# GNSS Precise Kinematic Positioning for Multiple Kinematic Stations Based on A Priori Distance Constraints

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Kinematic Positioning Based on A Priori Distance Constraints

#### 2.1. Classic Kalman Filter

#### 2.2. The Distance between Two Kinematic GNSS Antennas

_{1}and k

_{2}are the kinematic stations (antennas) and (x

_{i}, y

_{i}, z

_{i}) their respective position vectors at the epoch i. The precision of this distance was determined from the error estimates of the measurements performed in order to determine it.

#### 2.3. The A Priori Distance Constraints

## 3. Experiment and Analysis

^{−12}m

^{2}/s. The justifications for the selection of these values were the slow motion of the ship and the small changes in the height profile [9,23]. The two-way Kalman filter [24] was used for the parameter estimation, and the selected data containing GPS and GLONASS observations with a sampling rate of 1 Hz were used for the calculation of the trajectories for the multiple kinematic stations KIN1 and KIN3.

**Scheme 1**(Scenario for the nearby-located reference stations without distance constraints): The trajectories of the multiple kinematic stations KIN1 and KIN3 were calculated, where 0801 and 0775 served as multiple nearby-located reference stations.

**Scheme 2**(Scenario with the far-away reference stations without distance constraints): The trajectories of the multiple kinematic stations KIN1 and KIN3 were calculated, where WARN and POTS served as multiple reference stations which are located far away from the kinematic stations.

**Scheme 3:**Based on Scheme 2, the a priori distance constraint was applied when calculating the trajectories of KIN1 and KIN3.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The ship used in the Baltic Sea gravimetric campaign and the positions of the three Global Navigation Satellite System (GNSS) receiving antennas.

**Figure 3.**The trajectory of the ship (blue curve) and the positions of all reference stations (blue triangles) of the Baltic Sea shipborne gravimetric campaign on 18 June 2013.

**Figure 4.**Apparent distance (as a function of time) between two kinematic antennas KIN1 and KIN3 without applying distance constraints; the trajectories of these antennas were estimated by using two reference stations 0801 and 0775 (Scheme 1).

**Figure 5.**Apparent distance (as a function of time) between the two antennas KIN1 and KIN3 without distance constraints; the trajectories of these antennas were estimated by using two far-away-located reference stations, WARN and POTS (Scheme 2).

**Figure 6.**Differences between the trajectories of KIN1 for the scenario with far-away-located reference stations (Scheme 2) and those obtained for Scheme 1.

**Figure 7.**Total number of the visible satellites (GPS + GLONASS) during the measurement time span for this study.

**Figure 8.**Differences between the KIN1 positioning results obtained from Scheme 3 (far-away-located reference stations and distance constraints) and those of Scheme 1.

Station Name | Receiver Type | Antenna Type | With Radome |
---|---|---|---|

KIN1 | JAVAD TRE_G3TH DELTA | LEIAS10 | NONE |

KIN3 | JAVAD TRE_G3TH DELTA | ACCG5ANT_42AT1 | NONE |

0801 | TPS NET-G3A | TPSCR.G3 | TPSH |

0775 | TPS NET-G3A | TPSCR.G3 | TPSH |

WARN | JPS LEGACY | LEIAR25.R3 | LEIT |

POTS | JAVAD TRE_G3TH DELTA | JAV_RINGANT_G3T | NONE |

Scheme | Reference Stations | Min | Max | Mean | STD |
---|---|---|---|---|---|

1 | Nearby | 26.282 | 26.406 | 26.342 | 0.015 |

2 | Far away | 26.261 | 26.428 | 26.338 | 0.022 |

**Table 3.**Statistics for the differences between the positioning results for KIN1 obtained from Scheme 2 resp. 3 , and the corresponding results from Scheme 1 (Unit: mm).

Scheme | Direction | Min | Max | Mean | RMS |
---|---|---|---|---|---|

2 vs. 1 | North | –27.0 | 23.0 | –6.4 | 5.8 |

East | –23.5 | 14.8 | –6.4 | 4.5 | |

Up | –165.7 | 165.9 | 3.5 | 36.8 | |

3 vs. 1 | North | –27.7 | 15.8 | –6.4 | 5.8 |

East | –21.6 | 16.4 | –6.6 | 4.2 | |

Up | –161.6 | 145.4 | 0.0 | 33.1 |

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**MDPI and ACS Style**

He, K.; Xu, T.; Förste, C.; Petrovic, S.; Barthelmes, F.; Jiang, N.; Flechtner, F.
GNSS Precise Kinematic Positioning for Multiple Kinematic Stations Based on *A Priori* Distance Constraints. *Sensors* **2016**, *16*, 470.
https://doi.org/10.3390/s16040470

**AMA Style**

He K, Xu T, Förste C, Petrovic S, Barthelmes F, Jiang N, Flechtner F.
GNSS Precise Kinematic Positioning for Multiple Kinematic Stations Based on *A Priori* Distance Constraints. *Sensors*. 2016; 16(4):470.
https://doi.org/10.3390/s16040470

**Chicago/Turabian Style**

He, Kaifei, Tianhe Xu, Christoph Förste, Svetozar Petrovic, Franz Barthelmes, Nan Jiang, and Frank Flechtner.
2016. "GNSS Precise Kinematic Positioning for Multiple Kinematic Stations Based on *A Priori* Distance Constraints" *Sensors* 16, no. 4: 470.
https://doi.org/10.3390/s16040470