# Impact of GPS/BDS Satellite Attitude Quaternions on Precise Point Positioning with Ambiguity Resolution

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## Abstract

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## 1. Introduction

## 2. PPP Model at Network and User Ends

## 3. Data Processing

## 4. Impact on Integer Clock Products

## 5. Daily Kinematic Solutions of PPP-AR

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Distribution of used IGS stations. 184 stations (red dots) are used for integer clock products computation and 490 stations (blue dots) are used for PPP-AR experiments.

**Figure 2.**Time series of double-difference satellite clocks for satellites G08, G12, G27, G32, C14, C16, C30 and C34 in 2020. The reference satellites are G01 and C06 for GPS and BDS satellites, respectively. The red dots are the double-difference satellite clocks computed using the clocks based on nominal attitudes and attitude quaternions. The gray zones denote the eclipsing periods of corresponding satellites. The RMS error at the top part of a panel is for the eclipsing period. The average sun elevation angle for corresponding day is shown at the bottom part of a panel. Note that the different panels of the figure have different scales.

**Figure 3.**Distribution of GPS, BDS-2 and BDS-3 narrow-lane ambiguity residuals after correcting integer clock products from days 081 to 110 of 2020. The standard deviations of all residuals and the percentages of all within ±0.1 cycles are plotted at the top left and top right corners of each panel, respectively. The blue bar graphs are residuals after correcting integer clock products including the influence of the attitude quaternions; while the red bar graphs are those without the influence of the attitude quaternions or with the influence of nominal attitude.

**Figure 4.**Carrier-phase and pseudorange residuals for different satellite types. Red, cyan, green and blue dots denote the residuals based on strategies (a), (b), (c) and (d), respectively. The gray zones denote the eclipsing periods of corresponding satellites.

**Figure 5.**Narrow-lane ambiguity fixing rates (%) for daily kinematic solutions corresponding to four strategies on day 081 to 110, 2020.

**Figure 6.**Ambiguity-fixed mean standard deviations of position errors comparison in the east direction between the daily kinematic PPP-AR solutions based on strategies (a), (b), (c) and (d). For I-VI panels, each panel denotes the comparison of two strategies. Each red dot denotes a station. The standard deviations along the horizontal and vertical axes are the mean values of all stations.

**Figure 7.**Ambiguity-fixed mean standard deviations of position errors comparison in the north direction between the daily kinematic PPP-AR solutions based on strategies (a), (b), (c) and (d). For I-VI panels, each panel denotes the comparison of two strategies.

**Figure 8.**Ambiguity-fixed mean standard deviations of position errors comparison in the up direction between the daily kinematic PPP-AR solutions based on strategies (a), (b), (c) and (d). For I-VI panels, each panel denotes the comparison of two strategies.

Items | Strategies |
---|---|

Observables | Ionosphere-free combination |

Cut-off elevation | 7° |

Sampling rate | 30 s |

Weighting | Elevation-dependent for satellites;Equally weighted for GNSS |

Solid Earth tide, ocean tidal loading, pole tide | IERS conventions 2010 [34] |

Nadir-dependent pseudorange biases of BDS-2 | Corrected [35] |

Phase center offsets/variations | igs14.atx |

Phase wind-up | Corrected [36] |

Satellite attitude | Nominal attitude: Corrected by nominal attitude modelAttitude quaternions: WUM rapid products |

Earth rotation parameters | WUM rapid products |

Precise satellite orbits | WUM rapid products |

Precise satellite clocks | WUM rapid products |

Position coordinates | estimated as white-noise like parameter at each epoch |

Receiver clocks | estimated as white-noise like parameter per epoch for each GNSS system |

Zenith tropospheric delays | Piece-wise constants per hour |

Horizontal tropospheric gradients | Piece-wise constants per 12 h |

Ambiguities | Real-valued constants for each continuous arc |

Network End | User End | |
---|---|---|

Strategy (a) | Using nominal attitude | Using nominal attitude |

Strategy (b) | Using attitude quaternions | Using nominal attitude |

Strategy (c) | Using nominal attitude | Using attitude quaternions |

Strategy (d) | Using attitude quaternions | Using attitude quaternions |

**Table 3.**Mean standard deviations (mm) of position errors for kinematic solutions on day 081 to 110, 2020.

Strategy Names | Float Solutions in Different Directions (mm) | Fixed (mm) | Improvement (%) |
---|---|---|---|

E/N/U | E/N/U | E/N/U | |

Strategy (a) | 11.0/9.7/25.9 | 8.2/8.7/24.1 | 25.5/10.3/6.9 |

Strategy (b) | 13.4/11.1/29.4 | 10.5/10.0/28.0 | 21.6/9.9/4.8 |

Strategy (c) | 12.2/10.2/27.3 | 9.0/9.0/25.5 | 26.2/11.8/6.6 |

Strategy (d) | 10.3/9.3/24.6 | 7.8/8.4/23.2 | 24.3/9.7/5.7 |

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**MDPI and ACS Style**

Yang, S.; Zhang, Q.; Zhang, X.; Liu, D.
Impact of GPS/BDS Satellite Attitude Quaternions on Precise Point Positioning with Ambiguity Resolution. *Remote Sens.* **2021**, *13*, 3035.
https://doi.org/10.3390/rs13153035

**AMA Style**

Yang S, Zhang Q, Zhang X, Liu D.
Impact of GPS/BDS Satellite Attitude Quaternions on Precise Point Positioning with Ambiguity Resolution. *Remote Sensing*. 2021; 13(15):3035.
https://doi.org/10.3390/rs13153035

**Chicago/Turabian Style**

Yang, Songfeng, Qiyuan Zhang, Xi Zhang, and Donglie Liu.
2021. "Impact of GPS/BDS Satellite Attitude Quaternions on Precise Point Positioning with Ambiguity Resolution" *Remote Sensing* 13, no. 15: 3035.
https://doi.org/10.3390/rs13153035