# LEO Onboard Real-Time Orbit Determination Using GPS/BDS Data with an Optimal Stochastic Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Onboard RTOD Algorithm

#### 2.1. Dynamical Models

#### 2.2. GNSS Measurements

#### 2.3. Parameter Estimation

## 3. Pseudo-Ambiguity Optimal Stochastic Modeling

#### 3.1. Pseudo-Ambiguity Modeling

#### 3.2. Stochastic Model Setting

^{−3}–2.0 × 10

^{−3}m/s, so the values of three parameters $\left({\sigma}_{G}^{2},{\sigma}_{Cg}^{2},{\sigma}_{Cim}^{2}\right)$ are set to be ranging from 10

^{−9}–10

^{+1}m

^{2}/s along with the variation of change rate $\upsilon $ in the interval of 1.0×10

^{−5}─7.5×10

^{−1}m/s. In total, 20 sets of values are tested, which are listed in Table 2 and marked as S01─S20 for convenient expression in the following.

## 4. Orbit Results Analysis

#### 4.1. Numerical Search Tests

#### 4.2. Effect of Pseudo-Ambiguity

#### 4.3. Impact of GPS/BDS Fusion

## 5. Application Prospect Discussion

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wulder, M.A.; Coops, N.C. Satellites: Make Earth observations open access. Nature
**2014**, 513, 30–31. [Google Scholar] [CrossRef] [PubMed] - Su, Y.; Liu, Y.; Zhou, Y.; Yuan, J.; Cao, H.; Shi, J. Broadband LEO Satellite Communications: Architectures and Key Technologies. IEEE Wirel. Commun.
**2019**, 26, 55–61. [Google Scholar] [CrossRef] - Wang, L.; Lü, Z.; Tang, X.; Zhang, K.; Wang, F. LEO-Augmented GNSS Based on Communication Navigation Integrated Signal. Sensors
**2019**, 19, 4700. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hart, R.C.; Hartman, K.R.; Long, A.C.; Lee, T.; Oza, D.H. Global Positioning System (GPS) Enhanced Orbit Determination Experiment (GEODE) on the Small Satellite Technology Initiative (SSTI) Lewis Spacecraft. In Proceedings of the 9th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION-GPS-1996), Kansas City, MO, USA, 17–20 September 1996. [Google Scholar]
- Hart, R.C.; Long, A.C.; Lee, T. Autonomous Navigation of the SSTI/Lewis Spacecraft Using the Global Positioning System (GPS). In Proceedings of the Flight Mechanics Symposium, GSFC, Greenbelt, MD, USA, 19–21 May 1997. [Google Scholar]
- Goldstein, D.B. Real-Time, Autonomous Precise Satellite Orbit Determination Using the Global Positioning System. Ph.D. Thesis, University of Colorado, Boulder, CO, USA, October 2000. [Google Scholar]
- Reichert, A.; Meehan, T.; Munson, T. Toward Decimeter-Level Real-Time Orbit Determination: A Demonstration Using the SAC-C and CHAMP Spacecraft. In Proceedings of the ION-GPS-2002, Portland City, OR, USA, 24–27 September 2002. [Google Scholar]
- Gill, E.; Montenbruck, O.; Arichandran, K.; Tan, S.H.; Bretschneider, T. High-precision Onboard Orbit Determination for Small Satellites-the GPS-based XNS on X-SAT. In Proceedings of the 6th Symposium on Small Satellites Systems and Services, La Rochelle, France, 20–24 September 2004. [Google Scholar]
- Montenbruck, O.; Nortier, B.; Mostert, S. A miniature GPS Receiver for Precise Orbit Determination of the Sunsat 2004 Micro-Satellite. In Proceedings of the 2004 National Technical Meeting of The Institute of Navigation (ION NTM 2004), San Diego, CA, USA, 26–28 January 2004. [Google Scholar]
- Montenbruck, O.; Markgraf, M.; Naudet, J.; Santandrea, S.; Gantois, K.; Vuilleumier, P. Autonomous and Precise Navigation of the PROBA-2 spacecraft. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, HI, USA, 18–21 August 2008. [Google Scholar]
- Montenbruck, O.; Swatschina, P.; Markgraf, M.; Santandrea, S.; Naudet, J.; Tilmans, E. Precision spacecraft navigation using a low-cost GPS receiver. GPS Solut.
