Prediction of Beidou Satellite Orbit Maneuvers to Improve the Reliability of Real-Time Navigation Products

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Article Prediction of BeiDou Satellite Orbit Maneuvers to Improve the Reliability of Real-Time Navigation Products

Zhiwei Qin 1 , Le Wang 1,*, Guanwen Huang 1, Qin Zhang 1, Xingyuan Yan 2, Shichao Xie 1, Haonan She 1, Fan Yue 1 and Xiaolei Wang 3

1 School of Geology Engineering and Geomatics, Chang’an University, 126 Yanta Road, Xi’an 710054, China; [email protected] (Z.Q.); [email protected] (G.H.); [email protected] (Q.Z.); [email protected] (S.X.); [email protected] (H.S.); [email protected] (F.Y.) 2 School of Geospatial Engineering and Science, Sun Yat-sen University, No. 135, Xingang Xi Road, Guangzhou 510275, China; [email protected] 3 School of Earth Sciences and Engineering, Hohai University, 1 Xikang Road, Nanjing 210098, China; [email protected] * Correspondence: [email protected]; Tel.: +86-029-8233-9043

Abstract: The positioning, navigation, and timing (PNT) service of the Global Navigation Satellite System (GNSS) is developing in the direction of real time and high precision. However, there are some problems that restrict the development of real-time and high-precision PNT technology. Satellite orbit maneuvering is one of the factors that reduce the reliability of real-time navigation products, especially the high-frequency orbit maneuvering of geostationary earth orbit (GEO) and inclined geosynchronous orbit (IGSO) satellites. The BeiDou Navigation Satellite System (BDS) constellation is designed to contain GEO, IGSO, and medium earth orbit (MEO). These orbit maneuvers bring certain difﬁculties for data processing, especially for BeiDou satellites, such as decreased real-time service performance, which results in real-time navigation products including unusable maneuvered Citation: Qin, Z.; Wang, L.; Huang, satellites. Additionally, the performance of real-time navigation products will decrease because G.; Zhang, Q.; Yan, X.; Xie, S.; She, H.; the orbit maneuvers could not be known in advance, which diminishes the real-time PNT service Yue, F.; Wang, X. Prediction of BeiDou performance of BDS for users. Common users cannot obtain maneuvering times and strategies owing Satellite Orbit Maneuvers to Improve the Reliability of Real-Time to conﬁdentiality, which can lead to a decline in the BDS real-time service performance. Thus, we Navigation Products. Remote Sens. propose a method to predict orbit maneuvers. BDS data from the broadcast ephemeris were analyzed 2021, 13, 629. https://doi.org/ to verify the availability of the proposed method. In addition, the results of real-time positioning 10.3390/rs13040629 were analyzed by using ultra-rapid orbit products, demonstrating that their reliability is improved by removing maneuvered satellites in advance. This is vital to improve the reliability of real-time Academic Editor: Ali Khenchaf navigation products and BDS service performance. Received: 20 January 2021 Accepted: 5 February 2021 Keywords: BeiDou satellites; orbit maneuvers; prediction; real-time navigation products Published: 9 February 2021

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in 1. Introduction published maps and institutional affil- iations. The positioning, navigation, and timing (PNT) technology of the Global Navigation Satellite System (GNSS) grew out of nothing through development. The PNT service of GNSS is developing in the direction of real time and high precision. However, there are some problems restricting the development of real-time and high-precision PNT technology that decrease the reliability of real-time navigation products. Satellite orbit maneuvering is Copyright: © 2021 by the authors. one of the factors that reduce the reliability of real-time navigation products, especially the Licensee MDPI, Basel, Switzerland. high-frequency orbit maneuvers of geostationary earth orbit (GEO) and inclined geosyn- This article is an open access article chronous orbit (IGSO) satellites. It is necessary to improve the reliability of the PNT service distributed under the terms and of GNSS in order to satisfy the demands of high-reliability and high-precision navigation conditions of the Creative Commons Attribution (CC BY) license (https:// products for users. Thus, it is vital to enhance the reliability of real-time navigation prod- creativecommons.org/licenses/by/ ucts (such as orbit, clock, etc.). Since 27 December 2012, BeiDou-2 has provided continuous 4.0/). PNT service for users in the entire Asia-Pacific region, with a constellation of five GEO,

Remote Sens. 2021, 13, 629. https://doi.org/10.3390/rs13040629 https://www.mdpi.com/journal/remotesensing Remote Sens. 2021, 13, 629 2 of 18

five IGSO, and four MEO satellites. The second phase, BeiDou-3, will provide PNT service for global users at the end of 2020 [1–3]. The successful launch of the last satellite is a milestone of the construction completed for the BeiDou-3 constellation. At present, there are 30 BeiDou-3 satellites shining in the sky, consisting of three GEO, three IGSO, and 24 MEO satellites. BDS is the only GNSS that contains GEO and IGSO satellites in the nominal satellite constellation, which is the highlight of this GNSS. The MEO satellites offer complete global coverage, sharing numerous common features with GPS and GLONASS. Despite the higher orbit of the GEO and IGSO satellites as compared with the legacy MEO constellations, BDS achieves fully adequate signal strength and competitive code and car- rier tracking performance [4,5]. Together with GEO, IGSO and MEO satellites can provide fully global PNT service for worldwide users [6–10]. The GEO and IGSO satellites are considered relevant and valuable complements to GNSS services in the Asia-Pacific region, playing an important role [11–14]. They not only enhance satellite visibility and PNT availability for users in China and surrounding areas, but also improve PNT precision for global users. The estimation accuracy of the earth orientation parameters can be improved based on the different orbital altitudes of the BDS satellites [6]. However, due to gravitational perturbations from the Earth, Sun, and Moon, as well as other various perturbations to the satellites, orbit maneuvers are required at regular intervals to maintain predesignated positions. GEO and IGSO satellites are maneuvered more frequently than MEO satellites because they are geosynchronous and geostationary. When orbit maneuvering, the orbital elements are changed, which causes the status variable to change after maneuvering. The broadcast ephemeris includes the health status of GNSS satellites; their health identifier will be marked as 0 for healthy, and otherwise the satellite is unhealthy. The satellite health identifier from broadcast ephemeris is also marked as unhealthy for nonmaneuvered anomalies. Where information on the maneuvered satellites are unavailable owing to confidentiality, it brings several problems for GNSS data processing. With the development of Multi-GNSS, the position users can remove the unhealthy satellites directly, and the position accuracy could not be decreased. However, the orbit maneuvers and unmaneuvered abnormalities of satellites cannot be distinguished by using the SV health from broadcast ephemeris. They both would be marked as unhealthy. In addition, the unmaneuvered could not affect the precise orbit determination. However, the precise orbit of maneuvered satellites cannot be determined and predicted because of the extra maneuvered thrust. As the orbit maneuvers and unmaneuvered abnormalities cannot be distinguished, the ultra-rapid precise orbit products published by IGS analysis centers include the unusable maneuvered satellites. These problems decrease the reliability of high-precision real-time navigation products of BDS, such as the ultra-rapid precise orbit products published by the analysis center of the international GNSS Monitoring and Assessment System (iGMAS) and the Multi-GNSS Experiment (MGEX). The time information of maneuvered satellites is unavailable in advance, so the real-time navigation products contain the unusable orbits of maneuvered satellites. Unusable real-time navigation products that include maneuvered satellites will decrease the real-time PNT performance of BDS. In order to improve the reliability of the real-time navigation products, the time information of orbit maneuvers must be obtained in advance. The predicted information for orbit maneuvering can be provided for position strategy changes and precise orbit determination. Thus, it is vital to provide orbit maneuver information in order to remove unusable maneuvered satellites, which is crucial for real-time navigation product solutions and real-time positions. Previous studies have investigated orbit maneuvers, yielding certain useful results. For orbit maneuvering of an inertial device to rendezvous and dock with another spacecraft, Lemmens et al. [15] and Li et al. [16] presented a maneuver detection method for LEO satellites based on historical two-line element (TLE) data. LEO satellite maneuvers can be relatively easily handled with precise Global Positioning System (GPS) or TLE data. Compared to LEO satellites, the maneuvering of navigation satellites is significantly more difficult to investigate due to their high altitudes and the missing positions of maneuvered satellites. These methods cannot be used to analyze GNSS orbit maneuvers. Remote Sens. 2021, 13, 629 3 of 18

Huang et al. [17] and Cao et al. [18] used observations from the Chinese Area Positioning System (CAPS) to estimate the orbits of maneuvered GEO satellites. However, IGSO and MEO satellites cannot be continuously monitored via CAPS, so common users cannot obtain relevant observations. These methods bring certain difﬁculties when analyzing the characteristics of maneuvered orbits due to data limitations, especially for IGSO and MEO satellites. Huang et al. detected the start time of orbital maneuvers by using single point positioning (SPP) technology and residual values of a pseudo-range [19,20]. Qin et al. detected the orbit maneuvering time period by backward prediction of orbit and pseudo-range and determined the orbit for maneuvered satellites [21,22]. Numerous studies have investigated orbit maneuver detection and determination; however, there is a lack of studies speciﬁcally focusing on predicting orbit maneuvers. This is why we propose a prediction method for orbit maneuvers. In this study, we analyze BeiDou satellite orbit maneuvers to characterize orbital elements, including the relationship with the orbital semimajor axis. First, we analyzed long-term variations in the orbital semimajor axis and proposed a prediction method for east–west maneuvering. The experiment results based on broadcast ephemeris were analyzed to demonstrate the validity of the prediction method for orbit maneuvers. In addition, we analyzed the results of real-time positioning based on ultra-rapid precise orbit products from the German Research Centre for Geosciences (GFZ), and we present our conclusions and implications.

