Improving Low Earth Orbit (LEO) Prediction with Accelerometer Data

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Article Improving Low Earth Orbit (LEO) Prediction with Accelerometer Data

Haibo Ge 1,2, Bofeng Li 1,* , Maorong Ge 2,3, Liangwei Nie 2,3 and Harald Schuh 2,3

1 College of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China; [email protected] 2 Department of Geodesy, GeoForschungsZentrum (GFZ), Telegrafenberg, 14473 Potsdam, Germany; [email protected] (M.G.); [email protected] (L.N.); [email protected] (H.S.) 3 Institut für Geodäsie und Geoinformationstechnik, Technische Universität, 10963 Berlin, Germany * Correspondence: [email protected]

Received: 10 April 2020; Accepted: 15 May 2020; Published: 17 May 2020

Abstract: Low Earth Orbit (LEO) satellites have been widely used in scientific fields or commercial applications in recent decades. The demands of the real time scientific research or real time applications require real time precise LEO orbits. Usually, the predicted orbit is one of the solutions for real time users, so it is of great importance to investigate LEO orbit prediction for users who need real time LEO orbits. The centimeter level precision orbit is needed for high precision applications. Aiming at obtaining the predicted LEO orbit with centimeter precision, this article demonstrates the traditional method to conduct orbit prediction and put forward an idea of LEO orbit prediction by using onboard accelerometer data for real time applications. The procedure of LEO orbit prediction is proposed after comparing three different estimation strategies of retrieving initial conditions and dynamic parameters. Three strategies are estimating empirical coefficients every one cycle per revolution, which is the traditional method, estimating calibration parameters of one bias of accelerometer hourly for each direction by using accelerometer data, and estimating calibration parameters of one bias and one scale factor of the accelerometer for each direction with one arc by using accelerometer data. The results show that the predicted LEO orbit precision by using the traditional method can reach 10 cm when the predicted time is shorter than 20 min, while the predicted LEO orbit with better than 5 cm for each orbit direction can be achieved with accelerometer data even to predict one hour.

Keywords: Low Earth Orbit (LEO); orbit prediction; accelerometer data; calibration parameters; empirical coeﬃcient; initial conditions; dynamic parameters

1. Introduction More and more Low Earth Orbit (LEO) satellites have been launched to explore various phenomena on Earth [1,2], e.g., ocean altimetry [3,4], climate change, and Earth mass change [5]. In recent years, many research plans or commercial projects depending on a large number of LEO satellites have been proposed with the expectation of providing real time services on a global scale [6–8]. Meanwhile, the idea of a LEO enhanced Global Navigation Satellite System (LeGNSS) [9,10], where LEO satellites will transmit navigation signals to be used as navigation satellites, was put forward with such a large number of LEO satellites. With such a huge demand of real time services and diﬀerent kinds of real time applications, especially for the real time Precise Point Positioning (PPP) of LeGNSS, real time centimeter level LEO orbits are the prerequisite for real time users. Near real time LEO orbits are provided for the ﬁelds of satellite occultation for meteorological purposes [11,12] and the short-latency monitoring of continental, ocean, and atmospheric mass variations [13,14]. The European Space Agency (ESA) has deployed an operational system for routine Earth Observation named Copernicus

Remote Sens. 2020, 12, 1599; doi:10.3390/rs12101599 www.mdpi.com/journal/remotesensing Remote Sens. 2020, 12, 1599 2 of 18 since 2014, which is designed to support a sustainable European information network by monitoring, recording, and analyzing environmental data and events around the globe [15]. The Copernicus program consists of six different families of satellites being the first three missions Sentinel -1, -2, and -3. These missions have the requirements of 8-10 cm in less than 30 min for Near Real Time (NRT) applications to 2-3 cm in less than one month for Non-Time Critical (NTC) applications [16]. The orbit accuracy of Sentinel-2A NRT products is better than 5 cm, which is better than the requirement of 10 cm as well as better than the accuracy of previous NRT LEO orbits studies [17,18]. Though the accuracy of the NRT orbit is on the centimeter level, the latency of about 30 min is not acceptable for real time positioning service of LeGNSS. Thus, predicted LEO orbit with centimeter precision should be investigated. In this study, we try to find a proper way to conduct the prediction of LEO orbits with an orbit precision on the centimeter level for real time applications, especially for the case of LeGNSS. In order to obtain such highly accurate predicted LEO orbit for precise real time applications, precise initial conditions (position and velocity at reference time) as well as the dynamic parameters of LEO satellites used for LEO orbit prediction are required as fast as possible via LEO Precise Orbit Determination (POD). Thus, standard GNSS navigation message information is not adequate for LEO POD since the qualities of GNSS orbits and clocks are not good enough. The precision of the LEO orbit is usually at the decimeter or meter level by using the standard GNSS navigation message [12,19]. Under this circumstance, precise GNSS products should be used for LEO POD in order to get precise initial conditions and dynamic parameters of LEO satellites. Due to the Real Time Pilot Project (RTPP) launched by the International GNSS Service (IGS) in 2007 and officially operated since 2012 [20], real time precise satellite orbit and clock correction products can be obtained with Real Time Service (RTS). Currently, there are also many commercial providers, which deliver real time GPS/GNSS orbit and clock correction products, e.g., Veripos, magicGNSS, etc. Obviously, RTS can be applied to LEO POD both in the kinematic and dynamic mode. However, most onboard LEO receivers have not yet used these corrections, as it might not yet be mature enough for space applications. An alternative way is to conduct LEO POD at the analysis center with the RTS on the ground with powerful computational resources. As for the POD mode, though either kinematic or dynamic POD can achieve centimeter level orbit precision, we prefer to choose the dynamic method since the orbit of this method is consecutive and stable while that of the kinematic method is discrete. Above all, we conduct dynamic orbit determination, which estimates LEO initial conditions and dynamic parameters with real time precise GNSS orbit and clock products. Then, these dynamic parameters and initial conditions are used to do the orbit integration for times in the future, referred as orbit prediction. Here, two issues should be taken into consideration. The first one is the update rate of the precise LEO orbit. The update rate of the LEO orbit depends on the time gap of LEO onboard observation data collection as well as the calculation time of LEO POD. The LEO onboard observations cannot be transferred to the ground stations all the time and the LEO orbit results calculated at an analysis center cannot be always uploaded to the LEO satellites unless the LEO satellites fly over the uplink and downlink stations. It is reasonable to assume that the onboard data can be collected every one hour (more than half LEO orbit revolution, personal communication with Xiangguang Meng, who is in charge of Chinese FY3C near real time POD). The second issue is the length of the orbit prediction. An empirical force model with parameters is usually utilized in dynamic LEO precise orbit determination to absorb the unmodeled part of non-gravitational forces, such as atmosphere drag and solar radiation pressure [21–24]. These parameters are important for the precision of orbit prediction if they fit the real characteristics of the unmodeled part of non-gravitational forces for the prediction arc. Currently, some LEO satellites carry accelerometers to measure non-gravitational accelerations due to the surface forces acting on the LEO satellites, such as STAR instrument on Challenging Minisatellite Payload (CHAMP), SuperSTAR instruments on Gravity Recovery and Climate Experiment (GRACE), and GRADIO instruments on the Gravity field and steady-state Ocean Circulation Explorer (GOCE). These instruments have extreme sensitivity and unprecedented accuracy with 9 2 10 2 12 2 up to 10− m/s / √HZ, 10− m/s / √HZ, and 10− m/s / √HZ for STAR, SuperSTAR, and GRADIO, Remote Sens. 2020, 12, 1599 3 of 18 respectively. The measurement from the accelerometers has to be calibrated when processing the Remote Sens. 2018, 10, x FOR PEER REVIEW 3 of 19 measurement from their missions to produce the gravity field models [25]. Researchers proposed a calibrationprocessing method, the measurement which displays from approximatelytheir missions constantto produce scales theand gravity slowly field changing models biases[25]. for both GRACEResearchers A and proposed B satellites a calibration [26]. Numerous method, accelerationwhich displays spikes approximately related to theconstant switching scales activityand in the circuitsslowly of changing onboard biases heaters for are both identified GRACE andA and analyzed B satellites [27]. [26]. It is Numerous a possibility acceleration to use the spikes accelerometer related to the switching activity in the circuits of onboard heaters are identified and analyzed [27]. It data for LEO POD instead of empirical force models [28], [29]. The calibration parameters of the is a possibility to use the accelerometer data for LEO POD instead of empirical force models [28], accelerometer such as scale and bias should be estimated together with the LEO initial position and [29]. The calibration parameters of the accelerometer such as scale and bias should be estimated velocitytogether when with usingthe LEO accelerometer initial position data. and velocity Inspired when by the using stability accelerometer of thecalibration data. Inspired parameters by the of the accelerometer,stability of the we calibration try to use parameters accelerometer of the dataaccelerometer, to predict we the try LEO to use orbit accelerometer instead of data empirical to force models.predict Wethe LEO compare orbit instead two methods of empirical of accelerometer force models. We data compare calibrations two methods with of respect accelerometer to the empirical forcedata models calibrations in order with torespect develop to the the empirical potential fo LEOrce models orbit predictionin order to develop process the strategy. potential LEO orbitThe prediction article isprocess structured strategy. as follows. Different possible procedures of LEO orbit prediction are discussedThe inarticle the nextis structured section. as The follows. corresponding Different possible methods procedures of LEO of orbit LEO prediction orbit prediction are conducted are as discussed in the next section. The corresponding methods of LEO orbit prediction are conducted as well as the analysis of calibration parameters of accelerometers in the following sections. Then the well as the analysis of calibration parameters of accelerometers in the following sections. Then the extensiveextensive LEO LEO orbit orbit prediction prediction experiments experiments are are carried carried out, out, including including the the impact impact of of di ffdifferenterent integration intervalsintegration on theintervals LEO on orbit the precision.LEO orbit precision. Research Research findings findings and concluding and concluding remarks remarks are givenare in the followinggiven in section.the following Finally, section. practical Finally, issues practical about issues widespread about widespread use of use an accelerometerof an accelerometer are are discussed. discussed. 2. LEO Orbit Prediction Procedure 2. LEO Orbit Prediction Procedure The procedures of LEO orbit prediction for real time applications will be clarified. The simplified demonstrationThe procedures and flowchart of LEO areorbit shown prediction as Figure for re1.al time applications will be clarified. The simplified demonstration and flowchart are shown as Figure 1.

