic error in the carrier-phase-based solution is that the used GRAPHIC combination eliminates the first-order ionospheric error and the estimated pseudo-ambiguity could absorb the LOS error caused by the GNSS broadcast ephemeris or other error sources. In conclusion, a notably superior orbit results can be achieved if using the carrier-phase measurements, compared to the solution only using pseudo-range measurements. Remote Sens. 2019, 11, x FOR PEER REVIEW 14 of 23

The R/A/C/3D position and velocity errors of the pseudo-range-based and carrier-phase-based solutions on day 2015/71 are illustrated in Figure 7, and also their arithmetic mean and standard deviation statistics. As can be seen firstly, the variation magnitudes of the R/A/C/3D position and velocity errors of the carrier-phase-based solution are all notably smaller compared to those of the pseudo-range-based solution. Comparing the R/A/C error curves of these two solutions further, it can be observed that the R/A position and velocity errors of the pseudo-range-based solution all contains a notable overall deviation from the zero-axis, which indicates the real-time orbit results are affected by the systematic errors significantly. In the pseudo-range-based solution, the systematic errors include the residual ionospheric delay, broadcast ephemeris errors, group delay variation, phase center variation, and so on. According to the statistics, the arithmetic mean of the R/A position and velocity errors of the pseudo-range-based solution are −0.435 m, 0.368 m, −0.390 mm/s and 0.316 mm/s, respectively, while those of the carrier-phase-based solution are reduced to −0.071 m, −0.064 m, 0.041 mm/s and 0.073 mm/s dramatically, which means there is no obvious systematic error residua in the carrier-phase-based solution. The reason for the notably reduced systematic error in the carrier-phase-based solution is that the used GRAPHIC combination eliminates the first-order ionospheric error and the estimated pseudo-ambiguity could absorb the LOS error caused by the GNSS broadcast ephemeris or other error sources. In conclusion, a notably

Remotesuperior Sens. 2019 orbit, 11, 1391results can be achieved if using the carrier-phase measurements, compared15 ofto 23 the solution only using pseudo-range measurements.

GPS+BDS_C1 GPS+BDS_LH+C1 GPS+BDS_C1 GPS+BDS_LH+C1 1 -0.435+/-0.285 -0.071+/-0.126 2 -0.390+/-0.557 0.041+/-0.351 1 0 0 R [m]

R [mm/s] -1 -1 -2 2 0.368+/-0.633 -0.064+/-0.393 1 0.316+/-0.272 0.073+/-0.146 0 0 A [m] A [mm/s] -2 -1 1 0.016+/-0.271 0.006+/-0.083 1 -0.019+/-0.310 -0.005+/-0.102 0 0 C [m] C [mm/s] -1 -1 3 0.841+/-0.416 0.376+/-0.211 3 0.755+/-0.402 0.353+/-0.193 2 2

3D [m] 1 1 3D [mm/s] 0 0 024681012141618202224 0 2 4 6 8 10 12 14 16 18 20 22 24 Time [h] Time [h]

(a) (b)

FigureFigure 7. Comparison7. Comparison of real-timeof real-time orbit orbit errors errors of theof th pseudo-range-basede pseudo-range-based and and carrier-phase-based carrier-phase-based solutionssolutions for thefor FY3C the satelliteFY3C satellite on day 2015on /day71: (a2015/71:) The radial (a) (R),The along-track radial (R), (A), along-track cross (C), 1-dimensional (A), cross (C), (1D)1-dimensional and 3-dimensional (1D) and (3D) 3-dimens positionional errors; (3D) (b) position The R/A errors;/C/3D velocity(b) The R/A/C/3D errors. velocity errors.

ForFor the the YG30 YG30/SAT1/SAT1–SAT6−SAT6 satellites, satellites, only only the the pseudo-range pseudo-range measurement measurement is available. is available. Figure Figure8 8 presentspresents the the position position and and velocity velocity accuracies accuracies of pseudo-range-based of pseudo-range-based solution. solution. As can As be can observed, be observed, the R/Athe/C /R/A/C/3D3D position position accuracies accuracies of the SAT1–SAT6of the SAT1 satellites−SAT6 satellites are at 0.4–0.8 are at 0.4 m,− 0.7–1.30.8 m, m,0.7− 0.2–0.71.3 m, m0.2 and−0.7 m 0.8–1.6and m,0.8− respectively,1.6 m, respectively, and the Rand/A /Cthe/3D R/A/C/3D velocity accuraciesvelocity accuracies are about are 0.8–1.5 about mm 0.8/s,−1.5 0.4–0.7 mm/s, mm 0.4/s,−0.7 0.2–0.6mm/s, mm 0.2/s− and0.6 mm/s 0.9–1.7 and mm 0.9/s.−1.7 The mm/s. orbit accuraciesThe orbit foraccuracies all satellites for al arel satellites similar, are due similar, to identical due to space-borne GNSS receiver and altitude. The overall real-time orbit accuracies of FY3C and YG30 /SAT1–SAT6 satellites in the whole session are presented in Table3. For the FY3C satellite, the real-time orbit accuracy is 0.527 m for position and 0.479 mm/s for velocity with the carrier-phase-based solution and 0.847 m for position and 0.764 mm/s for velocity with the pseudo-range-based solution, respectively. Compared with the FY3C satellite, the inferior real-time orbit accuracy is achieved for the YG30/SAT1–SAT6 satellites, which is about 1.1–1.2 m for position and 1.2–1.3 mm/s for velocity. The reason of the poor performance is probably that the pseudo-range measurements from the single-frequency GNSS receiver onboard the YG30/SAT1–SAT6 satellites have a ranging noise higher than that from the dual-frequency GNSS receiver onboard the FY3C satellite. Furthermore, the reference orbit accuracies of YG30/SAT1–SAT6 satellites are at 20–25 cm level, which are not suﬃciently precise and perhaps also lead to worse real-time orbit accuracy assessment. Remote Sens. 2019, 11, x FOR PEER REVIEW 15 of 23