2. Orbital Maneuver Analysis and Prediction Method Orbital elements have changed continuously due to perturbations from multiple planets and spacecraft, which cause long-term satellite drift. Satellites tend to gradually deviate from their designed orbits due to various perturbations, including perturbations from solar radiation pressure, Earth’s nonspherical gravity, and the gravity of the Sun, Moon, Venus, Mars, and other planets. External forces are solar, and other forces come from places such as nonspherical bodies. The total forces are calculated by:

= + + Ft ∑ Fgi Fs ∑ Fni (1)

where Ft is total forces affecting satellites, Fgi is gravity from other planets, Fs is the force

from solar radiation pressure, and Fni is the force of nonspherical body [23]. There are two modes of orbit maneuvers: east–west (in plane) maneuvers can retain the shape of the orbit plane, and north–south (out of plane) maneuvers can retain the inclination of the orbit plane. Orbit maneuvers in the north–south direction are much less frequent than maneuvers in the east–west direction. Thus, orbit maneuvers in the north–south direction are not discussed in this study. A schematic diagram of the gravitational force from the Sun and other planets affecting satellites is shown in Figure1. In Figure1, the orange circle and triangle denote the Sun, the yellow circle denotes Venus, and the dotted yellow line is its orbit around the Sun. The red circle denotes Mars, and its orbit is illustrated by a dotted red line. Earth is denoted by a green circle, and the dotted green line denotes its orbit around the Sun. The blue line denotes the designed orbit for satellites around the Earth, and the blue object is a satellite. Other planets (Mercury, Jupiter, Saturn, Uranus, Neptune, and Moon) are not shown in Figure1. The effect of instantaneous gravity from planets on satellites is denoted by arrows. The dotted orange arrow denotes the joint forces of gravity and solar pressure from the Sun. The dotted yellow, red, and green arrows denote the gravity from Venus, Mars, and Earth, respectively. Supposing the force affecting satellites is only from the Earth and the density of the Earth is uniform, satellites could run perennially around the Earth in their designed orbits; the blue arrow denotes the velocity of satellites. In fact, there are various forces acting on satellites from the gravity of multiple planets, solar pressure, and nonspherical gravity from the Earth. Thus, the actual instantaneous joint forces acting on satellites are different with the gravity from the Earth, which is denoted by the dotted black arrow, and the actual instantaneous velocity is expressed by the solid black arrow. Satellite orbits are Remote Sens. 2021, 13, 629 4 of 18

Remote Sens. 2021, 13, x FOR PEER REVIEW 4 of 18 continuously changed due to perturbations from multiple planets and other factors, as shown in Figure2.

Remote Sens. 2021, 13, x FOR PEER REVIEW 5 of 18

Figure 1.Figure Schematic 1. Schematic diagram diagramof gravitational of gravitational force from force planets from affecting planets affecting satellites. satellites.

In Figure 1, the orange circle and triangle denote the Sun, the yellow circle denotes Venus, and the dotted yellow line is its orbit around the Sun. The red circle denotes Mars, and its orbit is illustrated by a dotted red line. Earth is denoted by a green circle, and the dotted green line denotes its orbit around the Sun. The blue line denotes the designed orbit for satellites around the Earth, and the blue object is a satellite. Other planets (Mer- cury, Jupiter, Saturn, Uranus, Neptune, and Moon) are not shown in Figure 1. The effect of instantaneous gravity from planets on satellites is denoted by arrows. The dotted orange arrow denotes the joint forces of gravity and solar pressure from the Sun. The dotted yellow, red, and green arrows denote the gravity from Venus, Mars, and Earth, respectively. Supposing the force affecting satellites is only from the Earth and the density of the Earth is uniform, satellites could run perennially around the Earth in their designed orbits; the blue arrow denotes the velocity of satellites. In fact, there are various forces acting on satellites from the gravity of multiple planets, solar pressure, and nonspherical gravity from the Earth. Thus, the actual instantaneous joint forces acting on satellites are different with the gravity from the Earth, which is denoted by the dotted black arrow, and the actual instantaneous velocity is expressed by the solid black arrow. Satellite orbits are continuously changed due to perturbations from multiple planets and other factors, as shown in Figure 2.

Figure 2. FigureSchematic 2. Schematic diagram ofdiagram satellite of orbit satellite drift orbit due todrift various due to perturbations various perturbations from multiple from planetsmultiple and spacecraft. planets and spacecraft. In Figure2, the green circle denotes the Earth, which is the focal point of satellite In Figure orbits.2, the green The initial circle orbit denotes of satellites the Earth, is shown which theis the elliptical focal point blue curve,of satellite line a is the orbital orbits. The initialsemimajor orbit of axis,satellites line isb isshown the orbital the elliptical semiminor blue axis,curve, and line line c isis the the orbital half focal length of semimajor axis,the line satellite is the orbit. orbital The semiminor light blue partaxis, denotesand line the is normal the half range focal of length satellite of orbits. When the satellite orbit.satellites The light deviate blue from part the denote normals the orbit normal range, range they of can satellite be adjusted orbits. by When orbit maneuvering, satellites deviateusing from their the propulsionnormal orbit system. range, they The greencan be arrow adjusted denotes by orbit the maneuvering, instantaneous gravity force using their propulsionacting on system. satellites The from green the arrow Earth. denotes The black the arrow instantaneous denotes the gravity instantaneous force joint forces acting on satellitesfrom from multiple the Earth. planets The acting black onarrow satellites, denotes which the instantaneous leads to satellites joint tendingforces to gradually from multiple deviateplanets fromacting their on normalsatellites, orbital which range. leads The to satellites shape of thetending satellite to gradually orbits also changes with deviate from their normal orbital range. The shape of the satellite orbits also changes with time. This can be shown by the pure curve, which denotes the satellite orbit that needs to be adjusted though orbit maneuvering; line ′ is the orbital semimajor axis, line ′ is the orbital semiminor axis, and line ′ is the half focal length of the satellite orbit. The variation in the orbital semimajor axis ∆ is the difference value of the norm of and ′, which is indicated by the dotted red line in Figure 2. Satellites periodically traverse the sky around the center of the Earth, which allows the perturbations from multiple planets to yield periodicity. The comprehensive perturbations from planets on satellites can be counteracted on short-term time scales [24]. The short-term periodical variation of the semimajor axis is not discussed for orbit maneuver prediction in this study. The comprehensive perturbations also cause long-term satellite drift, which leads to variation of the orbital semimajor axis. The orbital semimajor axis can be expressed as follows:

= + + , (2)

where is the initial value of the orbital semimajor axis; is the long-term variation function for the orbital semimajor axis; is the time of the epoch, i.e., the day of the year containing the hours used; is the error associated with the orbital semimajor axis, which includes short-term periodical variation, random error, and calculation error from broadcast ephemeris; is the value at epoch , for which we used data from the broadcast ephemeris. According to the effects of joint forces from multiple planets acting on satellites, the long-term variation in the orbital semimajor axis can be described by a polynomial function as follows:

Remote Sens. 2021, 13, 629 5 of 18

time. This can be shown by the pure curve, which denotes the satellite orbit that needs to be adjusted though orbit maneuvering; line a0 is the orbital semimajor axis, line b0 is the orbital semiminor axis, and line c0 is the half focal length of the satellite orbit. The variation in the orbital semimajor axis ∆a is the difference value of the norm of a and a0, which is indicated by the dotted red line in Figure2. Satellites periodically traverse the sky around the center of the Earth, which allows the perturbations from multiple planets to yield periodicity. The comprehensive perturbations from planets on satellites can be counteracted on short-term time scales [24]. The short-term periodical variation of the semimajor axis is not discussed for orbit maneuver prediction in this study. The comprehensive perturbations also cause long-term satellite drift, which leads to variation of the orbital semimajor axis. The orbital semimajor axis a can be expressed as follows: a xj = a0 + f xj + δj , (2) where a0 is the initial value of the orbital semimajor axis; f xj is the long-term variation function for the orbital semimajor axis; xj is the time of the epoch, i.e., the day of the year containing the hours used; δj is the error associated with the orbital semimajor axis, which includes short-term periodical variation, random error, and calculation error from broadcast ephemeris; a xj is the value at epoch xj, for which we used data from the broadcast ephemeris. According to the effects of joint forces from multiple planets acting on satellites, the long-term variation in the orbital semimajor axis can be described by a polynomial function as follows:

n p f xj = ∑ kaxj , (3) p=1

where n is the power of the polynomial function, which depends on the data fitting results, and ka is the multinomial coefficient. The remaining symbols have the same meaning as described above. The long-term rate of change for the orbital semimajor axis can be calculated using nonmaneuvered data based on Equation (3). The minimum values of the orbital semimajor axis before orbit maneuvers can be obtained from long-time historical data from broadcast ephemeris, which is defined as the warning value, aw. The warning time of maneuvers can be calculated as follows: aw − a0 tw = t0 + , (4) ka

where t0 is the time of the initial orbital semimajor axis value; aw is the warning value of the east–west orbit maneuvers, which can be obtained from the statistical results; a0 is the initial value of the orbital semimajor axis; ka is the long-term rate of the orbital semimajor axis, which can be calculated by data fitting; tw is the warning time of the maneuvers. This prediction method can be applied to GEO, IGSO, and MEO satellites. However, the start and end times are controlled by the satellite control center and common users cannot obtain orbit maneuvering strategies owing to confidentiality. Due to short-term perturbations to satellites from multiple planets, the warning values for the orbital semimajor axis are inaccurate. Thus, the predicted warning time is the reference time required to change the strategy of real-time positioning and orbit determination for users. Before orbit maneuvering, the maneuvered satellites would be marked as unhealthy in broadcast ephemeris, and nonmaneuvered satellites may also be marked as unhealthy. Thus, the proposed prediction method is combined with the SV health from the broadcast ephemeris to improve the prediction accuracy. The orbit maneuver prediction factor of orbital semimajor axis is defined as Pa: 1 True ai > aw Pa = , aTF = , , (5) 0 False ai ≤ aw Remote Sens. 2021, 13, 629 6 of 18

where Pa is the prediction factor of the orbital semimajor axis at epoch i; ai is the orbital semimajor axis value of epoch i; aw is the warning value of the orbital semimajor axis; aTF is the warning flag of the orbital semimajor axis, which is true when the value of ai is greater than aw, otherwise aTF is false. The orbit maneuver prediction factor of the satellite health identifier is defined as follows: 1 True HS 6= 0 PH = , HTF = , , (6) 0 False HS = 0

where PH is the prediction factor of the health identifier from broadcast ephemeris at epoch i, and HS is the health identifier from the broadcast ephemeris; 0 indicates that the satellite is healthy and the warning flag HTF is false, otherwise the satellite is unhealthy and HTF is true. The orbit maneuver prediction factor of the satellite health identifier is defined as follows: 1 True t > tw Pt = , tTF = , , (7) 0 False t ≤ tw

where Pt is the prediction factor of the warning time at epoch i; t is the epoch time and tw is the warning time predicted for the orbit maneuvers; tTF is the warning ﬂag of the epoch time for orbit maneuvering, which is true when t is greater than tw, otherwise tTF is false. The orbit maneuver prediction factor is deﬁned as follows:

Pi = Pa + PH + Pt , (8)

where Pi is the orbit maneuver prediction factor at epoch i, which is unrelated in every epoch; when Pi = 2, the satellite may be adjusted through orbit maneuvering several days later. Additionally, when Pi = 3, the satellite can be adjusted in 1–2 h later, at the same time, we suggest that the analysis centers should remove the satellite from real-time navigation products before being published. The prediction information of orbit maneuvers is signiﬁcant for calculating real-time navigation products, which dramatically improves the products’ reliability.

3. Experiment Analysis and Results To verify the proposed methods for analyzing the variation characteristics of orbital semimajor axis and orbit maneuver prediction, experimental data from the broadcast ephemeris were collected for processing. Due to the length limitation of the paper, we analyzed the variation characteristics of the orbital semimajor axes for GEO satellites and IGSO satellites using C01 in Sections 3.1 and 3.2, respectively. Considering the normal orbit determination precision of IGSO satellites is higher than that of GEO satellites, the reliability of ultra-rapid products for IGSO was analyzed in this study, but the prediction method can be applied to all GEO and IGSO satellites. In the first part, the variation characteristics of orbital semimajor axis proposed in this study are confirmed through the measured data from broadcast ephemeris. Next, the validity of the proposed orbit maneuver prediction method is verified in second part. Finally, we note that the reliability of ultra-rapid precise orbit products could be dramatically improved by using the prediction results of orbit maneuvering. The results of real-time positioning also indicate that the service performance of ultra-rapid precise orbit products can be enhanced. In order to demonstrate the applicability of GEO and IGSO satellites, their results are shown in the first and second part, respectively.

3.1. Validation for Variation Characteristics of Orbital Semimajor Axis For variation characteristics of the orbital semimajor axis, we analyzed the experimental results for BeiDou C01. The orbital semimajor axis from the broadcast ephemeris was selected to estimate the coefﬁcients of the functions, and data from 2–27 January 2015 were selected for the ﬁt; the results are shown in Figure3. Remote Sens. 2021, 13, x FOR PEER REVIEW 7 of 18

= + + , (8)

where is the orbit maneuver prediction factor at epoch i, which is unrelated in every epoch; when =2, the satellite may be adjusted through orbit maneuvering several days later. Additionally, when =3, the satellite can be adjusted in 1–2 h later, at the same time, we suggest that the analysis centers should remove the satellite from real-time navigation products before being published. The prediction information of orbit maneuvers is significant for calculating real-time navigation products, which dramatically improves the products’ reliability.

3. Experiment Analysis and Results To verify the proposed methods for analyzing the variation characteristics of orbital semimajor axis and orbit maneuver prediction, experimental data from the broadcast ephemeris were collected for processing. Due to the length limitation of the paper, we analyzed the variation characteristics of the orbital semimajor axes for GEO satellites and IGSO satellites using C01 in Sections 3.1 and 3.2, respectively. Considering the normal orbit determination precision of IGSO satellites is higher than that of GEO satellites, the reliability of ultra-rapid products for IGSO was analyzed in this study, but the prediction method can be applied to all GEO and IGSO satellites. In the first part, the variation characteristics of orbital semimajor axis proposed in this study are confirmed through the measured data from broadcast ephemeris. Next, the validity of the proposed orbit maneuver prediction method is verified in second part. Finally, we note that the reliability of ultra-rapid precise orbit products could be dramatically improved by using the prediction results of orbit maneuvering. The results of real-time positioning also indicate that the service performance of ultra-rapid precise orbit products can be enhanced. In order to demonstrate the applicability of GEO and IGSO satellites, their results are shown in the first and second part, respectively.

3.1. Validation for Variation Characteristics of Orbital Semimajor Axis For variation characteristics of the orbital semimajor axis, we analyzed the experi- Remote Sens. 2021, 13, 629mental results for BeiDou C01. The orbital semimajor axis from the broadcast ephemeris 7 of 18 was selected to estimate the coefficients of the functions, and data from 2–27 January 2015 were selected for the fit; the results are shown in Figure 3.

Figure 3. Time series values of orbital semimajor axis for C01 from 2–27 January 2015.