FigureFigure 1. 1.Procedures Procedures of of Low Low Earth Earth Orbit Orbit (LEO) (LEO) orbit orbit prediction. prediction. The LEO The onboar LEOd onboard Global Navigation Global Navigation SatelliteSatellite System System (GNSS) (GNSS) andand accelerometeraccelerometer data data are are downlinked downlinked and and transferred transferred to tothe the analysis analysis center andcenter processed and processed there. Thethere. estimated The estimated initial initial condition condition states states of LEO of LEO as wellas well as theas the dynamic dynamic parameters areparameters then uplinked are then to uplinked LEO satellites to LEO andsatellites the predictedand the predicted LEO orbit LEO isorbit integrated is integrated onboard. onboard.

TheThe downlink downlink stations stations receivereceive thethe LEOLEO onboardonboard GNSS GNSS observations observations and and accelerometer accelerometer data data when LEOwhen satellites LEO satellites pass over pass the over downlink the downlink stations. stations. LEO LEO POD POD can becan conducted be conducted in thein the analysis analysis center with realcenter time with GNSS real products time GNSS from products RTS. LEO from initial RTS. conditions LEO initial and conditions the corresponding and the corresponding dynamic parameters dynamic parameters can be obtained after LEO POD. Then these parameters could be uplinked to can be obtained after LEO POD. Then these parameters could be uplinked to the corresponding LEO satellites for orbit prediction and real time LEO orbits can be broadcast to the users for real time applications. As is well known, the length of LEO orbit prediction determines the precision of real time RemoteRemote Sens. Sens.2020 2018, ,12 10,, 1599x FOR PEER REVIEW 4 of 19 4 of 18

the corresponding LEO satellites for orbit prediction and real time LEO orbits can be broadcast to the LEOusers orbits. for real However, time applications. the length As of is orbit well know predictionn, the length is fully of dependent LEO orbit prediction on the time determines gap of LEO the onboard observationprecision of data real time collection LEO orbits. and computationHowever, the leng timeth of of LEO orbit POD.prediction It takes is fully about dependent 1.5–1.7 on h forthe one LEO satellitetime gap cycle of LEO with onboard the altitude observation from 500 data to 1000collection km. and For thecomputation computation time timeof LEO of POD. LEO PODIt takes with a 24 h arcabout length, 1.5–1.7 it takes hours about for one 10 minLEO for satellite one LEO cycle satellite with the at altitude the GFZ from server. 500 Into general,1000 km. LEO For satellitesthe ﬂy computation time of LEO POD with a 24 hour arc length, it takes about 10 minutes for one LEO overhead a ground station for 10–15 min. It is possible to download the observation data, calculate satellite at the GFZ server. In general, LEO satellites fly overhead a ground station for 10–15 minutes. LEO orbit and inject the data to the corresponding LEO satellites in 15 min. Then after a half cycle of It is possible to download the observation data, calculate LEO orbit and inject the data to the aboutcorresponding 50 min, this LEO procedure satellites canin 15 be minutes. done again Then at after another a half downlink cycle of andabout uplink 50 minutes, station. this Under this circumstance,procedure can one be done hour again is enough at another for LEOdownlink orbit and prediction uplink station. and we Under took this this circumstance, length for our one following analysis.hour is enough For the for sake LEO of insurance,orbit prediction we alsoand preparedwe took this two length hours for as our backup following if the analysis. orbital For elements the could notsake be uplinkedof insurance, in timewe also and prepared were delayed two hours to be as uplinked backup if at the the orbital nextuplink elements station. could Thenot be timeline of datauplinked process in istime shown and inwere Figure delayed2. If to the be time uplinked gap of at LEO the next onboard uplink observation station. The data timeline collection of data is shorter, theprocess length is of shown orbit in prediction Figure 2. If time the time can begap even of LEO shorter. onboard observation data collection is shorter, the length of orbit prediction time can be even shorter.

