Orbital Maneuver Analysis of BDS-3 Satellites Based on Time-Differenced Carrier Phase Velocity Measurement

Miao, Yage; Tu, Rui; Hong, Ju; Zhang, Shixuan; Li, Fangxin; Liu, Mingyue

doi:https://doi.org/10.1155/2023/1610393

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2023 | Article ID 1610393 | https://doi.org/10.1155/2023/1610393

Orbital Maneuver Analysis of BDS-3 Satellites Based on Time-Differenced Carrier Phase Velocity Measurement

Yage Miao,^1,2Rui Tu ,^1,2,3Ju Hong,^1,2Shixuan Zhang,^1,2Fangxin Li,^1,2and Mingyue Liu^1,2

Academic Editor: Ardashir Mohammadzadeh

Received31 May 2022

Revised29 Dec 2022

Accepted10 Jan 2023

Published20 Jan 2023

Abstract

The BeiDou-3 global navigation satellite system (BDS-3) has been widely used in various fields worldwide since its official launch on July 31, 2020. As of August 1, 2021, 33 satellite orbital maneuver events have occurred in BDS-3. The satellite orbits deviate owing to external factors, and orbital maneuvers are required to adjust the orbit to ensure the appropriate operation and service of the system. This study is the first systematic analysis of historical orbital maneuvers since the launch of BDS-3, eight stations that received BDS-3 signals in the Asia-Pacific region and India were selected, and 33 maneuver events of BDS-3 were detected and enumerated using the time-differenced carrier phase velocity measurement algorithm. The results revealed that the method can detect all orbital maneuvers of BDS-3, and the detection results are consistent with the maneuver dates marked by the broadcast ephemeris. In most maneuver events, the detection start and end times were approximately 10–60 min and 30–90 min earlier, respectively, than the marked time in the broadcast ephemeris. The orbital maneuvers of inclined geosynchronous orbit satellites significantly affected the velocity measurements, resulting in a mean error of approximately 0.06–0.73 m/s for each velocity component. Furthermore, the maneuvering of geostationary orbit satellites had a little effect on their velocity, and the mean velocity measurement error was mostly within 0.09 m/s. Among these satellites, the maneuvering of the C60 satellite had a smaller effect on the velocity measurement, and the mean velocity measurement error was within 0.03 m/s. The weak response at the ground level led to the weak detection sensitivity of this method to the maneuver of C60.

1. Introduction

The BeiDou-3 Global Satellite Navigation System (BDS-3), which provides PNT (positioning, navigation, and timing) services for several fields worldwide, was completed on July 31, 2020 [1, 2]. The BDS-3 satellite constellation comprises 24 medium-orbit (MEO) satellites, three geostationary orbit (GEO) satellites, and three inclined geosynchronous orbit (IGSO) satellites [3, 4]. Orbital maneuvering refers to the use of the propulsion system of the satellite to calibrate its orbit and return it to the preset orbit. As GEO and IGSO are more vulnerable to the effects of nonspherical gravity than MEO, they reveal more frequent orbital maneuvers of BDS [5, 6]. Previous statistics revealed that GEO and IGSO satellites may perform orbital maneuvers once a month and twice a year, respectively, [7]. The navigation message broadcast by the satellite was predicted using ground monitoring station data and a satellite orbit model. Therefore, during orbital maneuvers, the actual position of the satellite may be tens of kilometers away from the predicted orbit position [8]. Accurate detection of the PRN (pseudorandom noise code) and the maneuvering period of a satellite can improve the utilization rate of satellite effective data. Moreover, the actual orbit of the maneuvering satellite can be recovered as early as possible so that the satellite is still available during the maneuver, which can improve the performance of PNT services [9, 10].

Recently, researchers have proposed different methods for detecting orbital maneuvers. In 2021, Tu et al. [7] proposed a monitoring model based on the epoch difference velocity estimation principle and multistation observation of BDS-2, which can estimate the three-dimensional dynamic change in orbital maneuvers in real-time. In 2015, Sciré et al. [11] analyzed a spatial debris-orbit determination algorithm using space-based optical observation data. In 2018, Huang et al. [12] proposed an optimized robust detection method based on pseudorange observation, broadcast ephemeris, and known station coordinates. Cao et al. [13] established a maneuvering force model in 2014 to establish a continuous GEO satellite orbit during repositioning maneuvers. Cui et al. [14] used orbital residuals and mechanical models to detect the orbital maneuvers of space targets in 2016. Ye et al. [15] used the orbit difference before and after the maneuver to detect the orbital maneuver in 2017; however, its detection was only used after maneuver recovery. Nevertheless, these studies have only explored the detection models and methods of orbital maneuvers and have not systematically analyzed the orbital maneuvering events after the completion of the BDS-3 global system.

