# Comparison of Satellite Repeat Shift Time for GPS, BDS, and Galileo Navigation Systems by Three Methods

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## Abstract

**:**

## 1. Introduction

## 2. Two Models for Calculating the Satellite Repeat Period

_{SRP}. Since the ground track repeats for a circle in this process, it is also named the revisiting period [23]. As for the satellite operation period, it means the period when the satellite moves around the orbit of itself once in time, denoted as T

_{SOP}[19]. Lastly, the satellite repeat shift time is the interval between the precise satellite repeat period and the approximate satellite repeat period by solar days, denoted as T

_{SRST}[3].

#### 2.1. Orbit Offset Coefficient Method

^{14}m

^{3}/s

^{2}) is Earth’s gravitational constant, R denotes the mean radius of Earth, and H is the height of the satellite from the ground. Then, the number laps of the satellite around the Earth for one solar day can be calculated by:

_{SOP}is counted in minutes, and the 24 × 60 means the time of a solar day united in minutes. Thus, the approximate satellite repeat period united in solar days can be obtained by:

_{int}= Integer(n). D means the approximate days of the satellite repeat period and united in solar days. Additionally, d denotes the coefficient of orbit offset, which is the orbital offset relative to the previous day [24]. In the practical calculation, if the satellite orbital offset is westward, the value of d is negative. If the offset is eastward, the value of d is positive. Since the value of d is only given by satellite designers and is unknown for users, another method is adopted to calculate the approximate satellite repeat period.

#### 2.2. Simplest Fraction Method

_{SOP}. The calculation method is same as the description in Section 2.1. Then, based on the simplest fraction method, the simplest fractions, N and D, can satisfy the following equation:

_{e}is the time interval of the orbit revolving around the Earth one time, which is named the sidereal day. Since T

_{SOP}can be calculated by the broadcast ephemeris, the concrete values of N and D can be determined by the search and test process [23]. Similarly, D also means the approximate days of the satellite repeat period united in solar days.

## 3. Three Methods for Calculating Satellite Repeat Shift TTime

#### 3.1. Broadcast Ephemeris Method

_{SOP}, in inertial space can be expressed by:

_{SRFT}, can be defined as:

_{MSD}is the duration of a mean solar day in seconds. N denotes the number of laps of the satellite operations around the orbit of itself in D solar days. T

_{SOP}is the precise satellite repeat period in seconds, which is obtained by Equations (5) and (6).

_{SRP}, for GPS satellites is given by:

_{SRP}for Galileo satellites is determined by:

_{SRP}and TBM

_{SRP}denote the repeat period of BDS GEO/IGSO and MEO satellites, respectively.

_{SRFT}and TGA

_{SRFT}mean the repeat shift time of the GPS and Galileo satellite, respectively. Additionally, TB

_{SRFT}and TBM

_{SRFT}denote the repeat shift time of the BDS GEO/IGSO and BDS MEO satellite, respectively.

#### 3.2. Correlation Coefficient Method

_{t}is the sampling of the observations, and τ

_{m}is the epoch of the correlation coefficient that keeps the maximum. In order to calculate the repeat shift time accurately, the sample rate of the observation data must be 1 Hz. It is only in this way that the shift time can reach the seconds level.

#### 3.3. Aspect Repeat Time Method

_{0}(t

_{0}) on the epoch t

_{0}in the reference day for the first time it is observed, and the other vector is x

_{1}(t

_{1}) on the epoch t

_{1}in the subsequent day for the second time it is observed after a repetition period. It is noted that, the epochs t

_{0}and t

_{1}are counted from zero for every solar day. If the angle between the two unit vectors x

_{0}(t

_{0}) and x

_{1}(t

_{1}) reaches the minimum value, the satellite repeat shift time can be determined:

_{t}is the sampling of the observations and t

_{1}is the epoch of the second time it is observed. The process of the calculation involves the following steps and is presented in Figure 1:

_{0};

_{0}, and express it as x

_{0}(t

_{0});

_{1}, which has a similar unit vector x

_{1}(t

_{1}) to the initial vector x

_{0}(t

_{0}) during the repeat period; and

## 4. Experiments and Analysis

#### 4.1. Performance Analysis for the GPS System

#### 4.2. Performance Analysis for the BDS System

#### 4.3. Performance Analysis for the Galileo System

#### 4.4. Comparison and Analysis Among Systems

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The mean shift time for GPS satellites calculated by the three methods during DOY: 161–181, 2018.