Figure 2. Timeline of the data process. Figure 2. Timeline of the data process. With these procedures, we proposed three strategies to fulﬁll LEO prediction. We took GRACE-A With these procedures, we proposed three strategies to fulfill LEO prediction. We took dataGRACE-A for example data for with example its data with from its Daydata Offrom Year Day (DOY) Of Year 010, (DOY) 2007 to010, DOY 2007 360, to DOY 2007. 360, As 2007. aforementioned, As theaforementioned, LEO initial conditions the LEO initial and conditions dynamic and parameters dynamic parameters determinate determinate the precision the precision of predicted of LEO orbits.predicted The LEO accelerometer orbits. The accelerometer measures all measures non-gravitational all non-gravitational forces forces acting acting on the on satellite.the satellite. However, theHowever, measurement the measurement has to be calibratedhas to be calibrated in terms in of terms bias andof bias scale and when scale theywhen are they used are used to process to orbit determination.process orbit determination. The bias and The scale bias have and scale to be have estimated to be estimated together together with the with LEO the initialLEO initial position and velocity.position The and calibrationvelocity. The model calibration can bemodel written can be as written as

=+ (1) ang = B + Ka (1)

where denotes the non-gravitational acceleration. =Bh B B denotesiT the bias vector. where=diagangdenotesK K theK non-gravitational denotes the scale acceleration.matrix. denotesB = theB Xnon-gravitationalBY BZ denotes measurement. the bias vector. h i K Note= diag both KX andKY K areZ indenotes the GRACE the scale Science matrix. Referencea denotes Frame the (SRF), non-gravitational while LEO POD measurement. is Noteconducted both a ngin andthe inertiala are in system. the GRACE Thus, one Science needs Reference to transform Frame (SRF), from SRF while to LEOthe inertial POD isframe, conducted in which is the inertial system. Thus, one needs to transform ang from SRF to the inertial frame, which is = =+ (2) aCIS = Mang = M(B + Ka) (2) where is the transform matrix from SRF to the inertial frame. where M is the transform matrix from SRF to the inertial frame. The proposed three strategies are mainly focused on the determination of dynamic parameters in LEOThe POD proposed and they three are strategies as follows. are mainly focused on the determination of dynamic parameters in LEO POD and they are as follows. Strategy 1: Accelerometer data are used with the calibration parameters of biases and scale factors in each axis for one arc, which are named as BX,BY,BZ,KX,KY, and KZ. For simplicity and easy reading, ACC_B_K is used in the following study referring to this strategy. Strategy 2: Accelerometer data are used with the calibration parameters of biases in each axis every 60 min, which are named as BX,BY, and BZ and scale factors for three directions are equal to one. This strategy is called ACC_B in the following. Remote Sens. 2020, 12, 1599 5 of 18

Strategy 3: Empirical force models are used with the dynamic coeﬃcients of Ca, Sa, Cc, and Sc for along- and cross-track directions (cosine/sine), and a scale factor for the atmospheric drag every one cycle per revolution. EMP is used referring to this strategy in the following. Strategy 1 (ACC_B_K) and 2 (ACC_B) are diﬀerent at the estimation of calibration parameters of the accelerometer, which are based on the GFZ GRACE Level-2 processing standards document [30,31]. Strategy 3 (EMP) was conducted to show the precision of predicted LEO orbits with empirical force models. Actually, there is a shortcoming with the method using accelerometer data, for which orbit prediction must be conducted onboard, since real time accelerometer data should be acquired. The empirical method is simple that one can conduct the orbit prediction at the analysis center, then uplink the LEO orbits to the corresponding LEO satellites. The batch least square approach was used to estimate the LEO initial position and velocity as well as dynamic parameters. The force models, observation models, and the parameters to be estimated in LEO POD are listed in Table1.

Table 1. Force models, observation models, and estimated parameters of LEO POD with diﬀerent strategies.

Force Models Description ACC_B_K ACC_B EMP Earth gravity EIGEN-6C [32] 120 120 √ √ √ × N-body JPL DE405 [33] √ √ √ Solid earth tide IERS Conventions 2010 [34] √ √ √ Ocean tide EOT11a [35] √ √ √ Relativity effect IERS Conventions 2010 [32] √ √ √ Atmosphere drag DTM94 [36] √ × × Solar radiation Macro model [37] √ × × Observation models √ √ √ Observation Un-differenced ionosphere-free code and phase combination √ √ √ Arc length 24 h √ √ √ Sampling rate 30 s √ √ √ Cutoff elevation 3◦ √ √ √ Phase wind up Applied [38] √ √ √ LEO phase centre offset Applied [39] √ √ √ GPS phase centre offset igs08.atx [40] √ √ √ Relativity effect IERS Conventions 2010 [32] √ √ √ Tropospheric delay Not relevant × × × Parameters LEO orbits Initial conditions (positions and velocities) √ √ √ Receiver clock Estimated as white noise √ √ √ Atmosphere drag A scale factor estimated every one cycle per revolution (1.5 h) √ × × Ca, Sa, Cc, and Sc for along- and cross-track directions (cosine/sine) Empirical force √ every one cycle per revolution × × Accelerometer scale K ,K , and K , one set for one arc √ X Y Z × × Accelerometer bias B ,B ,B , one set for one arc for ACC_B_K, every 60 min for ACC_B √ √ X Y Z × Phase ambiguity One per satellite per pass (float solution) √ √ √

As recorded by the GRACE Science data system monthly report, the Disabling of Supplemental Heater Lines (DSHL) of GRACE-A would cause temperature control on accelerometer to be stopped. The cool down of the accelerometer caused the accelerometer biases to change and the reheating of the accelerometer returned the accelerometer biases to near nominal values after some days, which would aﬀect POD by using accelerometer data. For this reason, these days are excluded in the following analysis, such as the data from 17th to 21st January (DOY: 017 to 021, 2007) and from 22nd to 26th November (DOY: 326 to 330, 2007). We used the slide window process to simulate the real time process. The length of slide window is equal to the length of prediction time. So, we have 24 sets of results in one day if the length of prediction time is one hour.

3. Analysis of Dynamic Parameters In this section, we ﬁrst checked the precision of LEO orbits by using precise GNSS products with those three strategies. Here, we used CODE (Center for Orbit Determination in Europe) precise GPS orbit and 30 s clock products since there was no real time precise orbit and clock products in 2007 (ftp://cddis.gsfc.nasa.gov/pub/gps/products,[41]). Furthermore, one has to collect and store real time Remote Sens. 2018, 10, x FOR PEER REVIEW 6 of 19

Accelerometer scale KX, KY, and KZ, one set for one arc  × × BX, BY, BZ, one set for one arc for ACC_B_K, Accelerometer bias   × every 60 minutes for ACC_B Phase ambiguity One per satellite per pass (float solution)    As recorded by the GRACE Science data system monthly report, the Disabling of Supplemental Heater Lines (DSHL) of GRACE-A would cause temperature control on accelerometer to be stopped. The cool down of the accelerometer caused the accelerometer biases to change and the reheating of the accelerometer returned the accelerometer biases to near nominal values after some days, which would affect POD by using accelerometer data. For this reason, these days are excluded in the following analysis, such as the data from 17th to 21st January (DOY: 017 to 021, 2007) and from 22nd to 26th November (DOY: 326 to 330, 2007). We used the slide window process to simulate the real time process. The length of slide window is equal to the length of prediction time. So, we have 24 sets of results in one day if the length of prediction time is one hour.

3. Analysis of Dynamic Parameters Remote Sens. 2020, 12, 1599 6 of 18 In this section, we first checked the precision of LEO orbits by using precise GNSS products with those three strategies. Here, we used CODE (Center for Orbit Determination in Europe) precise orbitGPS and clockorbit and corrections 30 s clock by products themselves since since there no was public no real archive time isprecise available orbit forand the clock storage products of previous in real time2007 orbit(ftp://cddis.gsfc.nasa.gov/pub/ and clock products. Moregps/products, details about [41]). the Furthermore, impact on LEO one POD has to with collect real and time store products and finalreal time products orbit and can clock be found corrections in the by Appendix themselvesA since. The no Precise public archive Science is Orbits available (PSO) for the of storage GRACE-A of previous real time orbit and clock products. More details about the impact on LEO POD with real provided by Jet Propulsion Laboratory (JPL) was used as references for our orbit evaluation in this time products and final products can be found in the Appendix. The Precise Science Orbits (PSO) of article (https://podaac-tools.jpl.nasa.gov/drive/files/allData/grace/L1B/JPL). Then the accelerometer GRACE-A provided by Jet Propulsion Laboratory (JPL) was used as references for our orbit calibrationevaluation parameters in this article as well (https://podaac-tools.jpl.nasa.gov/drive/files/allData/grace/L1B/JPL). as the parameters of empirical models were also investigated Then for the first twothe accelerometer strategies. Figure calibration3 shows parameters the averaged as well Rootas the Mean parameters Square of (RMS)empirical values models for were along-track, also cross-track,investigated and radialfor the directions first two strategies. with the threeFigure strategies, 3 shows the respectively. averaged Root The correspondingMean Square (RMS) averaged RMSvalues values for of along-track, all days for cross-track, three directions and radial are showndirections in Tablewith the2. three strategies, respectively. The corresponding averaged RMS values of all days for three directions are shown in Table 2.

