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A Novel Method for Precise Onboard Real-Time Orbit Determination with a Standalone GPS Receiver Fuhong Wang 1,2 , Xuewen Gong 1,2, *, Jizhang Sang 1,2 and Xiaohong Zhang 1,2 Received: 9 October 2015; Accepted: 1 December 2015; Published: 4 December 2015 Academic Editor: Vittorio M. N. Passaro 1 2

*

School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China; [email protected] (F.W.); [email protected] (J.S.); [email protected] (X.Z.) Collaborative Innovation Center for Geospatial Technology, Wuhan 430079, China Correspondence: [email protected]; Tel.: +86-187-1716-7463

Abstract: Satellite remote sensing systems require accurate, autonomous and real-time orbit determinations (RTOD) for geo-referencing. Onboard Global Positioning System (GPS) has widely been used to undertake such tasks. In this paper, a novel RTOD method achieving decimeter precision using GPS carrier phases, required by China’s HY2A and ZY3 missions, is presented. A key to the algorithm success is the introduction of a new parameter, termed pseudo-ambiguity. This parameter combines the phase ambiguity, the orbit, and clock offset errors of the GPS broadcast ephemeris together to absorb a large part of the combined error. Based on the analysis of the characteristics of the orbit and clock offset errors, the pseudo-ambiguity can be modeled as a random walk, and estimated in an extended Kalman filter. Experiments of processing real data from HY2A and ZY3, simulating onboard operational scenarios of these two missions, are performed using the developed software SATODS. Results have demonstrated that the position and velocity accuracy (3D RMS) of 0.2–0.4 m and 0.2–0.4 mm/s, respectively, are achieved using dual-frequency carrier phases for HY2A, and slightly worse results for ZY3. These results show it is feasible to obtain orbit accuracy at decimeter level of 3–5 dm for position and 0.3–0.5 mm/s for velocity with this RTOD method. Keywords: real-time orbit determination; GPS carrier phase; pseudo-ambiguity; decimeter precision; random walk

1. Introduction The dynamic changes of Earth environments, and natural disasters, such as earthquakes and forest fires, have been monitored in real-time with spaceborne Earth observation systems (EOS) in the LEO region. The applications of the EOSs require provisions of near real-time position information of LEO satellite platforms with high precision. Therefore, accurate, autonomous and real-time orbit determination (RTOD) using global, abundant, and low-cost GPS measurements onboard LEO satellites has been widely applied to many LEO missions [1–12]. The Goddard Space Flight Center (GSFC) and Jet Propulsion Laboratory (JPL) of NASA began the development of algorithms and software about the RTODs using spaceborne GPS data in the 1990s. GSFC developed the GPS Enhanced Orbit Determination Experiment (GEODE) space flight navigation software [2]. The position and velocity accuracy (3D RMS) of GEODE were 7.8 m and 5.9 mm/s, respectively, when applied to TOPEX data collected in the absence of Selective Availability (SA) [3], and the position error (3D RMS) of the orbit results was reduced to 1.0 m after Goldstein’s improvements to GEODE’s algorithm [4]. JPL’s Real-time GIPSY (RTG) software was capable of real-time processing and suitable for embedded systems, such as GPS receivers [5]. Orbit results of 1.5 m for position accuracy were obtained when RTG was used to process SAC-C’s Sensors 2015, 15, 30403–30418; doi:10.3390/s151229805

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Sensors 2015, 15, 30403–30418

dual-frequency pseudo-range data with the GPS broadcast ephemeris [6]. In addition to the GSFC and JPL researches, the German Aerospace Center (DLR) has committed to the high-precision RTOD since 1999. A miniature single-frequency GPS receiver named “Phoenix-XNS” embedded with an integrated real-time navigation system for orbit determination of LEO satellites was developed by DLR [7]. “Phoenix-XNS” was equipped on X-SAT, a mini-satellite developed by the Satellite Engineering Centre of the Nanyang Technological University at Singapore in 2004 [8], as well as PROBA-2, another mini-satellite developed by the European Space Agency (ESA) in 2010 [9]. Tests demonstrated that the GPS-based navigation system of “Phoenix-XNS” could provide real-time navigation accuracy at 1.0 m level [10]. The onboard RTOD, also known as autonomous orbit determination, has no dependence on ground-based tracking assets, and all of its computations are completed onboard in the embedded system, which usually has very limited computing capacity. The results are required to be delivered within minutes, seconds, or even a fraction of a second after observations are made. GSFC, JPL, and DLR’s algorithm and software seem to use only pseudo-range data. They employ the extend Kalman filter (EKF) to estimate the LEO satellite’s orbit parameters with simplified dynamic models, and the errors of the determined position and velocity are usually at 1.0 m and 1.0 mm/s level, respectively, due to the use of the GPS broadcast ephemeris and large pseudo-range noises [13–15]. It appears that the errors of the GPS broadcast orbits and clock offsets were not dealt with in these algorithms. To improve the accuracy of the onboard spacecraft navigation, considering the requirements of real-time and autonomy at the same time, it is necessary to make full use of the available data, including carrier phases, from the standalone GPS receiver. Although the high-precision carrier phase data of GPS has been widely used in post-processing for precise orbit determination and the accuracy of orbit results has reached centimeter level [16–22], the post-processing algorithm is not suitable for onboard RTOD. For instance, one is not able to estimate the ambiguity of carrier phase measurements as a constant because only the GPS broadcast ephemeris is available in real-time onboard processing. HY2A is the first oceanographic satellite of China, and it is dedicated to measuring the wind field, height, and temperature of the sea surface [23]. ZY3 is China’s first civilian mapping satellite, and its central task is to investigate and monitor the land resources in China [24]. For the HY2A and ZY3 satellites, high-precision real-time navigation is essential for their scientific applications, and the real-time orbits at decimeter level or better are required. These two missions are equipped with dual-frequency GPS receivers developed by the China Academy of Space Technology (CAST). The onboard GPS receiver can offer a dual pseudo-range accuracy of 1.5–2.5 m and a dual carrier phase accuracy of 5–6 mm. If only the pseudo-range data is used for the RTOD, it is difficult to obtain real-time navigation accuracy at the decimeter level. Therefore, to obtain higher precise navigation, it is critical to make full use of the other available data, namely the carrier phases. This paper focuses on the algorithm of the real-time processing of high-precision carrier phase data, especially the mathematical method to deal with the ambiguity parameter of the carrier phases. A new parameter termed “pseudo-ambiguity”, as opposed to the traditional ambiguity parameter, will be defined to combine the phase ambiguity, the orbit, and clock offset errors of the GPS broadcast ephemeris together. A large part of the range error originated from the orbit and clock offset errors of the GPS broadcast ephemeris will be absorbed into the pseudo-ambiguity. By modeling the pseudo-ambiguity as a random walk process and estimating it in a filter process, orbits of higher accuracy are obtained. It will be shown that the introduction of this parameter is a key to achieve real-time orbit results with accuracies at decimeter level for position and sub-mm/s for velocity. In the following, the dynamical models of LEO satellite orbits and GPS measurement equations will be briefly discussed first. Then, the mathematical definition and properties of the pseudo-ambiguity will be presented in detail, followed by the descriptions of the estimation strategy and SATODS software development. Then, the results of using SATODS to process HY2A and ZY3 data are analyzed. Finally, some conclusions are made.