identical space-borne GNSS receiver and altitude. The overall real-time orbit accuracies of FY3C and YG30/SAT1−SAT6 satellites in the whole session are presented in Table 3. For the FY3C satellite, the real-time orbit accuracy is 0.527 m for position and 0.479 mm/s for velocity with the carrier-phase-based solution and 0.847 m for position and 0.764 mm/s for velocity with the pseudo-range-based solution, respectively. Compared with the FY3C satellite, the inferior real-time orbit accuracy is achieved for the YG30/SAT1−SAT6 satellites, which is about 1.1−1.2 m for position and 1.2−1.3 mm/s for velocity. The reason of the poor performance is probably that the pseudo-range measurements from the single-frequency GNSS receiver onboard the YG30/SAT1−SAT6 satellites have a ranging noise higher than that from the dual-frequency GNSS receiver onboard the FY3C satellite. Furthermore, the reference orbit accuracies of RemoteYG30/SAT1 Sens. 2019,−11SAT6, 1391 satellites are at 20−25 cm level, which are not sufficiently precise and 16perhaps of 23 also lead to worse real-time orbit accuracy assessment.

GPS+BDS_C1: R A C 3D GPS+BDS_C1: R A C 3D 2 2 SAT1 SAT1 1 1 0 0 2 2 SAT2 SAT2 1 1 0 0 2 2 SAT3 SAT3 1 1 0 0 2 2 SAT4 SAT4 1 1 0 0 Position Accuracy [m] 2 Velocity Accuracy [mm/s] 2 SAT5 SAT5 1 1 0 0 2 2 SAT6 SAT6 1 1

0 0 340 341 342 343 344 345 340 341 342 343 344 345 DOY [d] DOY [d]

(a) (b)

FigureFigure 8. Daily8. real-timeDaily real-time orbit accuracy orbit of theaccuracy YG30 /SAT1–SAT6of the YG30/SAT1 satellites for−SAT6 the pseudo-range-based satellites for the solution:pseudo-range-based (a) The radial (R), solution: along-track (a) The (A), radial cross (R), (C) andalong-track 3-dimensional (A), cross (3D) position(C) and accuracy;3-dimensional (b) The (3D) R/Aposition/C/3D velocity accuracy; accuracy. (b) The R/A/C/3D velocity accuracy.

TableTable 3. The3. The overall overall statistics statistics of real-timeof real-time orbit orbit accuracy. accuracy. Position/m Velocity/mm/s AccuracyAccuracy (RMS) Position/m Velocity/mm/s (RMS) RR AA CC 3D3D R R AA CC 3D3D FY3CFY3C (LH (LH+C1)+ C1) 0.1710.171 0.4830.483 0.1220.122 0.5270.527 0.419 0.419 0.186 0.186 0.1390.139 0.4790.479 FY3CFY3C (C1) (C1) 0.467 0.467 0.6720.672 0.2170.217 0.8470.847 0.618 0.618 0.385 0.385 0.2310.231 0.7640.764 SAT1SAT1 (C1) (C1) 0.561 0.561 0.9470.947 0.3070.307 1.1431.143 1.088 1.088 0.502 0.502 0.3190.319 1.2401.240 SAT2 (C1) 0.567 0.929 0.340 1.140 1.052 0.512 0.359 1.224 SAT2 (C1) 0.567 0.929 0.340 1.140 1.052 0.512 0.359 1.224 SAT3 (C1) 0.581 0.946 0.277 1.144 1.089 0.520 0.296 1.242 SAT4SAT3 (C1) (C1) 0.504 0.581 0.9810.946 0.3960.277 1.1711.144 1.061 1.089 0.491 0.520 0.4270.296 1.2451.242 SAT5SAT4 (C1) (C1) 0.572 0.504 0.9770.981 0.3990.396 1.2001.171 1.088 1.061 0.542 0.491 0.4380.427 1.2921.245 SAT6SAT5 (C1) (C1) 0.514 0.572 0.9770.977 0.4630.399 1.1971.200 1.061 1.088 0.490 0.542 0.4920.438 1.2681.292 SAT6 (C1) 0.514 0.977 0.463 1.197 1.061 0.490 0.492 1.268 3.5. Impact of GPS/BDS Fusion The most direct impact of the GPS/BDS fusion is to make more GNSS satellites observed to enhance the observation geometric condition of LEO satellite orbit determination. The number of tracked GNSS satellites by the FY3C and YG30/SAT1–SAT6 satellites for GPS/BDS fusion solutions are shown in Figure9, where the SAT1 satellite represents all YG30 series satellites due to their similar orbit configurations. There are four types of GPS/BDS combinations to be compared in total: (1) BDS alone; (2) GPS alone; (3) GPS and BDS IGSO/MEO fusion, which is shorted as “GPS + BDSN”, where BDSN means the BDS GEO satellites are not included; (4) GPS and BDS fusion, which is marked as “GPS + BDS”. For the YG30 series satellites, the sub-satellite points distribute at the latitude range of 35 , ± ◦ due to their low inclination angles of about 35◦. Only the BDS-2 regional navigation system is available for the FY3C and YG30 series satellites, so only in the Asia/Pacific region more than 4 BDS satellites can Remote Sens. 2019, 11, 1391 17 of 23 be tracked, and the numbers of tracked GNSS satellites in other regions are always less than 4. GPS is available for the global navigation service, thus in most places of the world more than 4 GPS satellites can be tracked. Compared to GPS-alone system, the GPS+BDSN and GPS+BDS combinations increase the number of tracked GNSS satellites notably, especially in the Asia/Pacific region where GEO/IGSO satellites of BDS are tracked. According to the statistics, the FY3C and YG30 series satellites can track an average of 8.4 and 10.7 GNSS satellites, while the average numbers of tracked GNSS satellites increase to 11.4 and 12.7, respectively, if the BDS satellites are added. The position dilution of precision (PDOP) can directly describe the observation geometry of LEO satellites. Table4 summarizes the PDOP statistics of the FY3C and YG30/SAT1–SAT6 satellites under four GPS/BDS combinations. As can Remote Sens. 2019, 11, x FOR PEER REVIEW 17 of 23 be observed clearly, the observation geometry is poor if only BDS is used and improves significantly if GPSis available, while the geometry improves further with the GPS/BDS fusion.