FigureFigure 3. Averaged 3. Averaged Root MeanRoot Mean Square Square (RMS) (RMS) values values for along-track, for along-track, cross-track, cross-track, and radialand radial directions withdirections the three strategies.with the three strategies.

In general,Table 2. theAveraged RMS values RMS are values smallest on all for days all directions for three directions when empirical with three force strategies. models are used. These empirical models can efficiently absorb the unmodeled part of the non-gravitational forces. Along-Track (mm) Cross-Track (mm) Radial (mm) The averaged RMS values for all days were 12.1 mm, 6.2 mm, and 6.8 mm for along-track, cross-track, andACC_B_K radial direction, respectively. 23.3 For radial direction, 14.3 the RMS values 6.8 of the two ACC_B 11.8 11.9 6.1

EMP 12.1 6.2 6.8

In general, the RMS values are smallest for all directions when empirical force models are used. These empirical models can efficiently absorb the unmodeled part of the non-gravitational forces. The averaged RMS values for all days were 12.1 mm, 6.2 mm, and 6.8 mm for along-track, cross-track, and radial direction, respectively. For radial direction, the RMS values of the two strategies using accelerometer data were nearly the same as using the empirical force models with about 6.8 mm, while the RMS values of cross-track direction were larger than that of empirical method, which were 14.3 mm and 11.9 mm for the ACC_B_K method and ACC_B method. This is mainly caused by the sensitivity of the accelerometer [42]. The more sensitive axes point in the flight and radial directions, the less sensitive axis points in the cross-track direction [26]. The precision of the sensitive axes is 10 2 9 2 specified to be 10− m/s and that of the less sensitive axis 10− m/s [27]. The RMS values of the along-track direction for the strategy of estimating the accelerometer bias every one hour and empirical forces were almost the same at about 12.0 mm while the strategy of estimating the accelerometer bias and scale for one day were worse than those two strategies with 23.3 mm. Such a difference may be caused by the number of estimated parameters. The number of estimated dynamic parameters of the three strategies is 6 (B ,B ,B ,K ,K , and K ), 72 (B ,B , and B every one hour for 24 h, 24 3 = 72), X Y Z X Y Z X Y Z × and 80 (Ca, Sa, Cc, Sc, and a scale factor every one cycle per revolution for 24 h, 24/1.5 5 = 80), × Remote Sens. 2018, 10, x FOR PEER REVIEW 7 of 19

strategies using accelerometer data were nearly the same as using the empirical force models with about 6.8 mm, while the RMS values of cross-track direction were larger than that of empirical method, which were 14.3 mm and 11.9 mm for the ACC_B_K method and ACC_B method. This is mainly caused by the sensitivity of the accelerometer [42]. The more sensitive axes point in the flight and radial directions, the less sensitive axis points in the cross-track direction [26]. The precision of the sensitive axes is specified to be 10-10 m/s2 and that of the less sensitive axis 10-9 m/s2 [27]. The RMS values of the along-track direction for the strategy of estimating the accelerometer bias every one hour and empirical forces were almost the same at about 12.0 mm while the strategy of estimating the accelerometer bias and scale for one day were worse than those two strategies with 23.3 mm. Such a difference may be caused by the number of estimated parameters. The number of estimated dynamic parameters of the three strategies is 6 (BX, BY, BZ, KX, KY, and KZ), 72 (BX, BY, and BZ every one hour for 24 hours, 24 × 3 = 72), and 80 (Ca, Sa, Cc, Sc, and a scale factor every one cycle per revolution for 24 hours, 24/1.5 × 5 = 80), respectively. In the least square estimator, the more parameters, the smaller the fitted residuals if all parameters are estimable.

Remote Sens. 2020, 12Table, 1599 2. Averaged RMS values on all days for three directions with three strategies. 7 of 18

Along-track (mm) Cross-track (mm) Radial (mm) respectively. In the leastACC_B_K square estimator,23.3 the more parameters, 14.3 the smaller 6.8 the fitted residuals if all parameters are estimable.ACC_B 11.8 11.9 6.1 The initial conditionsEMP and dynamic12.1 parameters estimated 6.2 from the POD 6.8 process are then used The initial conditions and dynamic parameters estimated from the POD process are then used for the LEO orbit prediction. It is of great importance to analyze dynamic parameters since they will for the LEO orbit prediction. It is of great importance to analyze dynamic parameters since they will have a direct effect on the orbit precision for orbit prediction. If only one set dynamic parameters have a direct effect on the orbit precision for orbit prediction. If only one set dynamic parameters is is estimatedestimated for for one one arc, arc, then then thesethese estimatedestimated parameters parameters are are used used for for LEO LEO orbit orbit prediction, prediction, such suchas as strategystrategy 1 (ACC_B_K). 1 (ACC_B_K). If there If there are are more more than than one one set setof of dynamicdynamic parameters parameters estimated estimated like like strategy strategy 2 (ACC_B)2 (ACC_B) and 3 and (EMP), 3 (EMP), one has one tohas firstly to firstly investigate investigate the the characteristics characteristics of of the the dynamic dynamic parametersparameters and thenand decide then the decide way the to way predict to predict the LEO the orbit.LEO orbit. Figure Figure4 shows 4 shows the the accelerometer accelerometer bias bias parametersparameters for strategyfor strategy 2. 2.

FigureFigure 4. Accelerometer 4. Accelerometer bias bias parameters parameters forfor strategystrategy 2. 2. The The data data between between black black dashed dashed lines linesare are excludedexcluded because because of theof the change change of of the the controllercontroller onboard or or abnormal abnormal behavior behavior of accelerometer. of accelerometer. BlackBlack lines lines are theare the linear linear ﬁtting fitting line. line. The The corresponding corresponding fitting ﬁtting results results are are also also shown shown in each in each panel. panel.