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2. RTOD Algorithm and Software 2.1. Dynamical Model of LEO Satellite Orbit The equation of the motion of a LEO satellite can be expressed as: ..

r“´

GMr ` ar ` T ¨ aw r3

(1)

` . ..˘ where r, r, r are the position, velocity, and acceleration vectors of the LEO satellite in a geocentric inertial coordinate frame, respectively, ar is the total perturbing acceleration, including the non-spherical part of the gravitational attraction of the Earth, lunisolar gravitational perturbations, atmospheric drag, solar radiation pressure, and earth tide. aw “ pa R , a T , a N q is the empirical acceleration in radial, tangential, and normal directions, which account for those unmodeled forces, and T is its transition matrix. The empirical accelerations are estimated in the Kalman filter. When the satellite has a complex surface geometry, the interactions of the upper atmosphere and solar radiation with the satellite are difficult to model. In such case, the atmospheric drag coefficient Cd and the solar radiation pressure coefficient Cr are estimated as scaling factors to account for the complex of the satellite surface. In summary, the parameters in the dynamic models to be estimated are paw , Cd , Cr q. Various simplifications to the force models in Equation (1) have to be made because of the computation capacity limit of the spaceborne processors. They are well documented in the astrodynamics literature [25]. Detailed settings about the dynamical models will be given in the description of the software development. Considering the accuracy and computational efficiency, a fourth-order Runge-Kutta integrator with a step size of 30 s will be used in the orbit propagation. 2.2. GPS Measurements The dual-frequency GPS receivers onboard the HY2A and ZY3 satellites, developed by CAST, produce pseudo-range and carrier phase data. Two ionosphere-free linear combinations Pα and Lα of the pseudo-ranges (C1 , P2 ) and carrier phase measurements (L1 , L2 ), respectively, can be formed: #

Pα “ ρ ` cδR ´ cδS ` ε Pα Lα “ ρ ` cδR ´ cδS ` B ` ε Lα

(2)

where, ρ is the geometric range between the receiver and GPS satellite, δS and δR are the GPS satellite and receiver clock offsets, respectively, c is the speed of light in vacuum. α “ f 1 { f 2 is the ratio of two ` ˘ ` ˘ frequencies. The measurements Pα and Lα can be expressed by Pα “ α2 ¨ C1 ´ P2 { α2 ´ 1 and ` 2 ˘ ` 2 ˘ Lα “ α ¨ L1 ´ L2 { α ´ 1 , respectively. The parameter B denotes the ambiguity of the ionosphere-free carrier phase measurements. ε Pα and ε Lα are the measurement noises, respectively for Pα and Lα , which include the multipath errors. 2.3. Pseudo-Ambiguity Parameter To use the ionosphere-free phase measurements, the ambiguity B has to be estimated. Although B is a constant in theory, it cannot be fixed if the broadcast ephemeris is used to calculate the orbits and clock offsets of GPS satellites in Equation (2), because the errors of the computed orbits and clock offsets are at the meter level. Without the appropriate processing of the ambiguity, it would be difficult to take full advantage of the high-precision carrier phases. Therefore, an alternative is proposed to consider the effects of the broadcast ephemeris errors on the orbit determination. Apparently, it is possible to express the geometric range between the receiver and GPS satellite as: ρ “ ρ˚ ` dρ (3)

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where ρ˚ is the geometric range between the receiver and GPS satellite, which is calculated using the broadcast ephemeris, and dρ is the range error in the line of sight (LOS) caused by the orbit error of the GPS broadcast ephemeris. Similarly, the GPS clock offset can be expressed as: δS “ δ˚ ` dδ

(4)

where δ˚ is the clock offset of GPS satellites calculated using the broadcast ephemeris, and dδ is the clock offset error. Using the above expressions for ρ and δS , Equation (2) is rewritten as: Lα “ pρ˚ ` dρq ` cδR ´ c pδ˚ ` dδq ` B ` ε Lα

(5)