(a)

(b)

Figure 9.9. The number of tracked GNSS satellites for didifferentﬀerent typestypes ofof GPSGPS/BDS/BDS combinations:combinations: (a)) forfor thethe FY3CFY3C satellite;satellite; ((bb)) forfor thethe YG30YG30/SAT1/SAT1 satellite.satellite.

Table 4. Comparison of position dilution of precisions (PDOPs) for different GPS/BDS combinations.

GPS + GPS + PDOP BDS GPS BDSN BDS FY3C 5.499 2.471 2.247 2.123 SAT1 5.148 1.261 1.244 1.221 SAT2 4.995 1.281 1.252 1.236 SAT3 5.854 1.268 1.247 1.222 SAT4 5.258 1.313 1.293 1.269 SAT5 5.630 1.320 1.295 1.275 SAT6 5.726 1.295 1.266 1.244 In order to realize the fusion of GPS and BDS data, the bias η between the GPS and BDS system is estimated as an unknown parameter in the onboard RTOD algorithm, which is modeled by a random walk process. Figure 10 shows the estimated bias sequences for the FY3C and YG30/SAT1 satellites. Since BDS is a regional system for the experiment times, there are apparently two scenarios. In the first one, no BDS satellites are trackable, and in the second one, both GPS and

Remote Sens. 2019, 11, 1391 18 of 23

Table 4. Comparison of position dilution of precisions (PDOPs) for diﬀerent GPS/BDS combinations.

PDOP BDS GPS GPS + BDSN GPS + BDS FY3C 5.499 2.471 2.247 2.123 SAT1 5.148 1.261 1.244 1.221 SAT2 4.995 1.281 1.252 1.236 SAT3 5.854 1.268 1.247 1.222 SAT4 5.258 1.313 1.293 1.269 SAT5 5.630 1.320 1.295 1.275 SAT6 5.726 1.295 1.266 1.244

In order to realize the fusion of GPS and BDS data, the bias η between the GPS and BDS system is estimated as an unknown parameter in the onboard RTOD algorithm, which is modeled by a random walk process. Figure 10 shows the estimated bias sequences for the FY3C and YG30/SAT1 satellites. Since BDS is a regional system for the experiment times, there are apparently two scenarios. In the first one, no BDS satellites are trackable, and in the second one, both GPS and BDS satellites are trackable. Comparing the middle and bottom panels, it can be observed that, for the arcs without BDS satellites, the bias varies drastically, because no BDS observations are available to measure the bias. According to the statistics, the standard deviations (STD) of the estimated biases are up to 7.0 ns and 24.1 ns for the FY3C and YG30/SAT1 satellites, respectively. Obviously, the arcs without BDS satellites occur periodically, when the LEO satellites fly over the no-Asia/Pacific regions as shown in Figure9. However, it should be noted that the inaccurate estimation of η on these arcs has no effect on the real-time orbit accuracy, because no BDS observations are used for the estimation of orbit parameters. On the other hand, considering the arcs with BDS satellites, as shown in the top panel, the variation amplitude of estimated bias reduces greatly where the STDs are only 3.8 ns and 7.9 ns for the FY3C and YG30/SAT1 satellites, respectively. It indicates that the system bias has been calibrated in the form of the estimated parameters with a certain precision on the arcs with GPS/BDS fusion. Of course, inevitably, the real-time estimation of η is affected by the error of GNSS broadcast ephemeris, the residual ionospheric error, the measurement noises and other uncorrectable errors, so the random jittering appears in the sequences of estimated bias. Compared to the FY3C satellite, the bias for YG30/SAT1 shows more notable variation. The reason of the poor performance is probably that the pseudo-range measurements noises of the YG30/SAT1 satellite are higher than those of the FY3C satellite. Furthermore, the YG30/SAT1 satellite at a lower altitude of 600 km is more susceptible to the ionospheric delays than the FY3C satellite at a higher altitude of 836 km, so the residual ionospheric error will be more obvious after the correction by Klobuchar model. After all, the calibration precision of several nanoseconds is enough for the onboard RTOD with the accuracy of sub-meter~meter level. The real-time orbit accuracies of the FY3C and YG30/SAT1–SAT6 satellites under different types of GPS/BDS combinations are shown in Figure 11, where “All” represents the overall statistics of the whole interval. For the FY3C satellites, if only using BDS measurements, the real-time orbit accuracy is respectively about 1.5–3.0 m for position and 1.5–2.5 mm/s for velocity with the carrier-phase-based solution and about 2.0–9.0 m for position and 2.0–8.0 mm/s for velocity with the pseudo-range-based solution. The poor performance with BDS-alone system is caused by a few tracked BDS satellites and the large orbit and clock errors of the BDS broadcast ephemeris. If using GPS measurements, a real-time orbit accuracy of 0.4–0.7 m for position and 0.4–0.7 mm/s for velocity is achieved with the carrier-phase-based solution, while that of the pseudo-range-based solution is 0.7–1.0 m for position and 0.7–0.9 mm/s for velocity, respectively. Compared to the GPS-alone system, the GPS + BDSN and GPS + BDS combinations only improve the overall orbit accuracy of the carrier-phase-based solution by 3.2 cm, but decrease that of the pseudo-range-based solution by 1.7 cm. No matter improvement or degradation, the slight change of 1.0–3.0 cm level is of little significance or even meaningless from the view of accuracy. It indicates that GPS/BDS fusion could not contribute to accuracy improvement notably. The similar conclusion can be obtained from the real-time orbit accuracies comparison of the Remote Sens. 2019, 11, x FOR PEER REVIEW 18 of 23

