# A Partial Carrier Phase Integer Ambiguity Fixing Algorithm for Combinatorial Optimization between Network RTK Reference Stations

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. GNSS Network RTK Data Solution Model

#### 2.2. Solving for Ambiguity

#### 2.3. Robust Extended Kalman Filter Algorithm

#### 2.4. Improving the Partial Ambiguity Fixed Solution

- Firstly, we try to fix all ambiguities, i.e., without rejection filtering of the ambiguity set, the search of the floating-point ambiguity parameters obtained by Kalman filtering using LAMBDA algorithm directly, and the search results are tested jointly by R-ratio and bootstrapping success rate indicators, and if they pass the test, the fixation is considered successful; otherwise, we proceed to the next step.
- The ambiguity subsets are divided at the satellite level, and the total set of satellite azimuths (0°~360°) is divided into one subset as long as it is over 90°, and subsets are denoted as ${Q}_{1}$, ${Q}_{2}$, ${Q}_{3}$, and ${Q}_{4}$ separately, and one satellite is removed from the divided subset to ensure that the GDOP value of the set is the smallest after removing a satellite. Then, the ambiguity of all remaining satellite sets ${H}_{1}$ is fixed.
- If the second step fails to be fixed, the set ${H}_{1}$ is then optimally filtered at the fuzzy degree level, and the three fuzzy degrees with larger ADOP values ${\alpha}_{max1}$, ${\alpha}_{max2}$, ${\alpha}_{max3}$ are selected from the subset. Then the GDOP1 value of the remaining satellite constellations can be calculated after eliminating the ${\alpha}_{max1}$ and ${\alpha}_{max2}$ ambiguities. The GDOP2 value of the remaining satellite constellations can be calculated after eliminating the ${\alpha}_{max1}$ and ${\alpha}_{max3}$ ambiguities, and the GDOP3 value of the remaining satellite constellations can be calculated after eliminating the ${\alpha}_{max2}$ and ${\alpha}_{max3}$ ambiguities. Comparing the size of the three GDOP values and eliminating the two ambiguities with large GDOP values from the set ${H}_{1}$ to obtain the optimal subset.
- The LAMBDA algorithm is employed to search the filtered optimal fuzzy degree subset and perform a joint R-ratio and bootstrapping success rate metric test, and if the fixation is successful, calculate the final fixed solution; if the fixation fails, the floating-point solution is saved.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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Related Parameters | Numerical Value |
---|---|

Elevation Threshold | 20 |

Consecutive epochs | 10 Epoch |

Signal-to-noise ratio | 35 dBHz |

GDOP | 2.0 |

Algorithm | FAR | PAR | N-PAR |
---|---|---|---|

successful epochs | 936 | 1983 | 273220 |

failed epochs | 1799 | 751 | 310 Epoch |

Fixed rate | 34.22% | 71.82% | 99.89 |

Algorithm | ΔE | ΔN | ΔU |
---|---|---|---|

FAR | 0.0601 | 0.0259 | 0.0915 |

PAR | 0.0459 | 0.0151 | 0.0493 |

N-PAR | 0.0060 | 0.0089 | 0.0152 |

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**MDPI and ACS Style**

Wang, S.; You, Z.; Sun, X.
A Partial Carrier Phase Integer Ambiguity Fixing Algorithm for Combinatorial Optimization between Network RTK Reference Stations. *Sensors* **2022**, *22*, 165.
https://doi.org/10.3390/s22010165

**AMA Style**

Wang S, You Z, Sun X.
A Partial Carrier Phase Integer Ambiguity Fixing Algorithm for Combinatorial Optimization between Network RTK Reference Stations. *Sensors*. 2022; 22(1):165.
https://doi.org/10.3390/s22010165

**Chicago/Turabian Style**

Wang, Shouhua, Zhiqi You, and Xiyan Sun.
2022. "A Partial Carrier Phase Integer Ambiguity Fixing Algorithm for Combinatorial Optimization between Network RTK Reference Stations" *Sensors* 22, no. 1: 165.
https://doi.org/10.3390/s22010165