Knowing the extreme difficulty to solve the correct ambiguity when only the broadcast ephemeris is available, it would be better to avoid its estimation. A close look of Equation (5) would suggest a new parameter to account for the combined effect of dρ, dδ, and B. In fact, this parameter can be defined as: A “ B ` dρ ´ c ¨ dδ (6) Here, the new parameter A is the termed “pseudo-ambiguity”. B is the true ambiguity. dρ ´ c ¨ dδ is the total LOS error caused by the broadcast ephemeris error. Obviously, the pseudo-ambiguity A is a varying parameter due to the variation of dρ ´ c ¨ dδ. 2.4. Properties of Pseudo-Ambiguity An analysis of the properties of the pseudo-ambiguity would reveal the rational for the definition of the parameter and the approach to estimate it. From the definition, it is only necessary to analyze dρ and c ¨ dδ, because B is a constant by its nature. The real data from HY2A is used in the analyses. Firstly, it is noted that the orbit elements and parameters in the broadcast ephemeris are a set of parameters determined through orbit arc fittings, and all the orbits are predicted using the dynamical models by the master station of the GPS system, so the range error dρ caused by the broadcast orbit error should be smooth, and vary slowly. Highly-stable atomic clocks are equipped on GPS satellites and the clock parameters in the broadcast ephemeris are predicted by a quadratic polynomial, so the clock offset error c ¨ dδ has a clear trend. These two error characteristics can be seen clearly in Figure 1, which shows the “true” orbit errors in the radial, along- and cross-track directions, and clock offset errors of GPS satellite PRN04. The “true” errors were computed using the International GNSS Service (IGS) precision products as a reference. There are another two important features presented in Figure 1: (1) because the orbit and clock parameters are released every two hours, the error curves are not continuous, which is a result from the use of new ephemeris, and this phenomenon is called “ephemeris switch” in the following text; (2) in addition to the clear changing trends, small random jittering appears in the curve of the clock offset error. More importantly, it is the combined effect of the GPS orbit and clock offset error of the broadcast ephemeris on the range, the total LOS error dρ ´ c ¨ dδ, between the GPS satellite and the LEO satellite, is of interest. Taking the HY2A satellite and GPS satellite PRN04 on day 2012/001 as an example, Figure 2 shows the range errors caused by the GPS orbit error and the clock offset error, the total LOS error and its inter-epoch difference. Since the trajectory of LEO and GPS satellites are periodic orbits, for every GPS satellite, only part of a GPS orbit arc is visible and tracked by the GPS receivers onboard the LEO satellites. Such a visible arc is called a “tracking arc” in the following text. The thick lines in Figure 2 represent tracking arcs, and the intervals between consecutive lines indicate the GPS satellite is not visible. As can be observed more clearly from Figure 3, if there is no ephemeris switch within a tracking arc, for example, the first arc in Figure 2, the curve of the range error dρ is smooth, and varies slowly, and the curve of the clock offset error dδ presents small random changes, so not

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ephemeris switch within a tracking arc, for example, the first arc in Figure 2, the curve of the range

ephemeris switch smooth, within a tracking arc, for example, arc it in has Figure 2, the random curve of the range only is the total LOS error and varies slowly, the butfirst also small changes up on the error d is smooth, and varies slowly, and the curve of the clock offset error d presents small error and varies slowly, and theswitch curve ofwithin the clocka offset error arc, small the fourth d is smooth, d presents smooth curve. However, if there is an ephemeris tracking for instance, random changes, so not only is the total LOS error smooth, and varies slowly, but also it has small random changes, so not only is the total LOS error smooth, and varies slowly, but also it has small changesis updiscontinuous on the smooth curve. However, if there is anof ephemeris switch within a tracking arc in Figure random 2, the curve and the processing the new curve should be reinitialized. random changes up on the smooth curve. However, if there is an ephemeris switch within a tracking arc, for instance, the fourth arc in Figure 2, the curve is discontinuous and the processing of the new Another important featuretheisfourth that the differences of theand total errors appear arc, for instance, arc ininter-epoch Figure 2, the curve is discontinuous the LOS processing of the new as random curve should be reinitialized. Another important feature is that the inter-epoch differences of the curve should be reinitialized. Another important featureto is zero. that the inter-epoch differences of the noises of a normal with its noises mean close total LOSdistribution errors appear as random ofvery a normal distribution with its mean very close to zero.

Clock Cross-track Along-track Clock [m][m]Cross-track [m][m]Along-track Radial [m][m]Radial [m][m]

total LOS errors appear as random noises of a normal distribution with its mean very close to zero. -1.2 -1.2 -1.4 -1.4 -1.6 day : 2012 001 -1.6 day : 2012 001 1 01 -10 -1 day : 2012 001 -2 -2 day : 2012 001 0.6 0.6 0.0 0.0 -0.6 -0.6 day : 2012 001 -1.2 -1.2 day : 2012 001 -0.4 -0.4 -0.8 -0.8 -1.2 day : 2012 001 -1.2 day : 2012 001 0 2 0 2

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Figure 1. The position and clock offset error of broadcast ephemeris for GPS satellite PRN04. Figure 1. The position and clock offset error broadcast ephemeris GPSPRN04. satellite PRN04. Figure 1. The position and clock offset errorof of broadcast ephemeris for GPSfor satellite -1.2 -1.2 -1.6 -1.6 HY2A : day2012 001

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Figure 2. The LOS error and its inter-epoch difference of HY2A and GPS PRN04.

Figure 2. The LOS error and its inter-epoch difference of HY2A and GPS PRN04. Figure 2. The LOS error and its inter-epoch difference of HY2A and GPS PRN04. Sensors 2015, 15, page–page The 1st Traking Arc without Ephemeris Switch

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Figure 3. The LOS error and its inter-epoch difference in a tracking arc. (a) The first tracking arc of

Figure 3. The Figure LOS2 error its inter-epoch difference in arc a of tracking arc.an ephemeris (a) The first tracking without and ephemeris switch; and (b) the fourth tracking Figure 2 with arc of Figure switch. 2 without ephemeris switch; and (b) the fourth tracking arc of Figure 2 with an ephemeris switch. On the basis of the theoretical and practical data analyses above, a random walk process [26] is used to represent the pseudo-ambiguity parameter. The stochastic model of the random walk process for the pseudo-ambiguity can be presented with a difference equation:

Ak30407 Ak wk 1

(7)

where Ak is the value of the pseudo-ambiguity parameter at epoch tk . Since the inter-epoch difference of the LOS error can be approximated by a normal distribution, as shown in Figures 2 and 3, wk denotes a random variable with Gaussian white noise.