BDS satellites are trackable. Comparing the middle and bottom panels, it can be observed that, for the arcs without BDS satellites, the bias varies drastically, because no BDS observations are available to measure the bias. According to the statistics, the standard deviations (STD) of the estimated biases are up to 7.0 ns and 24.1 ns for the FY3C and YG30/SAT1 satellites, respectively. Obviously, the arcs without BDS satellites occur periodically, when the LEO satellites fly over the no-Asia/Pacific regions as shown in Figure 9. However, it should be noted that the inaccurate estimation of η on these arcs has no effect on the real-time orbit accuracy, because no BDS observations are used for the estimation of orbit parameters. On the other hand, considering the arcs with BDS satellites, as shown in the top panel, the variation amplitude of estimated bias reduces greatly where the STDs are only 3.8 ns and 7.9 ns for the FY3C and YG30/SAT1 satellites, respectively. It indicates that the system bias has been calibrated in the form of the estimated parameters with a certain precision on the arcs with GPS/BDS fusion. Of course, inevitably, the real-time estimation of η is affected by the error of GNSS broadcast ephemeris, the residual ionospheric error, the measurement noises and other uncorrectable errors, so the random jittering

Remoteappears Sens. 2019 in the, 11, sequences 1391 of estimated bias. Compared to the FY3C satellite, the bias for YG30/SAT119 of 23 shows more notable variation. The reason of the poor performance is probably that the pseudo-range measurements noises of the YG30/SAT1 satellite are higher than those of the FY3C YG30satellite./SAT1–SAT6 Furthermore, satellites the with YG30/SAT1 diﬀerent GPSsatellite/BDS at combinations. a lower altitude As canof 600 be seenkm is from more sub-graphs susceptible (c) to andthe (d), ionospheric for the YG30 delays/SAT1–SAT6 than the satellites, FY3C thesatellite real-time at a orbit higher accuracy altitude is about of 836 3.0–10.0 km, mso forthe position residual andionospheric 3.0–10.0 mm error/s for will velocity be more with theobvious BDS-alone after system,the correction and that by of Klobuchar the GPS-alone model. solution After is aboutall, the 1.1–1.2calibration m for positionprecision and of several 1.2–1.3 nanoseconds mm/s for velocity, is enough while for the the GPS onboard+ BDSN RTOD and GPSwith+ theBDS accuracy fusions of onlysub-meter improve ~ ormeter reduce level. the orbit accuracy by 1.0–3.0 cm, which could be negligible, too.

90 Bias (not include the arcs without BDS) 200 Bias (not include the arcs without BDS) MEAN: 37.6 ns; STD: 3.8 ns 150 MEAN: 49.4 ns; STD: 7.9 ns 60 100 Bias [ns] Bias [ns] 30 50 0 90 Bias (include the arcs without BDS) 200 Bias (include the arcs without BDS) MEAN: 39.1 ns; STD: 7.0 ns 150 MEAN: 59.7 ns; STD: 24.1 ns 60 100 Bias [ns] Bias [ns] 30 50 0 6 6 4 4

2 2 BDS Number BDS Number 0 0 0 2 4 6 8 1012141618202224 024681012141618202224 Time [h] Time [h]

(a) (b)

FigureFigure 10. 10.Estimation Estimation of of the the system system bias bias between between GPS GPS and and BDS: BDS: (a) ( thea) the estimated estimated bias bias sequences sequences and and thethe corresponding corresponding number number of tracked of BDStracked satellites BDS for thesatellites FY3C satellitefor the with FY3C the pseudo-range-based satellite with the solutionpseudo-range-based on day 2015/70; solution (b) the estimatedon day 2015/70; bias sequences (b) the estimated and the number bias sequences of BDS for and the the YG30 number/SAT1 of satelliteBDS for with the the YG30/SAT1 pseudo-range-based satellite with solution the pseudo-range-based on day 2017/341. solution on day 2017/341.

It shouldThe real-time be noted orbit that accuracies the GPS/BDS of combinationthe FY3C and is of YG30/SAT1-SAT6 important significance satellites from under other aspects.different Firstly,types theof GPS/BDS fusion of combinations GPS/BDS data are could shown speed in Figure up the 11, convergence where “All” of represents the Kalman the filter, overall as shownstatistics inof the the sub-graph whole interval. (a) of Figure For the 12 .FY3C For thesatellites, FY3C satellite, if only using if only BDS using measurements, BDS measurements, the real-time the filter orbit isaccuracy in the state is ofrespectively prediction atabout most 1.5 epochs−3.0 m due for to position less than and 4 tracked 1.5−2.5 BDS mm/s satellites for velocity for most with of the the time,carrier-phase-based so it costs a long timesolution to achieve and about convergence. 2.0−9.0 m Iffor only position using and GPS 2.0 measurements,−8.0 mm/s for thevelocity filter with needs the aboutpseudo-range-based 1.2 h for the position solution. error toThe converge poor performance to less than 1.0 with m, whileBDS-alon the convergente system is time caused decreases by a tofew abouttracked 0.7 hBDS for satellites the GPS +andBDSN the large and GPS orbit+ andBDS clock solutions. errors Moreof the importantly,BDS broadcast the ephemeris. main advantage If using of the GPS/BDS fusion is not reduced convergence time, but the improved reliability of the onboard RTOD. Sub-graph (b) of Figure 12 shows the position errors of the GPS-alone and GPS/BDS solutions and the corresponding number of the tracked GPS and BDS satellites on day 2017/74, where there is an anomalous arc covering 16–24 h. In this abnormal arc, the space-borne GPS/BDS receiver is in an unstable state and cannot track the GPS satellites continuously. As can be observed clearly, if only using GPS measurements, the position error varies drastically to 3.2 m, while the position error is reduced to less than 1.5 m dramatically when the BDS satellites are included. Obviously, the inclusion of BDS satellites could make more GNSS satellites observed to suppress the local orbit error in some anomalous arc where GPS satellites are in a bad tracking. Although the anomalous arcs may not occur often, it can be generally concluded that the GPS/BDS fusion guarantees the orbit accuracy, improves the reliability and availability of onboard RTOD. RemoteRemote Sens. Sens. 2019 2019, 11, 11, x, FORx FOR PEER PEER REVIEW REVIEW 19 19 of of 23 23