In the present study, the carrier phase epoch difference velocity measurement algorithm and the BDS-3 observation data from multiple stations were used to measure the velocity of static stations, and the empirical threshold was set by the change in ionosphere-free combination carrier phase residuals to detect the orbital maneuver period of BDS-3. Based on the detection and analysis of all maneuver events within 1 year after the establishment of BDS-3, this study compares the different maneuver periods detected by this method using the broadcast ephemeris, and for the first time provides a systematic analysis of the historical maneuvering events of BDS-3, which contributes to the monitoring and evaluation of the BDS-3 system. The applicability and limitation of this method to different satellites in BDS-3 are also demonstrated. The influence of the BDS-3 satellite on the velocity of the receiver during maneuvering is analyzed, and valuable conclusions are drawn.

Section 2 describes the detection algorithm for orbital maneuvers, and detailed test data and results are presented in Section 3. Section 4 presents a discussion. Finally, Section 5 presents the conclusions and prospects of this study.

2. Methodology

The three-dimensional velocity of the station can be calculated in real-time using the principle of epoch differential velocity measurement based on a broadcast ephemeris. When the station is static and the satellite data are normal, the theoretical value of the measured station velocity should be zero. However, in the process of orbital maneuvers, owing to the large deviation between the actual position of the satellite and the ephemeris position and rapid adjustment process, the calculated station velocity may exhibit a large deviation [10]. Based on this principle, the orbital maneuvers of the participating satellites can be detected. Figure 1 illustrates the flow chart of the satellite orbit maneuver detection.

2.1. Time-Differenced Velocity Estimation

The observation equation of the carrier phase can be expressed as follows [16]:where is the wavelength of the carrier phase observation, is the carrier phase observation value, is the geometric distance from the satellite to the receiver, is the speed of light, is the receiver clock error, is the satellite clock error, and are the ionospheric and tropospheric delays, respectively, is the integer ambiguity, and comprises other modeling errors, nonmodeling errors, and measurement noise. The cycle slip is detected by the combination of phase geometry-free combination, Melbourne-Wübbena (MW) combination, and code-phase combination, with the thresholds all being empirical thresholds [17–20]. When there was no cycle slip between the two epochs, the difference between adjacent epochs was considered for the carrier phase observation as follows [21]:where is a single difference operator between the epochs. After a differential operation, the effect of integer ambiguity was eliminated, and the ionospheric and tropospheric delays were reduced. The Saastmoinen model [22] and the GMF [23, 24] mapping function were used to obtain the tropospheric dry component delay and partial wet component delay, and the residual zenith tropospheric wet component delay was calculated by piecewise constant estimation. Satellite clock errors and ionospheric delays were eliminated by broadcast ephemeris [22] and dual-frequency ionospheric-free models [25, 26], respectively. Ignoring the residual error after correction, the differenced equation can be rewritten as follows:where and are the coefficients of the dual-frequency ionosphere-free combination, and the values are as follows:

In equation (3), is the difference in geometric distance between the satellite and the receiver at and , which can be expressed as follows [27]:where and are the coordinate vectors of the satellite and the receiver at time and , respectively, and , is the unit vector. Let , then equation (5) can be rewritten as follows:

Combining equations (3) and (6), the observation equation can be written as follows:where and the unknown parameters include and .

When the number of available satellites is ≥4, the receiver velocity can be obtained using the least-squares method [28, 29]:where is the weight matrix and the other parameters are as follows:

According to the obtained , the receiver velocity from to can be expressed as follows:

2.2. Maneuver Period Determination

The change in the observation residual is sensitive to satellite maneuvers. Therefore, the observation residual can be used to determine the start and end times of satellite maneuvers. Based on the analysis of satellite orbital maneuver events within 1 year after the completion of BDS-3, the conditions for the maneuver start and end times can be set as follows:

In equation (11), is the standard deviation of satellite residuals at time ; is the residual of the satellite at time ; is the mean residual of all satellites at time ; indicates the judgment threshold of maneuvering start at ; and is a dynamic threshold whose value is the mean residual of the first 10 epochs per hour. Figure 2 shows the calculation and scope of the dynamic thresholds. If and lasts for 5 minutes, the orbital maneuver begins at . After identifying the start of the maneuver, if the condition in equation (13) is satisfied; that is, the standard deviation of 10 consecutive epochs is less than half of the mean standard deviation of the first 10 epochs, the end of the orbital maneuver for this satellite can be determined at .