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On the basis of the theoretical and practical data analyses above, a random walk process [26] is used to represent the pseudo-ambiguity parameter. The stochastic model of the random walk process for the pseudo-ambiguity can be presented with a difference equation: A k `1 “ A k ` w k

(7)

where Ak is the value of the pseudo-ambiguity parameter at epoch tk . Since the inter-epoch difference of the LOS error can be approximated by a normal distribution, as shown in Figures 2 and 3 wk denotes a random variable with Gaussian white noise. Within a tracking arc, the process noise is set-based on the change rate of the total LOS error. According to Figures 2 and 3 the inter-epoch difference of the LOS error is usually within ˘0.03 m for the epoch interval of 30 s, so the change rate is usually less than 1 mm/s. The variance of wk depends on the change rate of LOS error. Of course, if a new broadcast ephemeris is used, or a new GPS satellite is tracked, the pseudo-ambiguity parameter and its process noise should be reinitialized. For the former case, the variance of corresponding pseudo-ambiguity parameter in the estimated state vector should be enlarged appropriately. For the latter case, a new pseudo-ambiguity parameter should be introduced into the filter state and the corresponding variance and covariance should be reset simultaneously. In summary, a random walk process is proposed to model the pseudo-ambiguity. It can be estimated in the Kalman filtering process. On one hand, the use of the pseudo-ambiguity avoids the need to estimate the carrier phase ambiguity. On the other hand, the estimate of this parameter reduces the effects of the GPS broadcast orbit and clock offset errors on the range; thus, a higher accuracy of the onboard RTOD can be expected. In terms of the algorithm implementation, the number of estimated parameters in the filter remains the same. 2.5. Parameter Estimation An extend Kalman filter is used to estimate the unknown parameters in dynamic models and measurement equations. In total, the filter state ´ . ¯ . X “ r, r, aw , Cd , Cr , cδR , cδ R , A1 , A2 , ..., An

(8)

` .˘ consists of the spacecraft position and velocity vector r, r , the empirical accelerations aw “ pa R , a T , a N q in the radial, tangential, and normal directions, the of atmospheric ´ coefficients ¯ .

drag and solar radiation pressure pCd , Cr q, the clock offset and rate cδR , cδ R of the GPS receiver, and the pseudo-ambiguity parameters pA1 , A2 , ..., An q of n-dimension, n is the number of all tracked GPS satellites. Of all parameters, three empirical accelerations are modeled by three first-order Gauss-Markov processes, and the coefficients of atmospheric drag and solar radiation pressure are modeled by two random walk processes. Furthermore, each of the pseudo-ambiguity parameters is represented by a random walk process. 2.6. Software On the basis of the above key technique about the pseudo-ambiguity, a RTOD software SATODS is developed, and it is capable of processing both carrier phases and pseudo ranges. All real data from HY2A and ZY3 missions are processed using SATODS, which simulates the onboard operational scenarios. The model and parameter settings of SATODS take the autonomy, timing, and accuracy requirements of onboard RTOD into account. In order to reduce the computational load, at the same time without loss of orbit accuracy, the dynamical models of LEO satellites are simplified to the maximum extent. The neglected perturbations, including the earth radiation and relativistic effect, are small enough and have no notable effect on decimeter level RTOD. Similarly, simplified precession and nutation models are applied for transformation of coordinate systems. In onboard

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RTOD applications, the available Earth orientation parameters (EOP) are usually IERS bulletin-A EOP, which may be uploaded irregularly days or months ago. To simulate this scenario, the EOP used in the RTOD computations are obtained from the Bulletin-A files released every seven days by IERS, each of which can provide the predicted EOP for 365 days. In addition, the Dynamic Model Compensation (DMC) algorithm of empirical acceleration from the literature [4] is adopted. The main parameters of DMC algorithm are the correlation time of exponential decay (τ) and the standard deviation of the forcing noise in radial, along-track, and cross-track direction (σR , σT , σN ). All the specific model and parameter setting are listed in Table 1. Table 1. Models and Parameters Setting of SATODS. Model/Parameter Measurement model GPS data GPS orbit and clock Pseudo-ambiguity Dynamic model Earth Gravity Field Luni-solar gravitation Earth tides Atmosphere Drag Solar radiation pressure Empirical acceleration Other perturbations Reference frame Coordinate system Precession and nutation Earth rotation parameter

Relevant Setting Dual-frequency pseudo-range and carrier-phase data, sampling rate of 30 s Broadcast ephemeris Initial standard deviation σ0 “ 5.0 m, and the standard deviation of the process noise is σu “ 1.0 mm{s ˆ dt, where dt is the epoch interval. EGM 2008, adopt 45 ˆ 45 for HY2A and 60 ˆ 60 for ZY3 Low precision model, Moon and Sun’s position are computed via analytic method Low precision model, k20 solid only Modified Harris-Priester model (density), fixed effective area, estimates Cd parameter Cannonball model, fixed effective area, estimates Cr parameter Dynamic Model Compensation (DMC) [4] with a first-order Gauss-Markov model, τ=60 s, σR : σT : σN “ 5 : 40 : 200 nm/s2 Neglected WGS84 IAU1976/IAU 1980 simplified model Rapid predicted EOP in IERS Bulletin A

3. Flight Data Analyses 3.1. Datasets Experimental datasets covering two five-day intervals, from 2012/001 through 2012/005 and from 2012/032 through 2012/036, respectively, for both the HY2A and ZY3 missions, are processed. Both the ionosphere-free pseudo-range combination Pα and the ionosphere-free carrier phase combination Lα are used as measurements. A summary of the measurement data sets is listed in Table 2. Table 2. Information on missions and datasets.