GPSGPS measurements, measurements, a areal-time real-time orbit orbit accuracy accuracy of of 0.4 0.4−0.7−0.7 m m for for position position and and 0.4 0.4−0.7−0.7 mm/s mm/s for for velocityvelocity is is achieved achieved with with the the carrier-phase-base carrier-phase-based dsolution, solution, while while that that of of the the pseudo-range-based pseudo-range-based solutionsolution is is 0.7 0.7−1.0−1.0 m m for for position position and and 0.7 0.7−0.9−0.9 mm/s mm/s for for velocity, velocity, respectively. respectively. Compared Compared to to the the GPS-aloneGPS-alone system, system, the the GPS GPS + +BDSN BDSN and and GPS GPS + +BDS BDS combinations combinations only only improve improve the the overall overall orbit orbit accuracyaccuracy of of the the carrier-phase-based carrier-phase-based solution solution by by 3.2 3.2 cm, cm, but but decrease decrease that that of of the the pseudo-range-based pseudo-range-based solutionsolution by by 1.7 1.7 cm. cm. No No matter matter improvement improvement or or degradation, degradation, the the slight slight change change of of 1.0 1.0−3.0−3.0 cm cm level level is is ofof little little significance significance or or even even meaningless meaningless from from the the view view of of accuracy. accuracy. It It indicates indicates that that GPS/BDS GPS/BDS fusionfusion could could not not contribute contribute to to accuracy accuracy improvement improvement notably. notably. The The similar similar conclusion conclusion can can be be obtainedobtained from from the the real-time real-time orbit orbit accuracies accuracies comparison comparison of of the the YG30/SAT1 YG30/SAT1−SAT6−SAT6 satellites satellites with with differentdifferent GPS/BDSGPS/BDS combinations.combinations. AsAs cancan bebe seenseen fromfrom sub-graphssub-graphs (c)(c) andand (d),(d), forfor thethe YG30/SAT1YG30/SAT1−SAT6−SAT6 satellites, satellites, the the real-time real-time orbit orbit accuracy accuracy is is about about 3.0 3.0−10.0−10.0 m m for for position position and and 3.03.0−10.0−10.0 mm/s mm/s for for velocity velocity with with the the BDS-alone BDS-alone system, system, and and that that of of the the GPS-alone GPS-alone solution solution is is about about Remote Sens.1.11.1−20191.2−1.2 m, m11 for ,for 1391 position position and and 1.2 1.2−1.3−1.3 mm/s mm/s for for velocity, velocity, while while the the GPS GPS + +BDSN BDSN and and GPS GPS + +BDS BDS fusions fusions20 of 23 onlyonly improve improve or or reduce reduce the the orbit orbit accuracy accuracy by by 1.0 1.0−3.0−3.0 cm, cm, which which could could be be negligible, negligible, too. too.

LH+C1:LH+C1: C1:C1: BDS BDS GPS GPS GPS+BDSN GPS+BDSN GPS+BDS GPS+BDS BDS BDS GPS GPS GPS+BDSN GPS+BDSN GPS+BDS GPS+BDS 9 2.52.5 9 6 6 2.02.0 3 3 1.51.5 0.60.6 0.80.8 0.30.3 0.40.4 Position Accuracy [m] Accuracy Position Position Accuracy [m] Accuracy Position Position Accuracy [m] Position Accuracy Position Accuracy [m] Position Accuracy 0.00.0 0.00.0 2.52.5 6 6 2.02.0 3 3 1.51.5 0.60.6 0.80.8

0.30.3 0.40.4

0.00.0 0.0

Velocity Accuracy [mm/s] Accuracy Velocity 0.0 Velocity Accuracy [mm/s] Accuracy Velocity 6969 70 70 71 71 72 72 73 73 74 74 75 75 All All 69 70 71 72 73 74 75 All Velocity Accuracy [mm/s] Accuracy Velocity 69 70 71 72 73 74 75 All DOYDOY [d] [d] [mm/s] Accuracy Velocity DOY [d] DOY [d]