**2012**, 16, 519–529. [Google Scholar] [CrossRef] - Montenbruck, O.; Ramos-Bosch, P. Precision real-time navigation of LEO satellites using global positioning system measurements. GPS Solut.
**2008**, 12, 187–198. [Google Scholar] [CrossRef] - Gong, X.; Wang, F.; Liu, W. A Novel Algorithm on Sub-meter Level Real-Time Orbit Determination Using Space-Borne GPS Pseudo-Range Measurements. Lect. Notes Electr. Eng.
**2014**, 305, 89–100. [Google Scholar] - Wang, F.; Gong, X.; Liu, W. A novel decimeter-level real-time orbit determination algorithm using space-borne GPS measurements with separation and absorption of broadcast ephemeris error. Geomatics Inform. Sci. Wuhan Univ.
**2015**, 40, 1230–1236. [Google Scholar] - Wang, F.; Gong, X.; Sang, J.; Zhang, X. A Novel Method for Precise Onboard Real-Time Orbit Determination with a Standalone GPS Receiver. Sensors
**2015**, 15, 30403–30418. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gong, X.; Wang, F. Autonomous orbit determination of HY2A and ZY3 missions using space-borne GPS measurements. Geomatics Inform. Sci. Wuhan Univ.
**2017**, 42, 309–313. [Google Scholar] - China Satellite Navigation Office. Development of the BeiDou Navigation Satellite System (Version 4.0). Available online: http://www.beidou.gov.cn/xt/gfxz/201912/P020191227430565455478.pdf (accessed on 31 December 2019).
- Langley, R.B. Innovation: GLONASS—Past, present and future. 2017. Available online: https://www.gpsworld.com/innovation-glonass-past-present-and-future/ (accessed on 1 November 2019).
- Benedicto, J. Directions 2020: Galileo moves ahead. 2019. Available online: https://www.gpsworld.com/directions-2020-galileo-moves-ahead/ (accessed on 14 December 2019).
- Zhao, Q.; Wang, C.; Guo, J.; Yang, G.; Liao, M.; Ma, H.; Liu, J. Enhanced orbit determination for BeiDou satellites with FengYun-3C onboard GNSS data. GPS Solut.
**2017**, 21, 1179–1190. [Google Scholar] [CrossRef] [Green Version] - Li, M.; Li, W.; Shi, C.; Jiang, K.; Guo, X.; Dai, X.; Meng, X.; Yang, Z.; Yang, G.; Liao, M. Precise orbit determination of the Fengyun-3C satellite using onboard GPS and BDS observations. J. Geodesy
**2017**, 91, 1313–1327. [Google Scholar] [CrossRef] [Green Version] - Li, X.; Zhang, K.; Meng, X.; Zhang, W.; Zhang, Q.; Zhang, X.; Li, X. Precise Orbit Determination for the FY-3C Satellite Using Onboard BDS and GPS Observations from 2013, 2015, and 2017. Engineering
**2019**, 6, 904–912. [Google Scholar] [CrossRef] - Shi, C.; Zhao, Q.; Geng, J.; Lou, Y.; Ge, M.; Liu, J. Recent development of PANDA Software in GNSS Data Processing. In Proceedings of the International Conference on Earth Observation Data Processing and Analysis (ICEODPA), Wuhan, China, 28–30 December 2008. [Google Scholar]
- Montenbruck, O.; Gill, E. Satellite Orbits: Models, Methods and Application; Springer: Berlin, Germany, 2020. [Google Scholar]
- Wang, F. Theory and Software Development on Autonomous Orbit Determination Using Space-Borne GPS Measurements. Ph.D. Thesis, Wuhan University, Wuhan, China, November 2006. [Google Scholar]
- Gong, X. Key Technologies and Software toward Decimeter-LEVEL autonomous Orbit Determination with Space-Borne GPS Carrier-Phase Measurements. M.Sc. Thesis, Wuhan University, Wuhan, China, 2015. [Google Scholar]
- Liu, J.; Li, Z.; Wang, Y.; Sang, J. Theory and Applications of the GPS System; Beijing Surveying and drawing Press: Beijing, China, 1993. (In Chinese) [Google Scholar]
- Montenbruck, O.; Steigenberger, P.; Hauschild, A. Broadcast versus precise ephemeris—a multi-GNSS perspective. Gps Solut.
**2015**, 19, 321–333. [Google Scholar] [CrossRef] - Schutz, B.; Tapley, B.; Born, G.H. Statistical Orbit Determination; Elsevier Academic Press: Burlington, MA, USA, 2004. [Google Scholar]