In Figure3, the blue points indicate the time series values of the orbital semimajor axis for C01. The orbital semimajor axis trend is linear due to the long-term comprehensive influence from multiple planets. The red line indicates the function f + a0, which is fitted using the data from 2–27 January 2015. The fitting coefficients of function f are also shown in Figure3: a0 is 42,163,018.15 m and ka is 111.84 m/day. In addition, the measured orbital semimajor axis also includes short-term periodical variation of the orbital semimajor axis due to the short-term comprehensive influence from multiple planets, random error, and orbital semimajor axis calculation error, which is reflected in δj and root mean square error (RMSE) of 210.09 m. The coefficient of determination (R-square) is 0.9415, and the analyzed variation characteristics of the orbital semimajor axis is indicated by the data fitting results. In addition, time series values of the orbital semimajor axis for C01 from 11 February 2013 to 12 December 2019 were calculated. In Figure4, the blue points indicate the time series values of the orbital semimajor axis for C01. The orbital semimajor axis trend is linear due to the long-term comprehensive influence from multiple planets. The results demonstrate that the variation characteristics of orbital semimajor axis can be described as a linear function. From the results we can know that the slope may change after orbit maneuvers, which were consistent before maneuvers. Additionally, when the value of the orbital semimajor axis is greater than warning value, the long-term rate of the orbital semimajor axis and the warning time can be recalculated. Remote Sens. 2021, 13, x FOR PEER REVIEW 8 of 18

Figure 3. Time series values of orbital semimajor axis for C01 from 2–27 January 2015.

In Figure 3, the blue points indicate the time series values of the orbital semimajor axis for C01. The orbital semimajor axis trend is linear due to the long-term comprehensive influence from multiple planets. The red line indicates the function +, which is fitted using the data from 2–27 January 2015. The fitting coefficients of function are also shown in Figure 3: is 42,163,018.15 m and is 111.84 m/day. In addition, the measured orbital semimajor axis also includes short-term periodical variation of the orbital semimajor axis due to the short-term comprehensive influence from multiple planets, random error, and orbital semimajor axis calculation error, which is reflected in and root mean square error (RMSE) of 210.09 m. The coefficient of determination (R-square) is 0.9415, and the analyzed variation characteristics of the orbital semimajor axis is indicated by the data fitting results. In addition, time series values of the orbital semimajor axis for C01 from 11 February 2013 to 12 December 2019 were calculated. In Figure 4, the blue points indicate the time series values of the orbital semimajor axis for C01. The orbital semimajor axis trend is linear due to the long-term comprehensive influence from multiple planets. The results demonstrate that the variation characteristics of orbital semimajor axis can be described as a linear function. From the results we can know that the slope may change after orbit maneuvers, which were consistent before Remote Sens. 2021, 13maneuvers., 629 Additionally, when the value of the orbital semimajor axis is greater than 8 of 18 warning value, the long-term rate of the orbital semimajor axis and the warning time can be recalculated.

FigureFigure 4. Time 4. Time series series values values of orbitalof orbital semimajor semimajor axis ax foris for C01 C01 from from 11 February11 February 2013 2013 to 12 toDecember 12 Decem- 2019. ber 2019. 3.2. Validation of Orbit Maneuver Prediction Algorithm 3.2. Validation of Orbit Maneuver Prediction Algorithm In this section, we demonstrate the availability of the maneuver prediction method In this section,proposed we demonstrate in this study. the We availa analyzedbility of the the experimental maneuver prediction results for method BeiDou C07, and the proposed in thisorbital study. semimajor We analyzed axis and the SVexperimental health from results broadcast for ephemerisBeiDou C07, were and selected the for analysis. orbital semimajorBroadcast axis and ephemeris SV health published from broadcast by MGEX ephemeris and iGMAS were fromselected 2013 for to analysis. 2019 (7 y) was collected, Broadcast ephemeriswhich containspublished BeiDou by MGEX data sinceand 12iGMAS February from 2013. 2013 We to analyzed2019 (7 y) the was C07 col- satellite, and there lected, which containswere 13 BeiDou instances data of orbitsince maneuvers 12 February from 2013. 12 We February analyzed 2013 the to C07 10 March satellite, 2019. Before orbit and there weremaneuvering, 13 instances of the orbit minimum maneuvers value from of the 12 orbital February semimajor 2013 to axis10 March was 42,171,060 2019. m, which is Before orbit maneuvering,more appropriate the asminimum the warning value value of ofthe orbit orbital maneuvers semimajoraw. The axis orbital was semimajor axis 42,171,060 m, whichwas analyzed is more appropriate first, and the as results the warning are shown value in of Figure orbit 4maneuvers. . The orbital semimajor axisIn Figurewas analyzed5, the x-axis first, isand the the day results of the are year shown and thein Figure y-axis 4. is the value of the orbital In Figure 5,semimajor the x-axisaxis. is the The dayteal of the and ye blackar and lines the y-axis indicate is the the value graph of of the the orbital function. The func- semimajor axis.tion The was teal fitted and black using lines data indica from 21te the March graph to 13of Julythe function. 2019; a0 is The 42,149,416.488 function m and ka is was fitted using95.39 data m/day. from 21 In March addition, to 13 the July measured 2019; orbital is 42,149,416.488 semimajor axis m and also includes is short-term 95.39 m/day. Inperiodical addition, variationthe measured due to orbita the short-terml semimajor comprehensive axis also includes influence short-term from multiple planets, periodical variationrandom due error,to the andshort-term orbital comprehensive semimajor axis influence calculation from error, multiple which planets, is reflected in δj, and the root mean square error (RMSE) is 258.8 m. The coefficient of determination (R-square) is 0.9934. The value of aw is 42,171,060 m and the warning time of the maneuvers tw can be calculated as 15 August 2019 by Equation (4) The blue and purple points denote the

measured orbital semimajor axis before and after orbit maneuvering, respectively. The C07 satellite was adjusted through orbit maneuvering on 23 August 2019. The orbit maneuvering mechanism is unknown, and the measured orbital semimajor axis contains the short-term comprehensive inﬂuence from multiple planets, random error, and orbital semimajor axis calculation error, which brings difﬁculties in accurately calculating the warning time of the maneuvers. Therefore, orbit maneuver prediction factors were analyzed, and the results are shown in Figure6. RemoteRemote Sens. Sens. 2021 2021, 13, 13, x, xFOR FOR PEER PEER REVIEW REVIEW 9 9of of 18 18

randomrandom error, error, and and orbital orbital semimajor semimajor axis axis calculation calculation error, error, which which is is reflected reflected in in , ,and and thethe root root mean mean square square error error (RMSE) (RMSE) is is 258.8 258.8 m. m. The The coefficient coefficient of of determination determination (R-square) (R-square) isis 0.9934. 0.9934. The The value value of of is is 42,171,060 42,171,060 m m and and the the warning warning time time of of the the maneuvers maneuvers cancan bebe calculated calculated as as 15 15 August August 2019 2019 by by Equation Equation (4) (4) The The blue blue and and purple purple points points denote denote the the measuredmeasured orbital orbital semimajor semimajor axis axis before before and and after after orbit orbit maneuvering, maneuvering, respectively. respectively. The The C07C07 satellite satellite was was adjusted adjusted through through orbit orbit ma maneuveringneuvering on on 23 23 August August 2019. 2019. The The orbit orbit ma- ma- neuveringneuvering mechanism mechanism is is unknown, unknown, and and the the measured measured orbital orbital semimajor semimajor axis axis contains contains the the short-termshort-term comprehensive comprehensive influence influence from from mult multipleiple planets, planets, random random error, error, and and orbital orbital Remote Sens. 2021, 13, 629 semimajorsemimajor axis axis calculation calculation error, error, which which brings brings difficulties difficulties in in accurately accurately calculating calculating9 the ofthe 18 warningwarning time time of of the the maneuvers. maneuvers. Therefore, Therefore, orbit orbit maneuver maneuver prediction prediction factors factors were were ana- ana- lyzed,lyzed, and and the the results results are are shown shown in in Figure Figure 6. 6.

FigureFigureFigure 5. 5.5. Analysis AnalysisAnalysis results resultsresults of ofof orbital orbitalorbital semimajor semimajorsemimajor axis axisaxis for forfor C07 C07C07 satellite. satellite.satellite.

FigureFigureFigure 6. 6. Orbit Orbit 6. Orbit maneuver maneuver maneuver prediction prediction prediction factors factors factors for for C07 forC07 C07satellite. satellite. satellite.