The behaviorThe behavior of three of three biases biases was was diff differenterent for for the th threee three directions. directions. By By linear linear fitting,fitting, we could find that biasesfind that in thebiases y direction in the y showdirection a clear show smooth a clear trend smooth in thetrend year. in the It changed year. It changed about 10,450 about nm 10,450/s2 in the year.nm/s For the2 in the x direction, year. For therethe x direction, was also there a small was trend, also a whichsmall trend, wasabout which -9was nm about/s2 in -9 2007. nm/s2 Biases in 2007. in the z direction keep quite stable, which was only 1.3 nm/s2 in the year. Due to this increased or decreased trend of the biases, it may not be appropriate using the last set of accelerometer calibration parameters to conduct the real time orbit prediction and linear fitting and an extrapolation should be used when doing the orbit integration for real time LEO orbits. Figure5 shows the averaged RMS of calibration parameters’ differences between using the extrapolated calibration parameters for one hour and the estimated ones as well as the averaged RMS of calibration parameters’ differences between using the last set of calibration parameters and estimated ones for each arc. From Figure5, we can see that the extrapolated calibration parameters are much closer to the estimated ones, which means the precision of the predicted orbit by using extrapolated calibration parameters should be higher than that of using the last set of calibration parameters. The accelerometer bias in the y direction shows the largest difference since this direction is the least sensitive for the accelerometer instrument, which is pointing to the cross-track direction of the LEO orbit while the bias in the x direction shows the smallest difference. In the following LEO orbit prediction analysis, we will use the extrapolation calibration parameters for orbit prediction with strategy 2. Figure6 shows the dynamic parameters in the EMP method. It shows that the dynamic parameters are not stable, especially for the Sa and Ca terms, which are used for compensating the unmodeled part of the along-track component. They are quite different from day to day. This also implies that the non-gravitational force models used in the POD are not accurate enough and the dynamic parameters have to be estimated in order to absorb the unmodeled part of non-gravitation. It is not appropriate to Remote Sens. 2018, 10, x FOR PEER REVIEW 8 of 19

Biases in the z direction keep quite stable, which was only 1.3 nm/s2 in the year. Due to this increased or decreased trend of the biases, it may not be appropriate using the last set of accelerometer calibration parameters to conduct the real time orbit prediction and linear fitting and Remote Sens. 2020, 12, 1599 8 of 18 an extrapolation should be used when doing the orbit integration for real time LEO orbits. Figure 5 shows the averaged RMS of calibration parameters’ differences between using the extrapolated usecalibration the extrapolate parameters method for to obtainone hour dynamic and the parameters estimated for ones orbit as prediction well as the like averaged the ACC_B RMS method of sincecalibration these parameters parameters’ ﬂuctuated differences and arebetween hard tousin predict.g the So,last the set last of setcalibration of dynamic parameters parameters and was adoptedestimated to predict ones for the each orbit arc. in the following parts for the EMP method.

Figure 5. Averaged RMS of calibration parameters’ diﬀerences between using the extrapolated Figure 5. Averaged RMS of calibration parameters’ differences between using the extrapolated calibration parameters and the estimated ones as well as the averaged RMS of calibration parameters’ calibration parameters and the estimated ones as well as the averaged RMS of calibration diﬀerences between using the last set of calibration parameters and the estimated ones for each arc. parameters’ differences between using the last set of calibration parameters and the estimated ones Remote Sens. 2018, 10, x FOR PEER REVIEW 9 of 19 “W/Ofor Extro” each arc. means “W/O without Extro” means extrapolated without and extrap “Wolated Extro” and means “W Extro” with extrapolation.means with extrapolation.

From Figure 5, we can see that the extrapolated calibration parameters are much closer to the estimated ones, which means the precision of the predicted orbit by using extrapolated calibration parameters should be higher than that of using the last set of calibration parameters. The accelerometer bias in the y direction shows the largest difference since this direction is the least sensitive for the accelerometer instrument, which is pointing to the cross-track direction of the LEO orbit while the bias in the x direction shows the smallest difference. In the following LEO orbit prediction analysis, we will use the extrapolation calibration parameters for orbit prediction with strategy 2. Figure 6 shows the dynamic parameters in the EMP method. It shows that the dynamic parameters are not stable, especially for the Sa and Ca terms, which are used for compensating the unmodeled part of the along-track component. They are quite different from day to day. This also implies that the non-gravitational force models used in the POD are not accurate enough and the dynamic parameters have to be estimated in order to absorb the unmodeled part of non-gravitation. It is not appropriate to use the extrapolate method to obtain dynamic parameters for orbit prediction like the ACC_B method since these parameters fluctuated and are hard to predict. So, the last set of dynamic parameters was adopted to predict the orbit in the following parts for the EMP method.

Figure 6. Dynamic parameters estimated from Strategy 3. Figure 6. Dynamic parameters estimated from Strategy 3.

4. Results of LEO Orbit Prediction After conducting LEO POD, the initial conditions as well as the dynamic parameters can be uplinked to the corresponding LEO satellites. Then the processor on LEO satellites can do orbit prediction/propagation for real time applications. We used one hour for orbit prediction as mentioned before. One issue should be clarified here. If we used the EMP method in which empirical force models are applied, the orbit prediction could also be conducted at the analysis center in advance since no real time onboard observations are required whereas for the ACC_B_K and ACC_B method, it must be conducted onboard because real time accelerometer data must be used. The initial conditions and the dynamic parameters have great impact on the precision of LEO orbit. For the EMP method, the last set of empirical coefficients were chosen for real time orbit integration as usual and for the ACC_B_K method there was only one set of calibrations parameters for each direction including scales and biases for the orbit integration. For the ACC_B method, we used a linear fitting for the previous 24-hour arc since an obvious trend could be found in these biases, and then extrapolated the biases according to the integration time. The differences between PSO and predicted orbits were calculated for the orbit evaluation and their User Ranging Error (URE) were also calculated and compared. URE provides the average range error in the line-of-sight direction at a global scale. Its computation is related to the maximum satellite coverage on the Earth’s surface. The coverage depends on the angular range of the satellite, which is the angle of the emission cone with respect to the boresight direction. With the angular range of LEO satellites at the altitude of 500 km being about 68.02°, the URE can be approximately determined as [8, 9, 43]:

URE = 0.205RMS + 0.397RMS + RMS (3)

Remote Sens. 2020, 12, 1599 9 of 18

4. Results of LEO Orbit Prediction After conducting LEO POD, the initial conditions as well as the dynamic parameters can be uplinked to the corresponding LEO satellites. Then the processor on LEO satellites can do orbit prediction/propagation for real time applications. We used one hour for orbit prediction as mentioned before. One issue should be clarified here. If we used the EMP method in which empirical force models are applied, the orbit prediction could also be conducted at the analysis center in advance since no real time onboard observations are required whereas for the ACC_B_K and ACC_B method, it must be conducted onboard because real time accelerometer data must be used. The initial conditions and the dynamic parameters have great impact on the precision of LEO orbit. For the EMP method, the last set of empirical coefficients were chosen for real time orbit integration as usual and for the ACC_B_K method there was only one set of calibrations parameters for each direction including scales and biases for the orbit integration. For the ACC_B method, we used a linear fitting for the previous 24-h arc since an obvious trend could be found in these biases, and then extrapolated the biases according to the integration time. The differences between PSO and predicted orbits were calculated for the orbit evaluation and their User Ranging Error (URE) were also calculated and compared. URE provides the average range error in the line-of-sight direction at a global scale. Its computation is related to the maximum satellite coverage on the Earth’s surface. The coverage depends on the angular range of the satellite, which is the angle of the emission cone with respect to the boresight direction. With the angular range of LEO satellites at the altitude of 500 km being about 68.02◦, the URE can be approximately determined as [8,9,43]: q Remote Sens. 2018, 10, x FOR PEER REVIEW 2 2 2 10 of 19 URE = 0.205(RMSR) + 0.397 (RMSA) + (RMSC) (3) FigureFigure7 shows 7 shows the the averaged averaged RMS RMS values values of of a onea one hour hour predicted predicted orbit orbit in in three three directions directions with with the threethe strategies.three strategies. The corresponding The corresponding averaged averaged RMS valuesRMS values of all daysof all for days each for direction each direction with di withfferent strategiesdifferent are strategies shown are at the shown top leftat the of top each left panel. of each panel.