2.3. Determining PRN of Maneuverable Satellites

After the orbital maneuver is detected, we still require to determine the PRN of the satellite undergoing an orbital maneuver to accurately exclude the influence of orbital maneuvers. The satellites involved in the calculation were excluded individually, and the mean standard deviation of the residual of the next 10 epochs of the detected maneuvering start time was calculated. The satellite with the smallest mean value (obtained after removing it) was considered to have been maneuvered.

The JFNG station in Wuhan, China, was selected considering the maneuver event of the C59 satellite on March 11, 2021. As illustrated in Figure 3, after excluding the C59 satellite, the mean standard deviation after the maneuver began was the smallest, thereby indicating that the C59 satellite was maneuvered.

3. Datasets and Test Results

3.1. Datasets

To verify the applicability of this method to BDS-3 satellites and to analyze their maneuvers, the present study selected eight stations in the Asia-Pacific region and India, which received BDS-3 GEO and IGSO satellite signals and analyzed the orbit maneuver of the satellites collectively with station data. The observation data sampling interval was 30 s, the broadcast ephemeris and satellite phase center correction data were obtained from the International GNSS Service (IGS), and the Earth Rotation Parameters (ERP) file was obtained from the Center for Orbit Determination (CODE) in Switzerland, Europe. Figure 4 illustrates the location distribution of the eight selected stations, and Table 1 summarizes the station details.

3.2. Test Results

In total, 33 orbital maneuvers occurred between the completion of BDS-3 and August 1, 2021. Velocity analysis of the selected eight stations was carried out, and 264 detection results were obtained. Among these, 140 valid results were obtained and 124 results could not be used for the calculation because of the lack of files or the failure to receive the signal from the maneuvering satellite.

3.2.1. Comparison with Broadcast Ephemeris

Table 2 summarizes the maneuver periods detected by this method for the three IGSO satellites of BDS-3 at the eight selected stations. Excluding the special exception that the detected maneuver time occurs later than the time marked in the broadcast ephemeris (marked as a red flag in Table 2), the detected IGSO satellite maneuvering start time is approximately 15–56 min earlier than the broadcast ephemeris results and approximately 30–90 min earlier than the broadcast ephemeris recorded end time. On the 141st day of 2021, the maneuver period detected for C38 by the ULAB data differs significantly from that of other stations. A possible problem is the data quality of the ULAB station on that day.

Figure 5 illustrates the advancement of the detected maneuvering period of the two GEO satellites compared to the broadcast ephemeris results. The detected maneuver start and end times of GEO satellites were approximately 10–60 min and 90 min earlier than those of the broadcast ephemeris, respectively. Notably, in the detection of C60, as shown in Figure 6, the estimated velocity had a large bias before the detected maneuver start time, which was due to the poor detection sensitivity caused by the small residual variation during the maneuvering of the C60 satellite. In this case, the detection start time was significantly later than the marked time to avoid the influence of this situation on the C60 maneuver analysis. All results of the detected start time are later than that of the marked time. The data in the blue box in Figure 5 were not considered in the subsequent analysis.

Table 3 summarizes the detection of all maneuver events of the BDS-3 maneuvering satellites. Owing to the small number of IGSO satellite maneuvering samples, only partial analysis was performed. The detected start times of IGSO satellites (C38, C39, and C40) maneuvers were, on average, 30.67 minutes, 34.15 minutes, and 41.57 minutes ahead of the broadcast ephemeris, respectively. The detected end times were, on average, 90.50 minutes, 44.78 minutes, and 90.50 minutes ahead of the broadcast ephemeris, respectively. The detection results of two GEO satellites (C59 and C60) were relatively stable. With 95% confidence, the start and end time of the C59 satellite maneuvering were detected at approximately 32.03–44.45 min and 87.48–98.15 min earlier than the broadcast ephemeris, respectively; whereas that of the C60 satellite maneuvering was detected approximately 16.17–25.27 min and 88.89–96.40 min earlier than broadcast ephemeris, respectively. This method efficiently reduces the effect of abnormal data during the maneuvering period in practical applications and increases the utilization rate of normal data after the end of orbital maneuvering.