In Figure6, the x-axis is the day of the year, and the top half is the results of warning ﬂags of orbital semimajor axis and SV health and the warning time ﬂag for orbit maneu-

vering. The green vertical bar indicates that the corresponding value is false, and red vertical bars indicate true values. The ﬁrst line denotes the warning ﬂag of SV health; C07 was sometimes marked unhealthy in this period. Satellites marked as unhealthy are not necessarily adjusted by orbit maneuvering, which needs the measured orbital semimajor axis and the warning time for orbit maneuvers to calculate the orbit maneuver prediction factors. The results of orbit maneuver prediction factors are shown in the bottom half, and a partial detail of Figure6 is shown in Figure7. Remote Sens. 2021, 13, x FOR PEER REVIEW 10 of 18

In Figure 6, the x-axis is the day of the year, and the top half is the results of warning flags of orbital semimajor axis and SV health and the warning time flag for orbit maneuvering. The green vertical bar indicates that the corresponding value is false, and red vertical bars indicate true values. The first line denotes the warning flag of SV health; C07 was sometimes marked unhealthy in this period. Satellites marked as unhealthy are not necessarily adjusted by orbit maneuvering, which needs the measured orbital semimajor axis and the warning time for orbit maneuvers to calculate the orbit maneuver prediction Remote Sens. 2021, 13, 629 10 of 18 factors. The results of orbit maneuver prediction factors are shown in the bottom half, and a partial detail of Figure 6 is shown in Figure 7.

FigureFigure 7. Orbit 7. Orbit maneuver maneuver prediction prediction factors factors for C07 for C07satellite. satellite.

FromFrom Figure Figure 7,7 ,it it can can be be seen seen that that the the wa warningrning flag flag of the of theorbital orbital semimajor semimajor axis axisap- pearedappeared as true as true for forthe the first first time time on on11 11Augu Augustst 2019. 2019. Due Due toto the the short-term short-term comprehensive comprehensive influenceinfluence from from multiple multiple planetsplanets andand random random error, error, the the warning warning flag flag of theof the orbital orbital semimajor semi- majoraxis wasaxis truewas ortrue false or intermittentlyfalse intermittently until until 20 August. 20 August. The flagThe offlag warning of warning time time for orbit for orbitmaneuvering maneuvering changed changed to trueto true on on 15 15 August. August The. The flag flag of ofSV SV healthhealth changed to to true true at at 16:00:0016:00:00 on on 23 23 August. August. TheThe value value of of the the orbit orbit mane maneuveruver prediction prediction factor factor P isis determined determined by by the the above above flags, flags, whichwhich is is shown shown in inthe the bottom bottom part. part. The The green green vertical vertical bars barsshow show that the that value the valueof the oforbit the maneuverorbit maneuver prediction prediction factor is factor 0, indicating is 0, indicating that the thatsatellite the satelliteis healthy. is healthy.The purple The vertical purple barsvertical show bars that show the value that theof the value orbit of maneuver the orbit maneuverprediction predictionfactor is 1, factorindicating is 1, that indicating there arethat some there abnormalities are some abnormalities of the satellites, of theand satellites,this case is and defined this caseas attention is defined level. as The attention alert level. The alert level is defined when the value of the orbit maneuver prediction factor is 2, level is defined when the value of the orbit maneuver prediction factor is 2, which is de- which is denoted by vertical orange bars, indicating that the satellite was recently adjusted noted by vertical orange bars, indicating that the satellite was recently adjusted by orbit by orbit maneuvering. The vertical red bars show that the value of the orbit maneuver maneuvering. The vertical red bars show that the value of the orbit maneuver prediction prediction factor is 3 and the alarm level for maneuvers, which indicates the satellite would factor is 3 and the alarm level for maneuvers, which indicates the satellite would be ma- be maneuvered 1–2 h later. The orbit maneuver prediction factor of C07 satellite changed neuvered 1–2 h later. The orbit maneuver prediction factor of C07 satellite changed to 3 at to 3 at 16:00:00 on 23 August 2019. This indicates that the satellite would be maneuvered 16:00:00 on 23 August 2019. This indicates that the satellite would be maneuvered between between 19:00:00 and 21:00:00 on 23 August. In order to demonstrate that the C07 satellite 19:00:00 and 21:00:00 on 23 August. In order to demonstrate that the C07 satellite was was maneuvered on 23 August 2019, the final precise orbit products from iGMAS were maneuvered on 23 August 2019, the final precise orbit products from iGMAS were chosen. chosen. The final precise orbit cannot be estimated because of maneuvered thrust. The The final precise orbit cannot be estimated because of maneuvered thrust. The header of Remote Sens. 2021, 13, x FOR PEER REVIEWheader of the precise orbit products published by iGMAS and Wuhan University (WHU)11 of 18 the precise orbit products published by iGMAS and Wuhan University (WHU) was was shown in Figure8. shown in Figure 8.

FigureFigure 8. 8. TheThe headerheader of the precise orbit products products published published by by iGMAS iGMAS an andd WHU WHU on on 23 23 August August 2019.2019.

TheThe header header of of the the precise precise orbit orbit products products from from iGMAS iGMAS removed removed the C07 the satellite, C07 satellite, which iswhich the secondary is the secondary proof of proof the orbital of the maneuverorbital maneuver for C07. for The C07. results The canresults be demonstratedcan be demon- strated through the orbit maneuver detection methods proposed by Huang and Qin et al. [19–21]. In addition, in order to demonstrate the significance of orbit maneuver prediction for real-time navigation products and positioning, the real-time position results using ultra-rapid precise products are presented in the next section. Considering the time system of products from iGMAS is BeiDou Time (BDT), the ultra-rapid precise products from GFZ are analyzed in this study.

3.3. Validation of Reliability of Ultra-Rapid Products by Real-Time Position In order to demonstrate the performance of ultra-rapid products containing maneuvered satellites, the real-time position results were analyzed, and are shown in this section. The ultra-rapid precise orbit products published by MGEX and iGMAS include the unusable orbits of maneuvered satellites because the time information of orbit maneuvers cannot be acquired in advance. There are two schemes for real-time position. The differences between Scheme 1 and Scheme 2 were shown in Table 1.

Table 1. The differences between Scheme 1 and Scheme 2.

Items Orbit and Clock Products Ultra-rapid precise products of 18:00 on 23 August 2019 published by Scheme 1 GFZ Ultra-rapid precise products of 18:00 on 23 August 2019 published by Scheme 2 GFZ, which removed the maneuvered C07 satellite

In Scheme 1, the real-time position used the ultra-rapid precise products of 18:00 on 23 August 2019 published by GFZ. The GFZ ultra-rapid precise products were also chosen for Scheme 2, which removed the maneuvered C07 satellite in advance according to the prediction information. The carried phase and code measurements of single BDS from station ULAB, which is located at Ulaanbaatar, Mongolia, were collected for positioning. The C07 satellite was adjusted through orbit maneuvering between 19:00:00 and 21:00:00 on 23 August, which had to be observed by station ULAB. The station distribution and the ground track of the C07 satellite are shown in Figure 9.

Remote Sens. 2021, 13, 629 11 of 18

through the orbit maneuver detection methods proposed by Huang and Qin et al. [19–21]. In addition, in order to demonstrate the signiﬁcance of orbit maneuver prediction for real-time navigation products and positioning, the real-time position results using ultra- rapid precise products are presented in the next section. Considering the time system of products from iGMAS is BeiDou Time (BDT), the ultra-rapid precise products from GFZ are analyzed in this study.

3.3. Validation of Reliability of Ultra-Rapid Products by Real-Time Position In order to demonstrate the performance of ultra-rapid products containing maneuvered satellites, the real-time position results were analyzed, and are shown in this section. The ultra-rapid precise orbit products published by MGEX and iGMAS include the unusable orbits of maneuvered satellites because the time information of orbit maneuvers cannot be acquired in advance. There are two schemes for real-time position. The differences between Scheme 1 and Scheme 2 were shown in Table1.

Table 1. The differences between Scheme 1 and Scheme 2.

Items Orbit and Clock Products Scheme 1 Ultra-rapid precise products of 18:00 on 23 August 2019 published by GFZ Ultra-rapid precise products of 18:00 on 23 August 2019 published by GFZ, Scheme 2 which removed the maneuvered C07 satellite

In Scheme 1, the real-time position used the ultra-rapid precise products of 18:00 on 23 August 2019 published by GFZ. The GFZ ultra-rapid precise products were also chosen for Scheme 2, which removed the maneuvered C07 satellite in advance according to the prediction information. The carried phase and code measurements of single BDS from station ULAB, which is located at Ulaanbaatar, Mongolia, were collected for positioning. The C07 satellite was adjusted through orbit maneuvering between 19:00:00 and 21:00:00 Remote Sens. 2021, 13, x FOR PEER REVIEW 12 of 18 on 23 August, which had to be observed by station ULAB. The station distribution and the ground track of the C07 satellite are shown in Figure9.