Figure 7. Averaged RMS values for the three directions with three strategies for an orbit prediction of Figure 7. Averaged RMS values for the three directions with three strategies for an orbit prediction of one hour. one hour. It is easy to ﬁnd out that the ACC_B_K strategy shows the best results, especially for along-track It is easy to find out that the ACC_B_K strategy shows the best results, especially for direction at 4.8 cm while the ACC_B and EMP strategies were at 15.6 cm and 31.5 cm. For the radial along-track direction at 4.8 cm while the ACC_B and EMP strategies were at 15.6 cm and 31.5 cm. direction, the RMS values were 1.7 cm, 3.7 cm, and 8.6 cm for the ACC_B_K, ACC_B, and EMP strategy, For the radial direction, the RMS values were 1.7 cm, 3.7 cm, and 8.6 cm for the ACC_B_K, ACC_B, respectively. The RMS value of cross-track direction for the ACC_B_K strategy was 4.3 cm, which was and EMP strategy, respectively. The RMS value of cross-track direction for the ACC_B_K strategy was 4.3 cm, which was worse than the other two strategies with 1.6 cm and 1.7 cm. As Figure 8 shows, the averaged URE values for the three strategies were 4.4 cm, 10.1 cm, and 20.3 cm, respectively.

Figure 8. User Ranging Error (URE) of a real time LEO orbit for three different strategies.

Remote Sens. 2018, 10, x FOR PEER REVIEW 10 of 19

Figure 7 shows the averaged RMS values of a one hour predicted orbit in three directions with the three strategies. The corresponding averaged RMS values of all days for each direction with different strategies are shown at the top left of each panel.

Figure 7. Averaged RMS values for the three directions with three strategies for an orbit prediction of one hour.

It is easy to find out that the ACC_B_K strategy shows the best results, especially for along-track direction at 4.8 cm while the ACC_B and EMP strategies were at 15.6 cm and 31.5 cm. For the radial direction, the RMS values were 1.7 cm, 3.7 cm, and 8.6 cm for the ACC_B_K, ACC_B, Remote Sens. 2020, 12, 1599 10 of 18 and EMP strategy, respectively. The RMS value of cross-track direction for the ACC_B_K strategy was 4.3 cm, which was worse than the other two strategies with 1.6 cm and 1.7 cm. As Figure 8 worseshows, than the the averaged other two URE strategies values withfor the 1.6 three cm and strategies 1.7 cm. were As Figure 4.4 cm,8 shows, 10.1 cm, the averagedand 20.3 cm, URE valuesrespectively. for the three strategies were 4.4 cm, 10.1 cm, and 20.3 cm, respectively.

Figure 8. User Ranging Error (URE) of a real time LEO orbit for three different strategies. Figure 8. User Ranging Error (URE) of a real time LEO orbit for three different strategies. It is easy to understand that predicted LEO orbit precision by using accelerometer data were better than that of using empirical force models since accelerometer data could better reflect the non-gravitational forces than that of the empirical force models. We found that with the method of strategy 1 (ACC_B_K), the predicted LEO orbit precision shows a better result in the along and radial directions than with strategy 2 (ACC_B), though the calculation part (orbit determination part) of ACC_B_K of the along-track and radial directions was worse than those of ACC_B. We calculated the differences of the predicted non-gravitational acceleration between ACC_B_K and the corresponding non-gravitational acceleration calculated from estimated biases and the scale factor as well as the differences of predicted non-gravitational acceleration between ACC_B and the corresponding non-gravitational acceleration calculated from estimated biases for a one hour prediction, as shown in Figure9. From Figure9, we can see that the di fference of non-gravitational accelerations in the x direction (orbit along-track direction) and z direction (radial direction) with the ACC_B_K method were much smaller than those of the ACC_B method, which means that the along-track and radial direction precision of the prediction part with ACC_B_K should be better than those of the ACC_B method. For the cross-track direction (y direction of acceleration), the differences between the predicted acceleration and the calculated one with the ACC_B_K method were larger than that of the ACC_B method, which corroborated the worse results of the cross-track direction with the ACC_B_K method. In conclusion, one set of scales and biases for each direction reflected the calibration parameters of the accelerometer for the long term while one set of biases every hour reflected for the short term. For orbit prediction for less than 20 min, the precision of all orbit components with all strategies was better than 10 cm. This means that if the interval of the uplink and downlink was shorter than 20 min, all strategies could be used for real time LEO POD. For orbit prediction for more than 20 min, the best way to realize the real time LEO precise orbit determination is by using accelerometer data with the estimation of one set of scales and biases for each direction. In this way, less than a 5 centimeter level of LEO orbit precision can be achieved for the one hour LEO orbit prediction, which satisfies the demand of a centimeter level requirement mentioned in the Introduction section. Remote Sens. 2018, 10, x FOR PEER REVIEW 11 of 19

It is easy to understand that predicted LEO orbit precision by using accelerometer data were better than that of using empirical force models since accelerometer data could better reflect the non-gravitational forces than that of the empirical force models. We found that with the method of strategy 1 (ACC_B_K), the predicted LEO orbit precision shows a better result in the along and radial directions than with strategy 2 (ACC_B), though the calculation part (orbit determination part) of ACC_B_K of the along-track and radial directions was worse than those of ACC_B. We calculated the differences of the predicted non-gravitational acceleration between ACC_B_K and the corresponding non-gravitational acceleration calculated from estimated biases and the scale factor as well as the differences of predicted non-gravitational acceleration between ACC_B and the corresponding non-gravitational acceleration calculated from estimated biases for a one hour prediction, as shown in Figure. 9. From Figure 9, we can see that the difference of non-gravitational accelerations in the x direction (orbit along-track direction) and z direction (radial direction) with the ACC_B_K method were much smaller than those of the ACC_B method, which means that the along-track and radial direction precision of the prediction part with ACC_B_K should be better than those of the ACC_B method. For the cross-track direction (y direction of acceleration), the differences between the predicted acceleration and the calculated one with the ACC_B_K method were larger than that of the ACC_B method, which corroborated the worse results of the cross-track direction with the RemoteACC_B_K Sens. 2020 method., 12, 1599 11 of 18

Figure 9. Diﬀerences of predicted non-gravitational acceleration between the ACC_B_K method and theFigure calculated 9. Differences ones as of well predicted as the di non-gravitationalﬀerences of predicted acceleration non-gravitational between the acceleration ACC_B_K betweenmethod and the ACC_Bthe calculated method ones and as the well calculated as the differences ones for aof one predicted hour prediction non-gravitational in each direction.acceleration “Oneday between dif”the RemotemeansACC_B Sens. the 2018method ACC_B_K, 10, x andFOR methodPEERthe calculated REVIEW and “Onehour ones for dif”a one means hour the prediction ACC_B method.in each direction. “Oneday 12dif” of 19 means the ACC_B_K method and “Onehour dif” means the ACC_B method. Usually,Usually, most most precise precise positioning positioning users users care care more more about about the the orbit orbit errors errors for for every every epoch epoch while while not fornot theIn for averaged conclusion, the averaged ones one over setones oneof overscales hour. one and Figure hour. biases 10 Figure shows for ea 10ch RMS showsdirection values RMS reflected of values LEO orbits thofe LEOcalibration for threeorbits directions forparameters three of everyof directionsthe 10accelerometer min of over every a onefor 10 the hourminutes long prediction term over while a timeone one period.hour set predictionof Thebiases great every time increase hourperiod. reflected of along-trackThe great for the increase RMS short values term.of withForalong-track orbit the empirical prediction RMS methodvalues for less with can than bethe 20 easily empirical minutes, found, method the which pr ecisionca isn frombe easilyof about all orbitfound, 5.0 components cm which at 10 is min from with to about approximately all strategies 5.0 cm 49.4wasat cmbetter10 overminutes than one 10to hour. cm.approximately However,This means the 49.4 that method cm if the over withinterval one accelerometer hour. of the However, uplink data and the with downlinkmethod one set with was of biasesaccelerometer shorter and than scales 20 forminutes,data each with directionall one strategies set showsof biases could a stableand be scalesused RMS for of eachreal 4.9 cm,directiontime 4.0 LEO cm, shows POD. and a 1.2 Forstable cm orbit RMS in theprediction of along-track,4.9 cm, for4.0 cm,more cross-track, and than 1.2 20 andminutes,cm radial in the the directions along-track, best way all cross-track, theto time,realize respectively. and the ra dialreal directionstime LEO all precisethe time, orbit respectively. determination is by using accelerometer data with the estimation of one set of scales and biases for each direction. In this way, less than a 5 centimeter level of LEO orbit precision can be achieved for the one hour LEO orbit prediction, which satisfies the demand of a centimeter level requirement mentioned in the Introduction section.