(a()a ) ( b()b ) Remote Sens. 2019, 11, x FOR PEER REVIEW 20 of 23 9 9 9 C1:C1: BDS BDS GPS GPS 9 C1:C1: BDS BDS GPS GPS It should be GPS+BDSNnoted GPS+BDSN that the GPS+BDS GPS+BDSGPS/BDS combination is of important GPS+BDSN GPS+BDSN significance GPS+BDS GPS+BDS from other 6 6 6 6 aspects. Firstly, the fusion of GPS/BDS data could speed up the convergence of the Kalman filter, as shown3 3in the sub-graph (a) of Figure 12. For the FY3C3 satellite,3 if only using BDS measurements, the filter is in the state of prediction at most epochs due to less than 4 tracked BDS satellites for most of 1.01.0 1.01.0 the time, so it costs a long time to achieve convergence. If only using GPS measurements, the filter 0.5 Position Accuracy [m] Accuracy Position 0.5 0.5 needs [m] Accuracy Position 0.5 about 1.2 h for the position error to converge to less than 1.0 m, while the convergent time Velocity Accuracy [mm/s] Accuracy Velocity Velocity Accuracy [mm/s] Accuracy Velocity decreases0.00.0 to about 0.7 h for the GPS + BDSN and GPS0.00.0 + BDS solutions. More importantly, the main SAT1SAT1 SAT2 SAT2 SAT3 SAT3 SAT4 SAT4 SAT5 SAT5 SAT6 SAT6 SAT1SAT1 SAT2 SAT2 SAT3 SAT3 SAT4 SAT4 SAT5 SAT5 SAT6 SAT6 advantage of the GPS/BDS fusion is not reduced convergence time, but theSatellite improved reliability of SatelliteSatellite Satellite the onboard RTOD. Sub-graph (b) of Figure 12 shows the position errors of the GPS-alone and GPS/BDS solutions and the corresponding(c()c ) number of the tracked GPS and ( d()d )BDS satellites on day 2017/74, where there is an anomalous arc covering 16−24 h. In this abnormal arc, the space-borne FigureFigure 11. 11. Comparison Comparison of of real-time real-time orbit orbit accu accuraciesracies of of the the FY3C FY3C and and YG30/SAT1 YG30/SAT1−SAT6−SAT6 satellites satellites for for FigureGPS/BDS 11. receiverComparison is in an of real-timeunstable state orbit and accuracies cannot oftrack the the FY3C GPS and satellites YG30/SAT1–SAT6 continuously. satellites As can forbe fourfour types types of of GPS/BDS GPS/BDS combinations: combinations: (a ()a The) The position position and and velocity velocity accuracy accuracy of of the the FY3C FY3C satellite satellite for for fourobserved types clearly, of GPS /ifBDS only combinations: using GPS measurements, (a) The position the and position velocity error accuracy varies ofdrastically the FY3C to satellite 3.2 m, thethe carrier-phase-based carrier-phase-based solution; solution; (b ()b The) The position position and and velocity velocity accura accuracycy of of the the FY3C FY3C satellite satellite for for the the forwhile the the carrier-phase-based position error is solution;reduced (bto) Theless positionthan 1.5 andm dramatically velocity accuracy when of the the BDS FY3C satellites satellite are for pseudo-range-basedpseudo-range-based solution; solution; ( c()c )The The position position accura accuracycy of of the the YG30/SAT1 YG30/SAT1−SAT6−SAT6 satellites satellites for for the the c theincluded. pseudo-range-basedpseudo-range-basedpseudo-range-based Obviously, the solution; solution;inclusion solution; (( d )(of)d The)The BDSThe positionvelocity velocitysatellites accura accuracy accura couldcycy of ofmakeof the thethe YG30/SAT1 YG30moreYG30/SAT1/ SAT1–SAT6GNSS−SAT6−SAT6 satellites satellites satellites satellites observed for for forthe the the to pseudo-range-basedsuppresspseudo-range-basedpseudo-range-based the local orbit solution; errorsolution. solution. (ind) some The velocityanomalous accuracy arc where of the GPS YG30 satellites/SAT1–SAT6 are in satellites a bad tracking. for the pseudo-range-basedAlthough the anomalous solution. arcs may not occur often, it can be generally concluded that the GPS/BDS fusion guarantees the orbit accuracy, improves the reliability and availability of onboard RTOD.

10 BDS_LH+C1 3 GPS_LH+C1 GPS+BDS_LH+C1 8 GPS_LH+C1 2 GPS+BDSN_LH+C1 1 6 GPS+BDS_LH+C1 0 Position Error[m] 20 4 GPS BDS GPS+BDS 15 Position Error [m] 2 10 5 0

Satellite Number Satellite 0 1234560 2 4 6 8 1012141618202224 Time [h] Time [h]

(a) (b)

FigureFigure 12. Impact 12. Impact of GPS of GPS/BDS/BDS fusion: fusion: (a ()a) Position Position errors of of the the FY3C FY3C sa satellitetellite at the at theconvergent convergent stage stage with the carrier-phase-based solutions for different GPS/BDS combinations on day 2015/69; (b) with the carrier-phase-based solutions for diﬀerent GPS/BDS combinations on day 2015/69; (b) Position Position errors of the FY3C satellite with the carrier-phase-based solutions and the number of errors of the FY3C satellite with the carrier-phase-based solutions and the number of tracked GNSS tracked GNSS satellites on day 2015/74. satellites on day 2015/74. 4. Conclusions We have studied the onboard RTOD algorithm only using single-frequency GPS/BDS pseudo-range and carrier-phase measurements. The effects of different force models, different measurements and different GPS/BDS fusions are analyzed in detail through the experiments of processing single-frequency GPS/BDS data from the FY3C and YG30 series satellites. The experiments demonstrate that, with an Earth gravity model of higher order and degree, higher real-time orbit accuracy can be achievable, while the corresponding computation would be more time-consuming, therefore, an appropriate size needs to be determined to save the computational load and avoid the notable loss of orbit accuracy at the same time. This leads to the use of order/degree 45 × 45 for the FY3C satellite and 55 × 55 for the YG30 series satellites. The different stochastic model settings of empirical accelerations will result in vastly different real-time orbit results, so a reasonable stochastic model is essential to get the best real-time orbit accuracy. Under the optimal force model, different measurements also lead to different real-time orbit accuracies. If using the single-frequency carrier-phase and pseudo-range measurements, a real-time orbit accuracy of 0.4−0.7 m for position and 0.4−0.7 mm/s for velocity is achieved for the FY3C satellite, which is notably superior to those of the solution only using pseudo-range measurements, where