Figure 9. StationFigure distribution 9. Station and distribution ground track and of ground C07 satellite. track of C07 satellite.

In Figure 9, Inthe Figure blue line9, the denotes blue linethe grou denotesnd track the groundof the C07 track satellite, of the which C07 satellite, is dis- which is tributed in distributedthe Asia-Pacific in the region Asia-Paciﬁc as an 8 region shape. as The an 8red shape. point The denotes red point the denoteslocation theof location of station ULAB, which monitored the C07 satellite during the maneuvered period. There are two schemes for real-time positioning. The maneuvered C07 satellite contained in ultra-rapid orbit and clock products of GFZ for position in Scheme 1, which was removed in Scheme 2. The results of real-time precise point positioning (PPP) from 17:00:00– 23:59:59 on 23 August 2019 are shown in Figure 10.

Figure 10. Bias of real-time precise point positioning (PPP) of spatial coordinates.

In Figure 10, the x-axis is the epoch time and the y-axis is the value of the bias of real- time PPP of spatial coordinates. The red points denote the real-time PPP bias after the first estimation of Scheme 1, which shows the error of the ultra-rapid precise products including the maneuvered satellite. The black points denote the real-time PPP bias after iterative estimation of Scheme 1, which indicates the real-time positioning accuracy of PPP by using the ultra-rapid precise orbit and clock products including the maneuvered satellite, and the green points denote the bias of Scheme 2. The bottom part is a detail of the top

Remote Sens. 2021, 13, x FOR PEER REVIEW 12 of 18

Remote Sens. 2021, 13, 629 Figure 9. Station distribution and ground track of C07 satellite. 12 of 18

In Figure 9, the blue line denotes the ground track of the C07 satellite, which is distributed in the Asia-Pacific region as an 8 shape. The red point denotes the location of station ULAB, which monitored the C07 satellitesatellite duringduring thethe maneuveredmaneuvered period.period. ThereThere are two schemes schemes for for real-time real-time positioning. positioning. Th Thee maneuvered maneuvered C07 C07 satellite satellite contained contained in ul- in tra-rapidultra-rapid orbit orbit and and clock clock products products of of GFZ GFZ fo forr position position in in Scheme Scheme 1, 1, which which was removed in SchemeScheme 2. 2. The The results results of real-timeof real-time precise precise point point positioning positioning (PPP) from(PPP) 17:00:00–23:59:59 from 17:00:00– 23:59:59on 23 August on 23 2019August are 2019 shown are in shown Figure in 10 Figure. 10.

Figure 10.FigureBias 10. of real-timeBias of real-time precise pointprecise positioning point positioning (PPP) of (PPP) spatial of coordinates.spatial coordinates.

In Figure Figure 10,10 ,the the x-axis x-axis is is the the epoch epoch time time and and the the y-axis y-axis is the is thevalue value of the of bias the of bias real- of timereal-time PPP of PPP spatial of spatial coordinates. coordinates. The red The points red pointsdenote denotethe real-time the real-time PPP bias PPP after bias the after first estimationthe ﬁrst estimation of Scheme of Scheme1, which 1, shows which the shows error the of errorthe ultra-rapid of the ultra-rapid precise products precise products includ- ingincluding the maneuvered the maneuvered satellite. satellite. The black The points black denote points the denote real-time the real-time PPP bias PPP after bias iterative after Remote Sens. 2021, 13, x FOR PEER REVIEW 13 of 18 estimationiterative estimation of Scheme of 1, Scheme which indicates 1, which the indicates real-time the positioning real-time positioning accuracy of accuracy PPP by us- of ingPPP the by usingultra-rapid the ultra-rapid precise orbit precise and orbitclock and products clock productsincluding including the maneuvered the maneuvered satellite, andsatellite, the green and the points green denote points the denote bias of the Scheme bias of Scheme2. The bottom 2. The part bottom is a partdetail is aof detail the top of part.the top The part. biases The of biasesSchemes of Schemes1 and 2 are 1 andon the 2 are same on theorder same of magnitude order of magnitude before orbit before ma- neuveringorbit maneuvering of the C07 of thesatellite, C07 satellite, which is which less than is less 1.1 than m 1.1after m ambiguity after ambiguity resolution. resolution. The biasesThe biases of both of both schemes schemes changed changed differently differently after after orbit orbit maneuvering; maneuvering; the the bias bias of ofScheme Scheme 1 increased1 increased from from 19:00:00 19:00:00 to 19:31:30, to 19:31:30, when when the bias the biasof the of first the estimation ﬁrst estimation reached reached the high- the esthighest point point of 188.91 of 188.91 m, then m, thenit decreased it decreased because because the weight the weight of C07 of C07was wasreduced reduced with with the elevationthe elevation angle. angle. After After iterative iterative adjustments, adjustments, the bias the of bias Scheme of Scheme 1 was 1 less was than less than2 m. 2The m. positioningThe positioning biases biases in the in east, the north, east, north,and up and directions up directions are shown are shownin Figures in Figures11–13, respec- 11–13, tively.respectively.

Figure 11. Bias of real-time PPP in the east.

Figure 12. Bias of real-time PPP in the north.

Remote Sens. 2021, 13, x FOR PEER REVIEW 13 of 18 Remote Sens. 2021, 13, x FOR PEER REVIEW 13 of 18

part. The biases of Schemes 1 and 2 are on the same order of magnitude before orbit ma- part. The biases of Schemes 1 and 2 are on the same order of magnitude before orbit maneuvering of the C07 satellite, which is less than 1.1 m after ambiguity resolution. The neuvering of the C07 satellite, which is less than 1.1 m after ambiguity resolution. The biases of both schemes changed differently after orbit maneuvering; the bias of Scheme 1 biases of both schemes changed differently after orbit maneuvering; the bias of Scheme 1 increased from 19:00:00 to 19:31:30, when the bias of the first estimation reached the high- increased from 19:00:00 to 19:31:30, when the bias of the first estimation reached the highest point of 188.91 m, then it decreased because the weight of C07 was reduced with the est point of 188.91 m, then it decreased because the weight of C07 was reduced with the elevation angle. After iterative adjustments, the bias of Scheme 1 was less than 2 m. The elevation angle. After iterative adjustments, the bias of Scheme 1 was less than 2 m. The positioning biases in the east, north, and up directions are shown in Figures 11–13, respec- positioning biases in the east, north, and up directions are shown in Figures 11–13, respectively. tively.

Remote Sens. 2021, 13, 629 13 of 18

Figure 11. Bias of real-time PPP in the east. Figure 11. Bias of real-time PPP in the east.

Figure 12. Bias of real-time PPP in the north. Figure 12. Bias of real-time PPP in the north.

Figure 13. Bias of real-time PPP in the up direction.

In Figure 11, the x-axis is the epoch time and the y-axis is the value of the bias of real-time PPP in the east. The orange points denote the real-time PPP bias after the ﬁrst estimation of Scheme 1, which shows the error of the ultra-rapid precise products including the maneuvered satellite. The black points denote the real-time PPP bias after iterative estimation of Scheme 1, which indicates the real-time positioning accuracy of PPP by using the ultra-rapid precise orbit and clock products including the maneuvered satellite, and the blue points denote the bias of Scheme 2. The bottom part is a detail of the top part. The biases of Schemes 1 and 2 are on the same order of magnitude before orbit maneuvering for the C07 satellite, which is less than 0.3 m after ambiguity resolution. The biases of both schemes changed differently after orbit maneuvering; the east bias of Scheme 1 increased from 19:00:00 to 19:31:30, when the bias of the ﬁrst estimation reached the highest point of 116.26 m, then it decreased because the weight of C07 was reduced with the elevation angle. After iterative adjustments, the bias of Scheme 1 was less than 1 m. A comparison of the positioning biases between Schemes 1 and 2 is shown in Figure 14. Remote Sens. 2021, 13, x FOR PEER REVIEW 14 of 18

Figure 13. Bias of real-time PPP in the up direction.