Figure 10. RMS values of LEO orbits for the along-track, cross-track, and radial direction for every 10 Figure 10. RMS values of LEO orbits for the along-track, cross-track, and radial direction for every 10 minminutes over a over one houra one period.hour period. The corresponding URE for every 10 min is shown in Figure 11. The method with accelerometer The corresponding URE for every 10 minutes is shown in Figure 11. The method with data with one set of biases and scales for each direction shows the most stable results about 4.6 cm accelerometer data with one set of biases and scales for each direction shows the most stable results evenabout if the 4.6 integration cm even timeif the is integration one hour while time theis methodone hour of while empirical the andmethod accelerometer of empirical data and with accelerometer data with one-hour biases for each direction shows an increased trend, from about 4.6 cm at 0-10 minutes to 31.4 cm and 14.0 cm within 50-60 minutes.

Figure 11. URE for every 10 minutes over a one hour orbit prediction.

Remote Sens. 2018, 10, x FOR PEER REVIEW 12 of 19

Usually, most precise positioning users care more about the orbit errors for every epoch while not for the averaged ones over one hour. Figure 10 shows RMS values of LEO orbits for three directions of every 10 minutes over a one hour prediction time period. The great increase of along-track RMS values with the empirical method can be easily found, which is from about 5.0 cm at 10 minutes to approximately 49.4 cm over one hour. However, the method with accelerometer data with one set of biases and scales for each direction shows a stable RMS of 4.9 cm, 4.0 cm, and 1.2 cm in the along-track, cross-track, and radial directions all the time, respectively.

Figure 10. RMS values of LEO orbits for the along-track, cross-track, and radial direction for every 10 minutes over a one hour period.

The corresponding URE for every 10 minutes is shown in Figure 11. The method with Remoteaccelerometer Sens. 2020, 12 data, 1599 with one set of biases and scales for each direction shows the most stable results12 of 18 about 4.6 cm even if the integration time is one hour while the method of empirical and accelerometer data with one-hour biases for each direction shows an increased trend, from about 4.6 one-hour biases for each direction shows an increased trend, from about 4.6 cm at 0-10 min to 31.4 cm cm at 0-10 minutes to 31.4 cm and 14.0 cm within 50-60 minutes. and 14.0 cm within 50-60 min.

Figure 11. URE for every 10 min over a one hour orbit prediction. Figure 11. URE for every 10 minutes over a one hour orbit prediction.

As mentioned above in the “LEO orbit prediction procedure”, if the orbital elements do not uplink in time, one has to wait until the next uplink, which means a delay of nearly two hours. The RMS valuesRemote of Sens. LEO 2018 orbits, 10, x forFOR three PEER REVIEW directions with another one hour prediction and the corresponding 13 of URE 19 values are shown in Figure 12. As mentioned above in the “LEO orbit prediction procedure”, if the orbital elements do not We could easily ﬁnd that the method with accelerometer data with one set of biases and scales for uplink in time, one has to wait until the next uplink, which means a delay of nearly two hours. The each direction (ACC_B_K) still shows the most stable results while the method with empirical models RMS values of LEO orbits for three directions with another one hour prediction and the (EMP) ﬂuctuated, especially in the radial component. The RMS value for the along-track component corresponding URE values are shown in Figure 12. could be at one meter for the method with empirical models while it is still better than 10 cm for the method with one set of biases and scales.

Figure 12. Cont.

Figure 12. RMS values of LEO orbits for the along-track, cross-track, and radial direction for another one hour and corresponding URE values.

We could easily find that the method with accelerometer data with one set of biases and scales for each direction (ACC_B_K) still shows the most stable results while the method with empirical models (EMP) fluctuated, especially in the radial component. The RMS value for the along-track component could be at one meter for the method with empirical models while it is still better than 10 cm for the method with one set of biases and scales.

5. Conclusions

Remote Sens. 2018, 10, x FOR PEER REVIEW 13 of 19

Remote Sens. 2020, 12, 1599 13 of 18

Figure 12. RMS values of LEO orbits for the along-track, cross-track, and radial direction for another Figure 12. RMS values of LEO orbits for the along-track, cross-track, and radial direction for another oneone hour hour and and corresponding corresponding URE URE values. values. 5. Conclusions We could easily find that the method with accelerometer data with one set of biases and scales forWith each the direction development (ACC_B_K) of the still LEO shows related the missions most stable and results applications, while therethe method will be with a great empirical demand of realmodels time (EMP) LEO fluctuated, precise orbit. especially In this in study, the radi threeal component. strategies with The estimationRMS value offor di theﬀerent along-track dynamic parameterscomponent were could conducted be at one and meter the for conclusions the method are with summarized empirical models as follows while it is still better than 10 cm for the method with one set of biases and scales. (1) The precision of LEO satellites orbits could reach the centimeter level with estimation of diﬀerent 5. Conclusionskinds of dynamic parameters if precise GNSS products were adopted. However, under the current circumstances, RTS was not available for onboard processing. For the demand of the real time precise LEO orbit, at least a one-hour orbit prediction should be adopted. (2) For the current GNSS technology, LEO orbit prediction by using accelerometer data was feasible. Precision of one hour predicted LEO satellite orbits with initial conditions and dynamic parameters estimated with a 24 h arc could reach the centimeter level by using accelerometer data while that using empirical force models was at the decimeter level. (3) For the LEO orbit prediction over 20 min, the predicted LEO orbit precision with estimation of one set of biases and scales for each direction (strategy 1) was better than that of biases for each direction every 60 min (strategy 2), though the calculated orbit precision with the latter was better than the former method. (4) If there were enough uplink and downlink stations, which mean the interval of uplink and downlink could be shorter than 20 min, all these three strategies could be adopted for the real time applications.

On all accounts, the accelerometer data had great potential to do the LEO orbit prediction for real time applications in the near future.

6. Discussions This article aimed at investigating centimeter level precision of LEO orbit prediction for real time applications. It is a good replacement of using accelerometer data instead of empirical force models when conducting LEO orbit prediction. Three strategies with estimation of diﬀerent dynamic parameters were conducted. The results show that with our proposed method, the LEO orbit precision could reach the centimeter level and even predicting within two hours. Two more practical issues should be clariﬁed. Remote Sens. 2020, 12, 1599 14 of 18

The first one is the feasibility of widespread use of accelerometers. The idea of using accelerometers for orbit prediction is a good idea and it was already shown in this study. However, the accelerometers we used were of high quality for specific missions, such as a gravity mission. Such accelerometers may be too expensive for the mass market. Concerning this issue, experiments (one can add colored or white noise in the accelerometer data) could be conducted in the future to check the tolerance of accelerometer data with respect to the precision of orbit prediction. The second issue is the time gap of LEO onboard observation data collection. We took GRACE-A as example, which has an inclination of 89◦. It is quite easy to have a continuous coverage for sustained period (half cycle of the LEO orbital period) where the uplink and downlink station are located at high latitudes (north and south). However, more uplink and downlink stations are needed of LEO satellites with low orbital inclination to meet the max length of the LEO orbit prediction (such as about 20 min for GRACE-A to keep the centimeter precision of the orbit prediction with empirical models). The distribution of the uplink and downlink stations for different inclinations of LEO satellites should be investigated to guarantee the time gap of onboard data collection.