**Figure 1.**Line-of-sight (LOS) errors of the Global Positioning System (GPS)/BeiDou Navigation Satellite System (BDS) satellites: (

**a**) the LOS error curves of GPS (G01), geosynchronous Earth orbit (GEO) (C02), inclined geosynchronous Earth orbit (IGSO) (C10) and medium Earth orbit (MEO) (C14) satellites of BDS on DOY 71, 2015; (

**b**) the LOS error curves of GPS (G02), GEO (C03), IGSO (C09) and MEO (C12) satellites of BDS on DOY 71, 2015; (

**c**) the LOS error curves of the 2nd tracking arc of G01, the 1st one of C02, the 3rd one of C10 and the 7th of C14; (

**d**) the LOS error curves of the 3rd tracking arc of G02, the 2nd one of C03, the 2nd one of C09 and the 4th of C12.

**Figure 2.**The real-time orbit accuracies for different ${\sigma}^{2}$ in the GPS-alone and GPS+BDSN solutions.

**Figure 3.**The real-time orbit accuracies for different ${\sigma}^{2}$ in the GPS+BDS and BDS-alone solutions.

**Figure 4.**Comparison of the original true and estimated LOS errors: (

**a**) the LOS errors caused by the broadcast ephemeris and estimated in pseudo-ambiguity on DOY 71, 2015; (

**b**) the original and estimated LOS errors in a tracking arc.

**Figure 6.**Number of tracked GPS/BDS satellites: (

**a**) BDS-alone; (

**b**) GPS-alone; (

**c**) GPS and BDS IGSO/MEO fusion; (

**d**) GPS and BDS fusion.

**Figure 7.**Orbit accuracies comparison for different GPS/BDS fusion solutions: (

**a**) orbit accuracy statistics at DOY 69–75, 2015; (

**b**) orbit accuracy statistics at DOY 33─38, 2018; (

**c**) orbit accuracy statistics at DOY 275–300, 2013.

**Figure 8.**Position errors at the convergent stage for different GPS/BDS fusion solutions on DOY 69, 2015.

**Figure 9.**Position errors of GPS-alone and GPS/BDS fusion solutions and the number of tracked global navigation satellite system (GNSS) satellites on DOY 297, 2013.

**Table 1.**The statistics of the change rate $\upsilon $ and power spectral density ${\sigma}^{2}$ on DOY 69–75, 2015.

Satellite | $\mathit{\upsilon}$ (m/s) | ${\mathit{\sigma}}^{\mathbf{2}}$ (m^{2}/s) |
---|---|---|

GPS | 0.958 × 10^{−3} | 2.753 × 10^{−5} |

BDS | 1.720 × 10^{−3} | 8.875 × 10^{−5} |

BDS GEO | 1.986 × 10^{−3} | 1.183 × 10^{−4} |

BDS IGSO | 1.472 × 10^{−3} | 6.500 × 10^{−5} |

BDS MEO | 1.263 × 10^{−3} | 4.786 × 10^{−5} |

BDS IGSO/MEO | 1.373 × 10^{−3} | 5.655 × 10^{−5} |

Mark | $\mathit{\upsilon}$ (m/s) | ${\mathit{\sigma}}^{\mathbf{2}}$ (m^{2}/s) | Mark | $\mathit{\upsilon}$ (m/s) | ${\mathit{\sigma}}^{2}$ (m^{2}/s) |
---|---|---|---|---|---|