Mission

Altitude (km)

Data Arc Year/Days

HY2A ZY3

900 500

2012/001–005 2012/032–036

Noise (m) Pα Lα 2.172 2.157

ď0.006 ď0.006

In all the RTOD computations, the real-time broadcast ephemeris is used to calculate the GPS orbits and clock offsets. Reference orbits of HY2A and ZY3 for real-time accuracy assessment were generated by the French National Space Research Center (CNES) and Wuhan University (WHU), respectively, using the IGS high-precision GPS orbit and clock products. According to the literature [27,28], the accuracy of the reference orbits is at 2–5 cm.

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In all the RTOD computations, the real-time broadcast ephemeris is used to calculate the GPS orbits and clock offsets. Reference orbits of HY2A and ZY3 for real-time accuracy assessment were generated by the French National Space Research Center (CNES) and Wuhan University (WHU), respectively, using the IGS high-precision GPS orbit and clock products. According to the literature Sensors 2015, 15, 30403–30418 [27,28], the accuracy of the reference orbits is at 2–5 cm. 3.2. RTOD RTODAccuracy AccuracyAnalysis Analysis 3.2. Twosolutions solutionswere weregenerated generatedfrom fromSATODS SATODSto toexamine examinewhether whetherthe the pseudo-ambiguity pseudo-ambiguity isis able able Two to reduce reduce the the LOS LOS errors. errors. The The first first solution solution was was from from the the use use of of the the ionosphere-free ionosphere-free pseudo-range pseudo-range to P combination only, and it is abbreviated as “pseudo-range based” for convenient expression. The combination Pα only, and it is abbreviated as “pseudo-range based” for convenient expression. otherother solution was was obtained by processing both the pseudo-range and carrier phase The solution obtained by processing bothionosphere-free the ionosphere-free pseudo-range and carrier combinations, the pseudo-ambiguity was estimated in the filter process. To P and L , Land phase combinations, Pα and , and the pseudo-ambiguity was estimated in the process. To α emphasize emphasize the the use use of of the the carrier carrier phase phase data data and and the the effect effect of of the pseudo-ambiguity, pseudo-ambiguity, this this solution solution is is abbreviated abbreviatedas as“carrier-phase “carrier-phasebased”. based”. Figures the 3D 3D RMS RMSof ofthe theRTOD RTODposition positionand andvelocity velocityerrors, errors,where wherethe theformer former Figures 44 and 5 show the is is HY2Asatellite satelliteininthe theinterval interval2012/001–2012/005, 2012/001–2012/005, latter ZY3 satellite forforthetheHY2A andand thethe latter forfor thethe ZY3 satellite in in 2012/032–2012/036. “All” wholefive-day five-day interval. interval. Both Both the the 3D 3D RMS of 2012/032–2012/036. The The labellabel “All” is is forfor thethewhole of pseudo-range pseudo-range based based and andcarrier-phase carrier-phasebased basedsolutions solutionsare arepresented. presented. Pesudo-range Based

Carrier-phase Based

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Figure 4. 4. The The accuracy accuracy (3D (3D RMS) RMS) of of real-time real-time orbits orbitsfor forthe theHY2A HY2Asatellite. satellite. (a) (a) The The position position accuracy; accuracy; Figure and (b) the velocity accuracy. and (b)15, thepage–page velocity accuracy. Sensors 2015, 1.2

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Figure5.5. The The accuracy accuracy (3D (3D RMS) RMS) of of real-time real-timeorbits orbitsfor forZY3 ZY3satellite. satellite. (a) (a) The The position position accuracy; accuracy; and and Figure (b)the thevelocity velocityaccuracy. accuracy. (b)

For the HY2A satellite, the position and velocity accuracy of the pseudo-range-based solution For the HY2A satellite, the position and velocity accuracy of the pseudo-range-based solution are at 0.9–1.1 m and 0.8–1.0 mm/s, respectively, which are similar to GSFC, JPL and DLR’s accuracy are at 0.9–1.1 m and 0.8–1.0 mm/s, respectively, which are similar to GSFC, JPL and DLR’s accuracy level [2–9]. When the ionosphere-free carrier phases are used in the carrier-phase based solution, level [2–9]. When the ionosphere-free carrier phases are used in the carrier-phase based solution, much better accuracy of 0.2–0.4 m for position and 0.2–0.4 mm/s for velocity, respectively, is much better accuracy of 0.2–0.4 m for position and 0.2–0.4 mm/s for velocity, respectively, is achieved. achieved. Slightly worse results are obtained for the ZY3 satellite. A position accuracy of 0.8–1.0 m and Slightly worse results are obtained for the ZY3 satellite. A position accuracy of 0.8–1.0 m and velocity accuracy of 0.8–1.0 mm/s are achieved with the pseudo-range based solution. A superior velocity accuracy of 0.8–1.0 mm/s are achieved with the pseudo-range based solution. A superior accuracy of 0.3–0.5 m for position and 0.3–0.5 mm/s for velocity, respectively, is realized with the accuracy of 0.3–0.5 m for position and 0.3–0.5 mm/s for velocity, respectively, is realized with the carrier-phase-based solution. carrier-phase-based solution. The overall RMS statistics of the real-time orbit accuracy are summarized in Table 3. For the HY2A satellite, the radial (R), along-track (A),30410 cross-track (C), and total (3D) position accuracies of the carrier-phase based solution are better than those of the pseudo-range based solution by 73%, 64%, 64%, and 65%, respectively, and the 3D position accuracy of the carrier-phase based solution is