3.2.2. Effect of BDS-3 Satellite Maneuver on Velocity Measurement

To analyze the effect of the BDS-3 satellite maneuver on the results of the epoch differential velocity measurement, this study calculated the maximum and mean values of each velocity component of the velocity measurement error during the satellite maneuver, based on the detected maneuver period. Figure 7 illustrates the effect of the five maneuvering satellites on JFNG velocity during a maneuver event. Figure 8 depicts the maximum and mean values of the effect of IGSO orbital maneuvers on the velocity measurements at different stations. Furthermore, Figures 9 and 10 illustrate the maximum and mean values of the effect of the C59 and C60 satellite maneuvering on the velocity measurement of each station, respectively.

Considering velocity bias caused by some abnormal data and the mean velocity measurement bias during the orbit maneuver, it was observed that the velocity measurement bias caused by the C38 satellite orbit maneuver to the U component was highest (2.21 m/s), and the mean velocity measurement bias for the three components was between 0.09 and 0.51 m/s. Similarly, the orbit maneuver of the C39 satellite caused the largest velocity measurement bias to the U component (3.05 m/s), and the mean velocity measurement bias to the three velocity components was between 0.13 and 0.73 m/s. The bias of the three velocity components caused by the orbit maneuver of the C40 satellite differed slightly, and the mean error was approximately 0.06–0.37 m/s. The velocity measurement bias of the U component caused by the C59 and C60 satellite orbit maneuvers was slightly larger than that of the E and N components, and the maximum velocity measurement bias was 1.49 m/s and 0.26 m/s in the U component, respectively. Moreover, the mean velocity bias of each velocity component for the C59 and C60 satellite orbit maneuvers was <0.09 m/s and <0.03 m/s, respectively.

Figures 11 and 12 illustrate the residuals of the IGSO and GEO satellites during maneuvers, based on data from the eight stations selected in this study. It can be seen that the residuals of the five satellites remained at the same level when no maneuvering occurs, while during the maneuvering period, the residuals of C60 were much smaller than those of other satellites. That is, the residual value at the beginning of the orbital maneuver of C60 fluctuates slightly; therefore, this method is not sensitive to the maneuvering detection of C60. BDS-3 can be used in combination with other satellite navigation systems while considering the use of a type-3 fuzzy logic system [30–32] to reduce the influence of abnormal data on C60 residuals and improve the sensitivity of C60 orbital maneuver detection.

4. Discussion

Because the proposed maneuver detection method requires standard values to determine the start and end times of the maneuver period, the selection of the optimal reference value is also an interesting potential topic for future research. In addition, the accurate detection of cycle slips is significant and must be solved the problem in carrier phase measurement, which can avoid false detection of orbital maneuvers caused by the quality of carrier observations. The orbit maneuver of BDS-3 has different effects on each velocity component in the velocity measurement: when more satellites are visible and more real-time service is required, priority is given to the elimination of satellites with large errors or lower weights.

5. Conclusions

Based on the theory of carrier phase epoch differential velocity measurement, eight stations in the Asia-Pacific and India were selected to analyze the orbit maneuver events of BDS-3 within one year of its construction. The results revealed the following theories:(1)The maneuver time of the BDS-3 satellites detected by this method is only 5 min later than the actual maneuver time of the satellite, which has good real-time performance.(2)Because the orbital maneuver of the C60 satellite has little effect on the velocity measurement, this method is not sensitive to the orbital maneuver detection of the C60 satellite.(3)The maneuvering start and end times of the detected IGSO satellite were approximately 15–56 and 30–90 min earlier than that of the broadcast ephemeris. The detected GEO satellite maneuvering start and end times were approximately 10–60 min and 90 min earlier than that of the broadcast ephemeris.(4)The orbit maneuvers of the IGSO satellites in BDS-3 had a remarkable effect on the velocity measurement. The maximum velocity measurement bias was 3.05 m/s in the U-component, and the mean velocity component bias was approximately 0.06–0.73 m/s during the orbit maneuver. The effect of the GEO satellite maneuver on the velocity measurement was much smaller than that of IGSO, and the effect on the three-component velocity of ENU was also similar. The mean velocity measurement error was <0.09 m/s. The C60 satellite maneuver had the least effect on the velocity measurement, and the mean velocity measurement bias was <0.03 m/s.

In all, this method can detect all orbital maneuvers of BDS-3, and the detection results are consistent with the maneuver dates marked by the broadcast ephemeris. However, the orbital maneuver detection of the C60 is not sensitive; therefore, in the future, it is necessary to consider combining BDS-3 with other navigation systems and using type-3 fuzzy logic systems to improve this method.

Data Availability

The data are downloaded from the International GNSS Service Center.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The study was supported by the National Natural Science Foundation of China (Grant nos. 41974032 and 42274019).

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Copyright © 2023 Yage Miao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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