Remote Sens. 2019, 11, 1391 21 of 23

4. Conclusions We have studied the onboard RTOD algorithm only using single-frequency GPS/BDS pseudo-range and carrier-phase measurements. The effects of different force models, different measurements and different GPS/BDS fusions are analyzed in detail through the experiments of processing single-frequency GPS/BDS data from the FY3C and YG30 series satellites. The experiments demonstrate that, with an Earth gravity model of higher order and degree, higher real-time orbit accuracy can be achievable, while the corresponding computation would be more time-consuming, therefore, an appropriate size needs to be determined to save the computational load and avoid the notable loss of orbit accuracy at the same time. This leads to the use of order/degree 45 45 for the FY3C satellite and 55 55 for × × the YG30 series satellites. The different stochastic model settings of empirical accelerations will result in vastly different real-time orbit results, so a reasonable stochastic model is essential to get the best real-time orbit accuracy. Under the optimal force model, different measurements also lead to different real-time orbit accuracies. If using the single-frequency carrier-phase and pseudo-range measurements, a real-time orbit accuracy of 0.4–0.7 m for position and 0.4–0.7 mm/s for velocity is achieved for the FY3C satellite, which is notably superior to those of the solution only using pseudo-range measurements, where the real-time orbit accuracy is 0.7–1.0 m for position and 0.7–0.9 mm/s for velocity, respectively. For the YG30 series satellites, only the single-frequency pseudo-range measurements are available, the real-time orbit accuracy is about 0.8–1.6 m for position and 0.9–1.7 mm/s for velocity, respectively. No matter pseudo-range-based or carrier-phase-based solution, the different combination of GPS/BDS has different influence on onboard RTOD. If only using BDS measurements, the orbit accuracy is up to several meters due to only few tracked BDS regional navigation satellites and the large orbit and clock errors in the broadcast ephemeris. Of course, this poor performance will be improved if the BDS global navigation satellites are tracked in the near future. The GPS-alone solution could obtain a superior orbit results, but only slight accuracy improvement or even degradation is obtained when BDS satellites are included, even if the GPS/BDS fusion makes more GNSS satellites observed compared to GPS-alone system. However, the GPS/BDS fusion solution could speed up the convergence of Kalman filter effectively. Furthermore, the main advantage of the GPS/BDS fusion is to make up for the lack of tracked GPS satellites, suppress the local orbit errors and improve the reliability and availability of onboard RTOD, especially in some abnormal arcs. For small/micro satellite missions with low-cost single-frequency GPS/BDS receivers, the design of the onboard RTOD algorithm depends on the computational capacity of onboard micro-processers and the availability of GPS/BDS data from the receiver. If the onboard micro-processer has enough computational capability and the receiver could provide carrier-phase and pseudo-range measurements, the carrier-phase-based solution with more complex force models can be adopted, and then a real-time orbit accuracy of 0.4–0.7 m level is achievable. However, if only the pseudo-range measurements are available for some missions, such as the YG30 series satellites, the pseudo-range-based solution with simpler force models might be the only alternative, and thus only the inferior orbit accuracy at the 0.8–1.6 m level is obtained. Moreover, from the view of reliability, the GPS/BDS fusion can lead to more reliable real-time orbit results, if both GPS and BDS satellites are tracked. In conclusion, robust and real-time orbit results with accuracies at the sub-meter to meter level are feasible for onboard RTOD with a low-cost single-frequency GPS/BDS receiver, which is of important significance for small/micro satellites. A stable and accurate onboard RTOD algorithm and software has been developed in this paper, which will be an alternative choice for more space missions in the future.