**Figure 3.**The STD of the satellite repeat shift time for the three methods for each satellite during the period DOY:161–181, 2018.

**Figure 4.**The mean shift time for BDS GEO satellites calculated by the three methods during DOY: 161–181, 2018.

**Figure 5.**The STD of the satellite repeat shift time for the three methods for each satellite during the period DOY:161–181, 2018.

**Figure 6.**The mean shift time for BDS IGSO satellites calculated by the three methods during DOY: 161–181, 2018.

**Figure 7.**The STD of the satellite repeat shift time for the three methods for each satellite during the period DOY:161–181, 2018.

**Figure 8.**The mean shift time for BDS MEO satellites calculated by the three methods during DOY: 161–181, 2018.

**Figure 9.**The STD of the satellite repeat shift time for the three methods for each satellite during the period DOY:161–181, 2018.

**Figure 10.**The mean shift time for GAL(Galileo) satellites calculated by the three methods during DOY: 161–191, 2018.

**Figure 11.**The STD of the satellite repeat shift time for the three methods for each satellite during the period DOY:161–191, 2018.

**Table 1.**The mean shift time and STD for GAL E14 and E18 satellites, which was calculated by the three methods during DOY:161–191, 2018.

Satellite Number | BEM | CCM | ARTM | ||||
---|---|---|---|---|---|---|---|

Time(s) | STD(s) | Time(s) | STD(s) | Time(s) | STD(s) | ||

E14 | 4880.07 | 34.82 | 4885 | 33.56 | 4892 | 32.41 | |

E18 | 4875.22 | 35.59 | 4881 | 36.14 | 4878 | 34.13 |

**Table 2.**The STD of the GPS, BDS, and GAL systems, which was calculated by the three methods. BS denotes the STD between the satellites for the same system, and MS denotes the STD of the mean shift time for the same satellite.

System STD(S) Method | BEM | CCM | ARTM | ||||
---|---|---|---|---|---|---|---|

BS | MS | BS | MS | BS | MS | ||

GPS | 2.74 | 0.35 | 2.68 | 0.35 | 2.67 | 0.34 | |

BDS | GEO | 0.59 | 6.69 | 1.58 | 7.16 | 1.34 | 6.91 |

IGSO | 9.40 | 1.98 | 10.52 | 2.08 | 9.46 | 2.02 | |

MEO | 3.69 | 2.50 | 4.62 | 2.52 | 3.51 | 2.34 | |

GAL | 4.48 | 4.77 | 4.45 | 4.75 | 4.39 | 4.71 |

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**MDPI and ACS Style**

Yang, Y.; Jiang, J.; Su, M.
Comparison of Satellite Repeat Shift Time for GPS, BDS, and Galileo Navigation Systems by Three Methods. *Algorithms* **2019**, *12*, 233.
https://doi.org/10.3390/a12110233

**AMA Style**

Yang Y, Jiang J, Su M.
Comparison of Satellite Repeat Shift Time for GPS, BDS, and Galileo Navigation Systems by Three Methods. *Algorithms*. 2019; 12(11):233.
https://doi.org/10.3390/a12110233

**Chicago/Turabian Style**

Yang, Yanxi, Jinguang Jiang, and Mingkun Su.
2019. "Comparison of Satellite Repeat Shift Time for GPS, BDS, and Galileo Navigation Systems by Three Methods" *Algorithms* 12, no. 11: 233.
https://doi.org/10.3390/a12110233