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The overall RMS statistics of the real-time orbit accuracy are summarized in Table 3. For the HY2A satellite, the radial (R), along-track (A), cross-track (C), and total (3D) position accuracies of the carrier-phase based solution are better than those of the pseudo-range based solution by 73%, 64%, 64%, and 65%, respectively, and the 3D position accuracy of the carrier-phase based solution is 0.334 m. In particular, the radial accuracy of the HY2A satellite is 8.5 cm, which is better than 0.1 m. Similarly, for the ZY3 satellite, the performance improvements are 47%, 53%, 50%, and 51%, respectively, and the 3D position accuracy of the carrier-phase based solution is 0.407 m. Table 3. The overall statistics of real-time orbit accuracy. HY2A Pseudo-Range Carrier-Phase Based Based

Accuracy (RMS)

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0.085 0.256 0.197 0.334 0.226 0.113 0.181 0.311

ZY3 Pseudo-Range Carrier-Phase Based Based 0.348 0.714 0.255 0.835 0.741 0.357 0.304 0.877

0.186 0.339 0.128 0.407 0.360 0.209 0.142 0.440

The position errors of the pseudo-range based and carrier-phase based solutions on several days are shown in Figures 6 and 7 where the former is for the HY2A satellite on 2012/001 and 2012/005, and the latter for the ZY3 satellite on 2012/032 and 2012/035. As can be observed, the error curves for the radial, along-track and cross-track directions, and the 3D position error, for both solutions, present some periodicities. For each solution, the periodicities of orbit error curves are the results of combined effects of many periodic factors, such as the periodicity of the GPS orbit and clock offset error, and the periodical error of dynamic models. However, the variation amplitudes of radial, along-track, cross-track, and 3D position errors of the carrier-phase-based solutions are all notably reduced compared with those of the pseudo-range-based solutions. After the filter process is stabilized, the 3D position errors of the carrier-phase based solutions are all less than 1.0 m, with a RMS value of 0.2–0.5 m. The dynamical models for both solutions are exactly identical, so the results from Figures 6 and 7 demonstrate that the estimated pseudo-ambiguity in the carrier-phase based solution has absorbed a large part of the GPS orbit and clock offset errors and, thus, greatly reduce the orbit determination error. One should note that, the carrier-phase based solution is more dependent on the quality of GPS carrier-phase data. Some exceptions, such as frequent phase cycle-slips or frequent tracking-losses of GPS satellites, may affect the absorption effect of the pseudo-ambiguity parameter on the GPS broadcast ephemeris error, leading to less significant improvement in the orbit accuracy, when compared with the pseudo-range based solution. However, the HY2A and ZY3 experimental data sets only cover two five-day intervals, which may lead to an over-optimistic conclusion on the orbit accuracy of the carrier-phase based solution in this study. In order to further demonstrate the orbit accuracy improvement using this novel algorithm, more space-borne GPS data sets from other satellites are tested. This includes two 30-day intervals, from 2010/001 to 2010/030 and from 2013/030 to 2013/060, respectively, for Europe’s MetOp-A [18] and GRACE-A [20] missions. The altitudes of 820 km for MetOp-A and 460 km for GRACE-A are similar to those of China’s HY2A and ZY3, respectively.

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of radial,reduced along-track, cross-track, and of 3Dthe position errors of the carrier-phase-based are all notably compared with those pseudo-range-based solutions. After thesolutions filter process is notably reduced compared with those of the pseudo-range-based solutions. After the filter process stabilized, the 3D position errors of the carrier-phase based solutions are all less than 1.0 m, with is a stabilized, the 3D position errors of the carrier-phase based solutions are all less than 1.0 m, with a RMS value of 0.2–0.5 m. The dynamical models for both solutions are exactly identical, so the results RMS of 60.2–0.5 The dynamical for bothpseudo-ambiguity solutions are exactly identical, so the results from value Figures and 7m. demonstrate that models the estimated in the carrier-phase based from Figures 6 and 7 demonstrate that the estimated pseudo-ambiguity in the carrier-phase based Sensors 2015, 15, absorbed 30403–30418a large part of the GPS orbit and clock offset errors and, thus, greatly reduce solution has solution absorbed aerror. large part of the GPS orbit and clock offset errors and, thus, greatly reduce the orbit has determination the orbit determination error. HY2A: day2012/001

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Figure 6. 6. Position satellite. (a)(a) Position errors on 2012/001; and Figure Positionerrors errorsofofreal-time real-timeorbits orbitsfor forthe theHY2A HY2A satellite. Position errors on 2012/001; Figure 6. Position errors of real-time orbits for the HY2A satellite. (a) Position errors on 2012/001; and (b) position errors on 2012/005. and (b) position errors on 2012/005. (b) position errors on 2012/005. ZY3: day2012/035

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The daily position and velocity accuracies (RMS) of RTOD results in the radial, along-track and 10 cross-track directions, and 3D position of the carrier-phase-based solution for HY2A and MetOp-A 10 are shown in Figure 8. The accuracies of ZY3 and GRACE-A are shown in Figure 9. The label “All” is for the whole 30-day interval. Comparing subgraph (a) with (c), and (b) with (d) in Figure 8, it can be observed that daily 3D orbit accuracy (RMS) with China’s HY2A satellite is better than 0.4 m, while the daily 3D RMS with MetOp-A is between 0.3–0.5 m. Figure 9 shows that the orbit accuracies of ZY3 are only slightly worse than those of GRACE-A. These two figures demonstrate 30412