S01 | 1.0 × 10^{−5} | 3.000 × 10^{−9} | S11 | 5.0 × 10^{−3} | 7.500 × 10^{−4} |

S02 | 2.5 × 10^{−5} | 1.875 × 10^{−8} | S12 | 7.5 × 10^{−3} | 1.688 × 10^{−3} |

S03 | 5.0 × 10^{−5} | 7.500 × 10^{−8} | S13 | 1.0 × 10^{−2} | 3.000 × 10^{−3} |

S04 | 7.5 × 10^{−5} | 1.688 × 10^{−7} | S14 | 2.5 × 10^{−2} | 1.875 × 10^{−2} |

S05 | 1.0 × 10^{−4} | 3.000 × 10^{−7} | S15 | 5.0 × 10^{−2} | 7.500 × 10^{−2} |

S06 | 2.5 × 10^{−4} | 1.875 × 10^{−6} | S16 | 7.5 × 10^{−2} | 1.688 × 10^{−1} |

S07 | 5.0 × 10^{−4} | 7.500 × 10^{−6} | S17 | 1.0 × 10^{−1} | 3.000 × 10^{−1} |

S08 | 7.5 × 10^{−4} | 1.688 × 10^{−5} | S18 | 2.5 × 10^{−1} | 1.875 × 10^{0} |

S09 | 1.0 × 10^{−3} | 3.000 × 10^{−5} | S19 | 5.0 × 10^{−1} | 7.500 × 10^{0} |

S10 | 2.5 × 10^{−3} | 1.875 × 10^{−4} | S20 | 7.5 × 10^{−1} | 1.688 × 10^{1} |

Model | Relevant Setting |
---|---|

Measurement model | |

GNSS data | Dual-frequency GPS/BDS carrier-phase and pseudo-range measurements (interval 30s) |

GNSS orbit and clock | Broadcast ephemeris |

Receiver clock | A receiver clock offset and a system bias between GPS and BDS |

Ambiguity | Pseudo-ambiguity with a random walk process |

Dynamical model | |

Earth gravity field | EGM 2008 (45 × 45), neglect the time-varying part |

N-body gravitation | Moon and Sun only, low-precision analytic method (position) |

Earth and pole tide | Low-precision model, ${k}_{20}$ solid only |

Ocean and pole tide | Neglected |

Relativistic effects | Neglected |

Atmosphere drag | Modified Harris–Priester model (density), fixed effective area, drag coefficient with a random walk process |

Solar radiation pressure | Cannonball model, fixed effective area, radiation pressure coefficient with a random walk process |

Earth radiation pressure | Neglected |

Empirical acceleration | Three empirical accelerations in radial, along-track and cross with a first-order Gauss–Markov model |

Reference frame | |

Coordinate system | WGS84/CGCS2000 |

Precession and nutation | IAU1976/IAU 1980 simplified model |

Earth rotation parameter | Rapid predicted EOP in IERS Bulletin A |

Solutions | ${\mathit{\sigma}}_{\mathit{G}}^{\mathbf{2}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{g}}^{\mathbf{2}}$ | ${\mathit{\sigma}}_{\mathit{C}\mathit{i}\mathit{m}}^{\mathbf{2}}$ | Orbit Accuracy (m) | |||
---|---|---|---|---|---|---|---|

R | A | C | 3D | ||||

GPS-alone | S11 | - | - | 0.122 | 0.337 | 0.130 | 0.381 |

GPS+BDSN | S11 | - | S11 | 0.111 | 0.309 | 0.119 | 0.349 |

GPS+BDS | S11 | S13 | S11 | 0.110 | 0.308 | 0.118 | 0.348 |

BDS-alone | - | S12 | S09 | 0.264 | 0.771 | 0.379 | 0.899 |

- | S13 | S11 | 0.342 | 0.998 | 0.491 | 1.164 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gong, X.; Sang, J.; Wang, F.; Li, X.
LEO Onboard Real-Time Orbit Determination Using GPS/BDS Data with an Optimal Stochastic Model. *Remote Sens.* **2020**, *12*, 3458.
https://doi.org/10.3390/rs12203458

**AMA Style**

Gong X, Sang J, Wang F, Li X.
LEO Onboard Real-Time Orbit Determination Using GPS/BDS Data with an Optimal Stochastic Model. *Remote Sensing*. 2020; 12(20):3458.
https://doi.org/10.3390/rs12203458

**Chicago/Turabian Style**

Gong, Xuewen, Jizhang Sang, Fuhong Wang, and Xingxing Li.
2020. "LEO Onboard Real-Time Orbit Determination Using GPS/BDS Data with an Optimal Stochastic Model" *Remote Sensing* 12, no. 20: 3458.
https://doi.org/10.3390/rs12203458