In Figure 11, the x-axis is the epoch time and the y-axis is the value of the bias of real- time PPP in the east. The orange points denote the real-time PPP bias after the first estimation of Scheme 1, which shows the error of the ultra-rapid precise products including the maneuvered satellite. The black points denote the real-time PPP bias after iterative estimation of Scheme 1, which indicates the real-time positioning accuracy of PPP by using the ultra-rapid precise orbit and clock products including the maneuvered satellite, and the blue points denote the bias of Scheme 2. The bottom part is a detail of the top part. The biases of Schemes 1 and 2 are on the same order of magnitude before orbit maneuvering for the C07 satellite, which is less than 0.3 m after ambiguity resolution. The biases of both schemes changed differently after orbit maneuvering; the east bias of Scheme 1 increased from 19:00:00 to 19:31:30, when the bias of the first estimation reached the highest

Remote Sens. 2021, 13, 629 point of 116.26 m, then it decreased because the weight of C07 was reduced with the14 ofele- 18 vation angle. After iterative adjustments, the bias of Scheme 1 was less than 1 m. A comparison of the positioning biases between Schemes 1 and 2 is shown in Figure 14.

Figure 14. Average value of real-time PPPPPP biasesbiases byby usingusing ultra-rapidultra-rapid preciseprecise products.products.

In Figure Figure 12,12, the the x-axis x-axis is is the the epoch epoch time time and and the the y-axis y-axis is the is thevalue value of the of bias the biasof real- of real-timetime PPP in PPP the in north. the north. The purple The purplepoints denote points denotethe real-time the real-time PPP bias PPP after bias the afterfirst esti- the firstmation estimation of Scheme of Scheme1, the black 1, the points black denote points the denote real-time the real-time PPP bias PPP after bias iterative after iterative estima- estimationtion of Scheme of Scheme 1, and 1, the and teal the points teal points denote denote the thebias bias of Scheme of Scheme 2. The 2. The bottom bottom part part is isa adetail detail of of the the top top part. part. The The biases biases of of Scheme Schemess 1 1 and and 2 2 are are on on the the same same order of magnitude before orbit maneuvering for the C07 satellite, which is less than 0.3 mm afterafter ambiguityambiguity resolution. The biases of both schemes changedchanged differently after orbitorbit maneuvering;maneuvering; thethe north-direction bias bias of of Scheme Scheme 1 1increased increased from from 19:00:00 19:00:00 to to19:31:30, 19:31:30, when when thethe bias bias of the of thefirst firstestimation estimation reached reached the highest the highest point pointof 42.86 of m, 42.86 then m, it then decreased it decreased because because the weight the weightof C07 ofwas C07 reduced was reduced with the with elevation the elevation angle. angle. After After iterative iterative adjustments, adjustments, the thebias bias of ofScheme Scheme 1 was 1 was less less than than 0.8 0.8 m. m. A Acomparison comparison of ofthe the positioning positioning biases biases between between Schemes Schemes 1 1and and 2 2is isshown shown in in Figure Figure 14. 14 . In Figure Figure 13,13, the the x-axis x-axis is is the the epoch epoch time time and and the the y-axis y-axis is the is thevalue value of the of bias the biasof real- of real-timetime PPP PPPin the in theup updirection. direction. The The crimson crimson po pointsints denote denote the the real-time real-time PPP PPP bias bias after the firstfirst estimation of Scheme 1, the black points denote the real-time PPP biasbias afterafter iterativeiterative estimation of Scheme 1, and the dark blue points denote the bias of Scheme 2. The bottom estimation of Scheme 1, and the dark blue points denote the bias of Scheme 2. The bottom part is a detail of the top part. The biases of Schemes 1 and 2 are on the same order of magnitude before orbit maneuvering for the C07 satellite, which was less than 1 m after ambiguity resolution. The biases of both schemes changed differently after orbit

maneuvering; the up bias of Scheme 1 increased from 19:00:00 to 19:31:30, when the bias of the ﬁrst estimation reached the highest point of 142.60 m, then it decreased because the weight of C07 was reduced with the elevation angle. After iterative adjustments, the bias of Scheme 1 was less than 1.6 m. A comparison of the positioning biases between Schemes 1 and 2 is shown in Figure 14. Figure 14 shows the average positioning biases in space and in the east, north, and up directions of station ULAB. The red columns denote the results of Scheme 1 and the blue columns are the results of Scheme 2. In the east direction, the average bias of Scheme 1 is 0.67 m and of Scheme 2 is 0.18 m, which was improved by 0.50 m (73.8%). In the north direction, the average bias of Scheme 1 is 0.48 m and of Scheme 2 is 0.21 m, which was improved by 0.27 m (55.9%). In the up direction, the average bias of Scheme 1 is 0.88 m and of Scheme 2 is 0.31 m, which was improved by 0.57 m (64.9%). In the three directions, the improvement in the north is more obvious when using the ultra-rapid products and removing the maneuvered C07 satellite, and the improvement in the east is imperceptible. Remote Sens. 2021, 13, x FOR PEER REVIEW 15 of 18

part is a detail of the top part. The biases of Schemes 1 and 2 are on the same order of magnitude before orbit maneuvering for the C07 satellite, which was less than 1 m after ambiguity resolution. The biases of both schemes changed differently after orbit maneuvering; the up bias of Scheme 1 increased from 19:00:00 to 19:31:30, when the bias of the first estimation reached the highest point of 142.60 m, then it decreased because the weight of C07 was reduced with the elevation angle. After iterative adjustments, the bias of Scheme 1 was less than 1.6 m. A comparison of the positioning biases between Schemes 1 and 2 is shown in Figure 14. Figure 14 shows the average positioning biases in space and in the east, north, and up directions of station ULAB. The red columns denote the results of Scheme 1 and the blue columns are the results of Scheme 2. In the east direction, the average bias of Scheme 1 is 0.67 m and of Scheme 2 is 0.18 m, which was improved by 0.50 m (73.8%). In the north direction, the average bias of Scheme 1 is 0.48 m and of Scheme 2 is 0.21 m, which was improved by 0.27 m (55.9%). In the up direction, the average bias of Scheme 1 is 0.88 m Remote Sens. 2021, 13, 629 and of Scheme 2 is 0.31 m, which was improved by 0.57 m (64.9%). In the three directions,15 of 18 the improvement in the north is more obvious when using the ultra-rapid products and removing the maneuvered C07 satellite, and the improvement in the east is imperceptible. It is related to the positional correlation of the orbit maneuvering direction and the station position.It is related In tothe the three positional dimensions, correlation the average of the orbitbias of maneuvering Scheme 1 is direction1.22 m and and of the Scheme station 2 isposition. 0.43 m, Inwhich the three was improved dimensions, by the 2.17 average m (64.7%). bias of Scheme 1 is 1.22 m and of Scheme 2 is 0.43From m, which the above was improved results, it by is 2.17obvious m (64.7%). that the reliability of ultra-rapid orbit products can obviouslyFrom the be above improved results, with it is obviousout maneuvered that the reliability satellites. of Th ultra-rapide unusable orbit orbits products of maneu- can veredobviously satellites be improved in these without products maneuvered can bring satellites.some problems The unusable for data orbits preprocessing of maneuvered and satellites in these products can bring some problems for data preprocessing and ambiguity ambiguity resolution, which causes the phase measurements to be unusable. The real-time resolution, which causes the phase measurements to be unusable. The real-time positioning positioning results using the ionosphere-free code observations of station ULAB are results using the ionosphere-free code observations of station ULAB are shown in Figure 15. shown in Figure 15.

Figure 15.FigureResults 15. of Results real-time of positioningreal-time positioning by using ionosphere-free by using ionosphere-free code observations. code observations.

In Figure 15, the x-axis is the epoch time and the y-axis is the value of real-time positioning bias by using the ionosphere-free code observations, which includes the results

in space and in the east, north, and up directions. The red points denote the spatial positioning biases of Scheme 1 and the black points denote the results of Scheme 2. The biases of both schemes decreased modestly before 7:00:00, and the bias of Scheme 1 is less than Scheme 2 due to more redundant observations. The biases of both schemes changed differently after orbit maneuvering; the spatial positioning bias of Scheme 1 increased from 19:00:00 to 19:31:30, when it reached the highest point of 13.54 m, then it decreased because the weight of C07 was reduced with the elevation angle. The orange points denote the east positioning biases of Scheme 1 and the blue points denote the results of Scheme 2. The results show that there were no more differences between the two schemes, because the positional correlation of the orbit maneuvering direction and the station position were randomized. The purple points denote the north direction positioning biases of Scheme 1 and the teal points denote the results of Scheme 2. The spatial positioning biases of Scheme 1 were less than Scheme 2 before 7:15:30 due to more redundant observations. The biases of both schemes changed differently after orbit maneuvering; the spatial positioning bias of Scheme 1 increased from 19:00:00 to 19:31:30, when it reached the highest point of Remote Sens. 2021, 13, x FOR PEER REVIEW 16 of 18