The daily position and velocity accuracies (RMS) of RTOD results in the radial, along-track and cross-track directions, and 3D position of the carrier-phase-based solution for HY2A and MetOp-A are shown in Figure 8. The accuracies of ZY3 and GRACE-A are shown in Figure 9. The label “All” is for the whole 30-day interval. Comparing subgraph (a) with (c), and (b) with (d) in Figure 8, it can be observed daily 3D orbit accuracy (RMS) with China’s HY2A satellite is better than 0.4 m, while Sensors 2015,that 15, 30403–30418 the daily 3D RMS with MetOp-A is between 0.3–0.5 m. Figure 9 shows that the orbit accuracies of ZY3 are only slightly worse than those of GRACE-A. These two figures demonstrate that the RTOD that the RTOD with and China’s andare ZY3 at the level with thoseand of accuracies withaccuracies China’s HY2A ZY3HY2A missions at missions the sameare level withsame those of MetOp-A MetOp-A and GRACE-A. Although the quality of China’s space-borne GPS receiver is different from GRACE-A. Although the quality of China’s space-borne GPS receiver is different from that of that of Europe’s receivers, it is believed pseudo-ambiguity parameter playsa amajor major role role in Europe’s receivers, it is believed that that the the pseudo-ambiguity parameter plays in achieving accurate carrier-phase based orbit solutions. Therefore, it can be generally concluded that achieving accurate carrier-phase based orbit solutions. Therefore, it can be generally concluded that the the method method presented presented in in this this paper paper can can provide provide the the carrier-phase carrier-phase based based orbit orbit accuracy accuracy of of 0.3–0.5 0.3–0.5 m m for for velocity. velocity. for position position and and 0.3–0.5 0.3–0.5 mm/s mm/s for R

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(d) Figure 8. 8. Orbit for HY2A HY2A and and MetOp-A. (a) Position Position Figure Orbit accuracy accuracy of of the the carrier-phase carrier-phase based based solutions solutions for MetOp-A. (a) accuracy of HY2A; (b) velocity accuracy of HY2A; (c) position accuracy of MetOp-A; and (d) velocity accuracy of HY2A; (b) velocity accuracy of HY2A; (c) position accuracy of MetOp-A; and (d) velocity accuracy of MetOp-A. accuracy of MetOp-A.

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Figure 8. Orbit accuracy of the carrier-phase based solutions for HY2A and MetOp-A. (a) Position of HY2A; (b) velocity accuracy of HY2A; (c) position accuracy of MetOp-A; and (d) velocity Sensorsaccuracy 2015, 15, 30403–30418 accuracy of MetOp-A. 0.5 A

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3.3. Effect of Pseudo-Ambiguity 3.3. Effect of Pseudo-Ambiguity Orbit Orbit results results with with higher higher precision precision are are obtained obtained using using the the carrier-phases carrier-phases compared compared with with those those using only the pseudo-ranges. On one hand, this noteworthy performance benefits from the fact that using only the pseudo-ranges. On one hand, this noteworthy performance benefits from the fact that the ionosphere-free dual-frequency carrier phases L have a ranging noise far lower than that of the ionosphere-free dual-frequency carrier phases L have a ranging noise far lower than that of the α

the ionosphere-free dual-frequency pseudo-ranges On the other hand, credits should be P . other ionosphere-free dual-frequency pseudo-ranges Pα . On the hand, credits should be attributed to attributed to the introduction and estimation of the pseudo-ambiguity, which absorbs a large part of the LOS error caused by the GPS broadcast orbit and clock offset error. Figure 10 shows two LOS error curves on30414 day 2012/001 between HY2A and GPS satellite. The green curve is the “true” total LOS error curve computed using the GPS broadcast orbit and clock offset, and the IGS high-precision GPS orbit and clock products. The red one is the estimated LOS error in the form of the pseudo-ambiguity (excluding the true ambiguity) using the carrier-phases.

3.3. Effect of Pseudo-Ambiguity Orbit results with higher precision are obtained using the carrier-phases compared with those using only the pseudo-ranges. On one hand, this noteworthy performance benefits from the fact that Sensors 2015, 15, 30403–30418 the ionosphere-free dual-frequency carrier phases L have a ranging noise far lower than that of the ionosphere-free dual-frequency pseudo-ranges P . On the other hand, credits should be attributed to the introduction and of the pseudo-ambiguity, absorbs large part of the introduction and estimation of estimation the pseudo-ambiguity, which absorbswhich a large part ofathe LOS error the LOSby error by the GPS and clock offset error. caused the caused GPS broadcast orbitbroadcast and clockorbit offset error. Figure 10 day 2012/001 between HY2A and and GPS GPS satellite. The 10 shows showstwo twoLOS LOSerror errorcurves curvesonon day 2012/001 between HY2A satellite. green curvecurve is the is “true” total LOS error using theusing GPS broadcast orbit and clock The green the “true” total LOScurve errorcomputed curve computed the GPS broadcast orbit offset, and offset, the IGSand high-precision GPS orbit and clock products. Theproducts. red one is The the estimated and clock the IGS high-precision GPS orbit and clock red one isLOS the error in the form of the pseudo-ambiguity (excluding the true ambiguity) using the carrier-phases. estimated LOS error in the form of the pseudo-ambiguity (excluding the true ambiguity) using the To separate theToestimated LOS error from the pseudo-ambiguity, the true the ambiguity in the carrier-phases. separate the estimated LOS error from the pseudo-ambiguity, true ambiguity pseudo-ambiguity is computed and excluded using the IGS high-precision orbit and clock products in the pseudo-ambiguity is computed and excluded using the IGS high-precision orbit and clock via post-processing. As can beAs observed, the two curves follow a similar that the products via post-processing. can be observed, the two curves follow apattern, similar indicating pattern, indicating LOS error caused the GPS broadcast orbit and offset offset errorserrors is well absorbed through the that the LOS error by caused by the GPS broadcast orbitclock and clock is well absorbed through estimate of the pseudo-ambiguity. Comparisons of the two LOS error curves in two tracking arcs are the estimate of the pseudo-ambiguity. Comparisons of the two LOS error curves in two tracking arcs presented in Figure 11. 11. In the 1st1st arc,arc, thethe two curves dodonot are presented in Figure In the two curves notagree agreewell wellbecause becausethe thefilter filterprocess process is is not not stabilized. stabilized. This This means means the the LOS LOS error error is is not not well well absorbed, absorbed, resulting resulting in in less less accurate accurate RTOD RTOD orbits. orbits. Within follow a Within the the fourth fourth tracking tracking arc, arc, although althoughthere thereisisan anephemeris ephemerisswitch, switch,the thetwo twoerror errorcurves curves follow similar pattern, because the filter process a similar pattern, because the filter processhas hasbeen beenstabilized. stabilized.