Author Contributions: All authors contributed to the writing of this manuscript. H.G., B.L., and M.G. conceived the idea of employing accelerometer data to conduct LEO orbit prediction. H.G. and L.N. processed the data and investigated the results. M.G., B.L., and H.S. validated the data. All authors have read and agreed to the published version of the manuscript. Funding: This study is sponsored by National Natural Science Foundation of China (41874030), The Scientific and Technological Innovation Plan from Shanghai Science and Technology Committee (18511101801), and The National Key Research and Development Program of China (2017YFA0603102), and the Fundamental Research Funds for the Central Universities. Acknowledgments: Thanks to Physical Oceanography Distributed Active Archive Center (PO. DAAC) of JPL for providing the raw GNSS observations and post-processed reduced dynamic orbit of GRACE-A and GRACE-C. The GNSS observations and orbit products are available from https://podaac-tools.jpl.nasa.gov/drive/files/allData/ grace/L1B/JPL and https://podaac-tools.jpl.nasa.gov/drive/files/allData/gracefo/L1B/JPL/RL04/ASCII for GRACE-A and GRACE-C, respectively. We are also grateful to CODE and GFZ real time group for providing precise GPS final products and real time products, respectively. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations and Acronyms The following acronyms and abbreviations are used in this manuscript:

CHAMP Challenging Minisatellite Payload CODE Center for Orbit Determination in Europe DOY Day of Year DSHL Disabling of Supplemental Heater Lines ESA European Space Agency GFZ GeoForschungsZentrum GNSS Global Navigation Satellite System GOCE Gravity ﬁeld and steady-state Ocean Circulation Explorer GPS Global Positioning System GRACE Gravity Recovery and Climate Experiment GRACE-FO Gravity Recovery and Climate Experiment Follow-On IGS International GNSS Service JPL Jet Propulsion Laboratory LEO Low Earth Orbit LeGNSS Leo enhanced Global Navigation Satellite System NRT Near Real Time NTC Non Time Critical POD Precise Orbit Determination PPP Precise Point Positioning RMS Root Mean Square RTPP Real Time Pilot Project Remote Sens. 2020, 12, 1599 15 of 18

RTS Real Time Service SRF Science Reference Frame ACC_B_K Strategy of estimating bias and scale of accelerometer ACC_B Strategy of estimating bias of accelerometer EMP Strategy of estimation empirical dynamic parameters

Appendix A The LEO data used in the article is GRACE-A data in 2007 when no real time precise orbit and clockRemote products Sens. 2018 are, 10,available. x FOR PEER UntilREVIEW now, there is no public archive available for the storage of 16 real of time 19 orbit and clock products. Thanks to the GFZ real time working group, they have stored these real time productsended in since October, April 2017. 26th, 2018,GRACE-FO which mission make it is possible the successor for us toof demonstratethe GRACE mission, the orbit the di fforbiterence of of LEOGRACE-FO POD by using satellite precise (GRACE-C GPS products and GRACE-D) and real is time nearly products. the same We as updatedthat of GRACE our software with the recently orbital to conductinclination GRACE of 89 Follow degree. On Moreover, (GRACE-FO) GRACE-FO POD since observatory GRACE builds mission on wasthe design ended inof October,the original 2017. GRACE-FOGRACE, but mission incorporates is the successor a number of of the impr GRACEovements mission, based the on orbit lessons of GRACE-FO learned [44]. satellite (GRACE-C and GRACE-D)In this part, is nearly we firstly the same evaluate as that one of GRACEweek GFZ with real the time orbital products inclination (2019, of DOY 89 degree. 216-222) Moreover, with GRACE-FOrespect to observatoryCODE final products. builds on Then the designGRACE-C of the PO originalD with empirical GRACE, butforce incorporates model is conducted a number to of improvementsdemonstrate how based much on lessons difference learned is expected [44]. due to the use of real time products instead of precise CODEIn this products. part, we firstly evaluate one week GFZ real time products (2019, DOY 216-222) with respect to CODE final products. Then GRACE-C POD with empirical force model is conducted to demonstrate how much difference is expected due to the use of real time products instead of precise CODE products.

FigureFigure A1. A1.RMS RMS of GPSof GPS orbit orbit diﬀ differenceerence (top ()top and) and STD STD of GPS of GPS clock clock diﬀerence difference (bottom (bottom) between) between CODE productsCODE products and GFZ and real GFZ time real products. time products.

Figure A1 shows the RMS values of GPS orbit difference and STD of GPS clock difference between CODE products and GFZ real time products. The averaged RMS values of all GPS satellites for along-track, cross-track, and radial directions are 3.5 cm, 2.7 cm, and 2.4 cm, respectively. The STD values of GPS clock difference are generally smaller than 0.2 ns for all GPS satellites. The averaged STD value of all satellites is 0.14 ns, which is about 4.0 cm in distance. With these products, GRACE-C satellite POD is conducted. Since GRACE-FO is built on the design of GRACE, all the force models used for GRACE can be adopted for GRACE-FO as shown in Table 1. The PSO of GRACE-C provided by JPL are used as references for our orbit evaluation

Remote Sens. 2020, 12, 1599 16 of 18

Figure A1 shows the RMS values of GPS orbit difference and STD of GPS clock difference between CODE products and GFZ real time products. The averaged RMS values of all GPS satellites for along-track, cross-track, and radial directions are 3.5 cm, 2.7 cm, and 2.4 cm, respectively. The STD values of GPS clock difference are generally smaller than 0.2 ns for all GPS satellites. The averaged STD value of all satellites is 0.14 ns, which is about 4.0 cm in distance. With these products, GRACE-C satelliteRemote Sens. POD 2018, is 10 conducted., x FOR PEER REVIEW Since GRACE-FO is built on the design of GRACE, all the force models 17 of 19 used for GRACE can be adopted for GRACE-FO as shown in Table1. The PSO of GRACE-C provided by(https://podaac-tools.jpl.nasa.gov/drive/files/allData/gracefo/L1B/JPL/RL04/ASCII JPL are used as references for our orbit evaluation (https://podaac-tools.jpl.nasa.gov). The/ driveaveraged/files/ allDataRMS of/ gracefoGRACE-C/L1B orbits/JPL/RL04 with/ ASCIIrespect). to The the averaged PSO is shown RMS of in GRACE-C Figure A2. orbits with respect to the PSO is shown in Figure A2.

Figure A2.A2. RMSRMS ofof GRACE-CGRACE-C orbit orbit with with respect respect to to JPL JPL PSO PSO orbits orbits by usingby using CODE CODE ﬁnal final products products and GFZand GFZ real timereal time products. products.

From Figure A2A2,, we can see that the RMS values of orbit by using GFZ realreal time products are slightly larger than those of CODECODE ﬁnalfinal products. The differences diﬀerences are about 8 mm, 3 mm, and 2 mm for along-track,along-track, cross-track,cross-track, and and radial radial directions, directions respectively., respectively. By By using using real real time time products, products, the orbit the precisionorbit precision for 3D for will 3D degrade will degrade about about 1 cm. 1 cm.

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