In Figure 15, the x-axis is the epoch time and the y-axis is the value of real-time positioning bias by using the ionosphere-free code observations, which includes the results in space and in the east, north, and up directions. The red points denote the spatial positioning biases of Scheme 1 and the black points denote the results of Scheme 2. The biases of both schemes decreased modestly before 7:00:00, and the bias of Scheme 1 is less than Scheme 2 due to more redundant observations. The biases of both schemes changed differently after orbit maneuvering; the spatial positioning bias of Scheme 1 increased from 19:00:00 to 19:31:30, when it reached the highest point of 13.54 m, then it decreased because the weight of C07 was reduced with the elevation angle. The orange points denote the east positioning biases of Scheme 1 and the blue points denote the results of Scheme 2. The results show that there were no more differences between the two schemes, because the positional correlation of the orbit maneuvering direction and the station position were randomized. The purple points denote the north direction positioning biases of Scheme 1 and the teal points denote the results of Scheme 2. The spatial positioning biases of Scheme Remote Sens. 2021, 13, 629 16 of 18 1 were less than Scheme 2 before 7:15:30 due to more redundant observations. The biases of both schemes changed differently after orbit maneuvering; the spatial positioning bias of Scheme 1 increased from 19:00:00 to 19:31:30, when it reached the highest point of 13.49 m,13.49 then m, it then decreased it decreased because because the weight the weight of C07 of was C07 reduced was reduced with the with elevation the elevation angle. angle. The crimsonThe crimson points points denote denote the up the direction up direction biases biasesof Scheme of Scheme 1 and the 1 and dark the blue dark points blue denote points thedenote results the of results Scheme of Scheme2. The biases 2. The of biases both schemes of both schemeschangedchanged differently differently after orbit after maneu- orbit vering;maneuvering; the spatial the positioning spatial positioning bias of Scheme bias of Scheme1 decreased 1 decreased before 19:31:30, before 19:31:30,when it reached when it thereached lowest the point lowest of 0.72 point m. of As 0.72 the m. positional As the positional correlation correlation of the orbit of maneuvering the orbit maneuvering direction anddirection the station and the position station positionwere randomized, were randomized, the positioning the positioning bias in bias the in up the direction up direction of Schemeof Scheme 1 was 1 was less less than than Scheme Scheme 2 for 2 for station station ULAB ULAB from from 19:00:00 19:00:00 to to 21:00:00 21:00:00 on on 23 23 August August 2019.2019. The The average average value value and and standard standard deviation deviation (STD) (STD) of of positioning positioning biases biases from from 18:00:00 18:00:00 toto 23:59:30 23:59:30 on on 23 23 August August 2019 2019 are are shown shown in in Figure Figure 16.16.

Figure 16. Average value andFigure standard 16. Average deviation value (STD) and of standard positioning deviation biases by(STD) using of positioning ionosphere-free biases code by observations.using ionosphere- free code observations. In Figure 16, the graph on the left shows the average positioning bias of station ULAB. The redIn Figure columns 16, denotethe graph the resultson the ofleft Scheme shows 1 the and average the blue positioning columns are bias the of results station of ULAB.Scheme The 2. Inred the columns east, the denote average the biasresults of Schemeof Scheme 1 is 1 0.51 and mthe and blue of columns Scheme 2are is 0.47the re- m, sultswhich of was Scheme improved 2. In the by 0.04east, m the (7.5%). average The bias average of Scheme bias of 1 Scheme is 0.51 1m is and 3.61 of m Scheme in the north 2 is 0.47and m, for which Scheme was 2 isimproved 1.01 m, which by 0.04 was m improved(7.5%). The by average 2.60 m bias (71.9%). of Scheme The average 1 is 3.61 bias m in of theScheme north 1 and is 4.61 for Scheme m in the 2 upis 1.01 direction m, which and was of Scheme improved 2 is by 4.19 2.60 m, m which (71.9%). was The improved average biasby 0.43 of Scheme m (9.3%). 1 is In 4.61 the m three in the directions, up direction the improvementand of Scheme in 2 the is 4.19 north m, is which more was obvious im- provedwhen using by 0.43 the m ultra-rapid (9.3%). In products the three and directions, removing the the improvement maneuvered C07in the satellite, north is and more the obviousimprovement when inusing the eastthe ultra-rapid is imperceptible. products This an isd related removing to the the positional maneuvered correlation C07 satellite, of the andorbit the maneuvering improvement direction in the east and is the imperceptibl station position.e. This In is the related three to dimensions, the positional the correla- average tionbias of of the Scheme orbit maneuvering 1 is 6.62 m and direction of Scheme and 2th ise station 4.50 m, position. which was In the improved three dimensions, by 2.17 m the(32.8%). average The bias graph of Scheme on the shows 1 is 6.62 the m standard and of Scheme deviation 2 is of 4.50 positioning m, which biases. was improved The orange by 2.17columns m (32.8%). denote The the graph results on of the Scheme shows 1 andthe standard the green deviation columns areof positioning the results of biases. Scheme The 2. The results of Scheme 1 are greater than Scheme 2 in all directions, which is obvious in the north. The spatial positioning biases of Scheme 1 are less than Scheme 2 before 7:15:30, but the standard deviation is greater. The above results show that the real-time service performance of ultra-rapid precise products is improved by removing maneuvered satellites in advance. The orbit maneuver prediction method proposed in this study can provide alarm information of orbit maneuvers to remove maneuvered satellites from ultra-rapid precise orbit products. The reliability of ultra-rapid orbit products can be improved obviously without maneuvered satellites.

4. Conclusions and Discussion In this study, we propose a prediction method for orbit maneuvers that can provide alarm information required to change the strategies for real-time navigation products and real-time positioning of maneuvered satellites. We analyzed the variation characteristics of the orbital semimajor axis of the satellites. Satellites tend to gradually deviate from their designed orbits due to various perturbations. The long-term variation in the orbital semimajor axis changes based on a linear trend, which can be described by a linear function. Based on the long-term variation characteristics of the orbital semimajor axis for satellites, a Remote Sens. 2021, 13, 629 17 of 18

prediction method for orbit maneuvers is proposed, combined with SV health in broadcast ephemeris. BDS data from the broadcast ephemeris were analyzed to verify the availability of the proposed method. In addition, in order to demonstrate the signiﬁcance of orbit maneuver prediction for real-time navigation products and positioning, we analyzed the results of real-time PPP and positioning using ionosphere-free code observations with ultra-rapid precise orbit products, demonstrating that the reliability of the products is improved by removing maneuvered satellites in advance. The orbit maneuver prediction method proposed in this study can provide alarm information of orbit maneuvers to remove maneuvered satellites from real-time navigation products. It is vital to improve the reliability of real-time navigation products and BDS service performance, and the method can also be applied to other GNSS. However, there are some questions need to be discussed for removing the maneuvered satellites of real-time navigation products. The analysis centers of IGS and iGMAS can removed the maneuvered before publishing. However, the satellites may be maneuvered after products published. In this case, does the publishing strategy for ultra-rapid products need to be modiﬁed? At present, most of the analysis centers publish ultra-rapid products every three or six hours. Wuhan university publishes the ultra-rapid products once an hour, which can remove the maneuvered satellites timely to ensure the reliability of real-time navigation products. We suggest that the other analysis centers change original publishing strategy to provide the reliable real-time products for users.

Author Contributions: Z.Q., L.W., and G.H. provided the initial idea for this study; Z.Q., L.W., X.Y., and S.X. conceived and designed the experiments; G.H., Q.Z., Z.Q., and F.Y. analyzed the experimental results; Z.Q., G.H., H.S., and X.W. wrote the paper. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the Fundamental Research Funds for the Central Universities, CHD (300102260707), the Chinese Scholarship Council Studentship (ref. 202006560033), programs of the National Natural Science Foundation of China (No. 41774025, 41731066), the Special Fund for Technological Innovation Guidance of Shaanxi Province (2018XNCGG05), the Special Fund for Basic Scientific Research of Central Colleges (grant No. CHD300102268305, Chang’an University), the program of National Key Research and Development Plan of China (2018YFC1505102), and the Grand Projects of the Beidou-2 System (GFZX0301040308). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data can be downloaded from WHU (ftp://igs.gnsswhu.cn/) and Crustal Dynamics Data Information System (CDDIS, https://cddis.nasa.gov/). Acknowledgments: The authors would like to thank the MGEX, iGMAS, and IGS data center of Wuhan University for the data support. Conflicts of Interest: The authors declare no conflict of interest.

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