Author Contributions: In this study, X.G. and L.G. designed the algorithm, performed experiments and wrote this paper. F.W. and W.Z. designed the software, performed experiments, analyzed data and edited the manuscript. J.S., M.G. and H.S. supervised its analysis and edited the manuscript. Funding: This research was funded by the National Natural Science Foundation of China (Nos. 91638203). Acknowledgments: We are very grateful to the IGS MGEX to provide the GPS/BDS precise orbit and clock products. We would like to thank four reviewers for their valuable comments and suggestions. Special thanks to the Meteorological Administration/National Satellite Meteorological Center (CMA/NSMC) of China for providing Remote Sens. 2019, 11, 1391 22 of 23 the space-borne GPS/BDS data of the FY3C satellite and Wuxi Kalman Navigation Technology Co., Ltd. for providing the GPS/BDS data of the YG30 series satellites. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Fang, B.T.; Seifert, E. An evaluation of Global Positioning System data for Landsat-4 orbit determination. In Proceedings of the AIAA 23rd Space Sciences Meeting, Reno, NV, USA, 14–17 January 1985. 2. Montenbruck, O.; Ramos-Bosch, P. Precision real-time navigation of LEO satellites using global positioning system measurements. GPS Solut. 2008, 12, 187–198. [CrossRef] 3. Wang, F.; Gong, X.; Sang, J.; Zhang, X. A novel method for precise onboard real-time orbit determination with a standalone GPS receiver. Sensors 2015, 15, 30403–34018. [CrossRef][PubMed] 4. Cerri, L.; Berthias, J.P.; Bertiger, W.I.; Haines, B.J.; Lemoine, F.G.; Mercier, F.; Ries, J.C.; Willis, P.; Zelensky, N.P.; Ziebart, M. Precision orbit determination standards for the Jason series of altimeter missions. Mar. Geod. 2010, 33, 379–418. [CrossRef] 5. Li, M.; Li, W.; Shi, C.; Jiang, K.; Guo, X.; Dai, X.; Meng, X.; Yang, Z.; Yang, G.; Liao, M. Precise orbit determination of the Fengyun-3C satellite using onboard GPS and BDS observations. J. Geod. 2017, 91, 1313–1327. [CrossRef] 6. Sweeting, M.N. Modern small satellites-changing the economics of space. Proc. IEEE 2018, 106, 343–361. [CrossRef] 7. Collins, J.T.; Conger, R.E. MANS: Autonomous navigation and orbit control for communications satellites. In Proceedings of the 15th AIAA International Communications Satellites Systems Conference, San Diego, CA, USA, 28 February–3 March 1994. 8. Guinn, J.R.; Williams, B.G.; Wolﬀ, P.J.; Fennessey, R.; Wiest, T. TAOS orbit determination results using Global Positioning Satellites. Adv. Astronaut. Sci. 1995, 89, 95–146. 9. Wertz, J.R. Autonomous navigation and autonomous orbit control in planetary orbits as a means of reducing operations cost. In Proceedings of the 5th International Symposium on Reducing the Cost of Spacecraft Ground Systems and Operations, Pasadena, CA, USA, 8–11 July 2003. 10. Plam, Y.; Allen, R.V.; Wertz, J.; Bauer, T. Autonomous orbit control experience on TacSat-2 using Microcosm’s Orbit Control Kit (OCK). In Proceedings of the 31st Annual AAS Guidance and Control Conference, Breckenridge, CO, USA, 1–6 February 2008. 11. Hart, R.C.; Hartman, K.R.; Long, A.C.; Lee, T.; Oza, D.H. Global Positioning System (GPS) Enhanced Orbit Determination Experiment (GEODE) on the small satellite technology initiative (SSTI) lewis spacecraft. In Proceedings of the ION-GPS-1996, Kansas City, MO, USA, 17–20 September 1996. 12. Hart, R.C.; Long, A.C.; Lee, T. Autonomous navigation of the SSTI/Lewis spacecraft using the Global Positioning System (GPS). In Proceedings of the Flight Mechanics Symposium, GSFC, Greenbelt, MD, USA, 19–21 May 1997. 13. Gill, E.; Montenbruck, O.; Brieß, K. GPS-based autonomous navigation for the BIRD satellite. In Proceedings of the 15th International Symposium on Spaceﬂight Dynamics, Biarritz, France, 26–30 June 2000. 14. Gill, E.; Montenbruck, O.; Brieß, K. Flight experience of the BIRD onboard navigation system. In Proceedings of the 16th International Symposium on Space Flight Dynamics, Pasadena, CA, USA, 3–7 December 2001. 15. Gill, E.; Montenbruck, O.; Kayal, H. The BIRD satellite mission as a milestone toward GPS-based autonomous navigation. Navigation 2001, 48, 69–75. [CrossRef] 16. Montenbruck, O.; Gill, E.; Markgraf, M. Phoenix-XNS—A miniature real-time navigation system for LEO satellites. In Proceedings of the NAVITEC’2006, Noordwijk, The Netherlands, 11–13 December 2006. 17. Gill, E.; Montenbruck, O.; Arichandran, K.; Tan, S.H.; Bretschneider, T. High-precision onboard orbit determination for small satellites-the GPS-based XNS on X-SAT. In Proceedings of the 6th Symposium on Small Satellites Systems and Services, La Rochelle, France, 20–24 September 2004. 18. Montenbruck, O.; Nortier, B.; Mostert, S. A miniature GPS receiver for precise orbit determination of the Sunsat 2004 micro-satellite. In Proceedings of the 2004 National Technical Meeting of The Institute of Navigation (ION NTM 2004), San Diego, CA, USA, 26–28 January 2004. Remote Sens. 2019, 11, 1391 23 of 23

19. Montenbruck, O.; Markgraf, M.; Naudet, J.; Santandrea, S.; Gantois, K.; Vuilleumier, P. Autonomous and precise navigation of the PROBA-2 spacecraft. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, HI, USA, 18–21 August 2008. 20. Montenbruck, O.; Swatschina, P.; Markgraf, M.; Santandrea, S.; Naudet, J.; Tilmans, E. Precision spacecraft navigation using a low-cost GPS receiver. GPS Solut. 2012, 16, 519–529. [CrossRef] 21. Zhang, Q.; Guo, X.; Qu, L.; Zhao, Q. Precise orbit determination of FY-3C with calibration of orbit biases in BeiDou GEO satellites. Remote Sens. 2018, 10, 382. [CrossRef] 22. Shi, C.; Zhao, Q.; Geng, J.; Lou, Y.; Ge, M.; Liu, J. Recent development of PANDA software in GNSS data processing. In Proceedings of the International Conference on Earth Observation Data Processing and Analysis (ICEODPA), Wuhan, China, 28–30 December 2008. 23. Wang, F. Theory and Software Development on Autonomous Orbit Determination Using Space-Borne GPS Measurements. Ph.D. Thesis, Wuhan University, Wuhan, China, 2006. 24. Gong, X. Key Technologies and Software toward Decimeter-Level Autonomous Orbit Determination with Space-Borne GPS Carrier-Phase Measurements. Master’s Thesis, Wuhan University, Wuhan, China, 2015. 25. Odijk, D.; Nadarajah, N.; Zaminpardaz, S.; Teunissen, P.J.G. GPS, Galileo, QZSS and IRNSS diﬀerential ISBs: Estimation and application. GPS Solut. 2017, 21, 439–450. [CrossRef] 26. Nicolini, L.; Caporali, A. Investigation on Reference Frames and Time Systems in Multi-GNSS. Remote Sens. 2018, 10, 80. [CrossRef] 27. Torre, A.D.; Caporali, A. An analysis of intersystem biases for multi-GNSS positioning. GPS Solut. 2015, 19, 297–307. [CrossRef] 28. Montenbruck, O.; Gill, E. Ionospheric correction for GPS tracking of LEO satellites. J. Navig. 2002, 55, 293–304. [CrossRef]

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