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Figure 12 shows the RMS of the original total LOS error of HY2A about every GPS satellite in Figure 12 shows the RMS of the original total LOS error of HY2A about every GPS satellite in the whole interval 2012/001–2012/005 and the RMS of its residual errors after the pseudo-ambiguity the whole interval 2012/001–2012/005 and the RMS of its residual errors after the pseudo-ambiguity is estimated. The label “All” represents the overall statistics about all tracked GPS satellites. As can is estimated. The label “All” represents the overall statistics about all tracked GPS satellites. As can be seen clearly, for every GPS satellite, the original LOS error is at 0.5–2.0 m level, but it is reduced to be seen clearly, for every GPS satellite, the original LOS error is at 0.5–2.0 m level, but it is reduced 0.1–0.4 m level due to the absorption effect of the pseudo-ambiguity. Apparently, a large part of the to 0.1–0.4 m level due to the absorption effect of the pseudo-ambiguity. Apparently, a large part of LOS error has been absorbed by the pseudo-ambiguity. In fact, according to overall statistics labeled the LOS error has been absorbed by the pseudo-ambiguity. In fact, according to overall statistics “All”, the residual error after the estimate of the pseudo-ambiguity is only about 25% of the original LOS errors , which is a primary contributing factor for the realization of the decimeter level 30415 (0.3–0.5 m) RTOD. 2.0

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Figure 12 shows the RMS of the original total LOS error of HY2A about every GPS satellite in the whole interval 2012/001–2012/005 and the RMS of its residual errors after the pseudo-ambiguity is estimated. The label “All” represents the overall statistics about all tracked GPS satellites. As can be seen clearly, for every GPS satellite, the original LOS error is at 0.5–2.0 m level, but it is reduced to Sensors 2015, 15, 30403–30418 0.1–0.4 m level due to the absorption effect of the pseudo-ambiguity. Apparently, a large part of the LOS error has been absorbed by the pseudo-ambiguity. In fact, according to overall statistics labeled “All”, the residual error after theafter estimate of the pseudo-ambiguity is only about of the labeled “All”, the residual error the estimate of the pseudo-ambiguity is only25% about 25%original of the LOS errors which is a primary contributing factor forfor thethe realization level original LOS ,errors , which is a primary contributing factor realizationofofthe the decimeter level (0.3–0.5m) m)RTOD. RTOD. (0.3–0.5 2.0

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4. Conclusions 4. Conclusions Many of of the the spaceborne spaceborne Earth Earth observation observation systems, systems, such such as as China’s China’s HY2A HY2A and and ZY3 ZY3 satellites, satellites, Many require onboard onboard RTOD RTOD with with decimeter decimeter position positionaccuracy. accuracy. Although Although these these satellites satellites are are equipped equipped require dual-frequency carrier phase GPS receivers, the limited computing capacity and system autonomy dual-frequency carrier phase GPS receivers, the limited computing capacity and system autonomy requirement prevent prevent the the implementation implementation of of complicated complicated orbit orbit determination determination software, software, such such as as the the requirement post-processingprecise preciseorbit orbitdetermination determinationsoftware. software. post-processing This paper presents some of key techniques in developing SATODS software, an onboard This paper presents some of key techniques in developing SATODS software, an onboard RTOD RTOD system using dual-frequency carrier phase measurements. Experiment results using tracking system using dual-frequency carrier phase measurements. Experiment results using tracking data data China’s from China’s HY2A ZY3 missions are presented. of the key developments is the from HY2A and ZY3and missions are presented. One of theOne key developments is the introduction introduction of the “pseudo-ambiguity” parameter, which is aof combination of the phase of the “pseudo-ambiguity” parameter, which is a combination the phase ambiguity, theambiguity, orbit and the orbit and clockfrom offset from the GPS broadcast ephemeris. parameter modeledwalk as a clock offset errors theerrors GPS broadcast ephemeris. This parameterThis is modeled as ais random random walk process, and estimated in the extend Kalman filter. It has been shown to be able to process, and estimated in the extend Kalman filter. It has been shown to be able to absorb a large absorb a large part of the LOS error originated from the GPS broadcast ephemeris, which is a key part of the LOS error originated from the GPS broadcast ephemeris, which is a key factor to achieve factorresults to achieve orbit results with accuracies at for the position decimeter level position and sub mm/s for orbit with accuracies at the decimeter level and sub for mm/s for velocity. The results velocity. The results demonstrate that the onboard RTOD accuracy of 0.2–0.4 m for position and 0.2– demonstrate that the onboard RTOD accuracy of 0.2–0.4 m for position and 0.2–0.4 mm/s for velocity 0.4achieved mm/s for velocity is achieved the GPS ephemeris and dual-frequency carrier is with the GPS broadcast with ephemeris andbroadcast dual-frequency carrier phase measurements for the HY2A satellite. A slightly inferior accuracy of 0.3–0.5 m for position and 0.3–0.5 mm/s for velocity 14 is obtained for the ZY3 satellite. The robust results presented here show that the onboard RTOD for LEO space missions can achieve orbit accuracy at 3–5 dm with a standalone dual-frequency GPS receiver and the GPS broadcast ephemeris. Using the algorithm presented in this paper, a better and more accurate onboard RTOD system with higher data utilization has been developed. The system is expected to fly on other missions to enhance real-timeliness of China’s Earth observation systems. Acknowledgments: This work was supported by the National Natural Science Foundation of China (Nos. 41374035 and 41474024). Author Contributions: In this study, Fuhong Wang designed the algorithms, performed experiments and wrote this paper. Xuewen Gong designed the software, performed experiments, analyzed data and edited the manuscript. Jizhang Sang and Xiaohong Zhang supervised its analysis and edited the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

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