# Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Combined GPS and BDS Observation Model

#### 2.1. Undifferenced Observation Model

#### 2.2. Single-Differenced Observation Model

#### 2.3. Intra-System Double-Differenced Observation Model

#### 2.4. Inter-System Double-Differenced Observation Model

## 3. F-ISPB Estimation by Improved Particle Swarm Optimization Algorithm

#### 3.1. Relationship between Ratio and F-ISPB

#### 3.2. Standard Particle Swarm Optimization Algorithm

Algorithm 1 standard PSO to estimate F-ISPB |

1: Calculate $(\mathsf{\Delta}\nabla {N}_{ij}^{G1Ci}+ISPB)$ by the Formula (6).2: Initialize 10 particles in search space.3: Use the value of each particle to correct the value of $(\mathsf{\Delta}\nabla {N}_{ij}^{G1Ci}+ISPB)$.4: Use the LAMBDA method to fix the corrected $(\mathsf{\Delta}\nabla {N}_{ij}^{G1Ci}+ISPB)$ values.5: Calculate the value of ratio corresponding to each particle.6: Record pbest and gbest.7: For 1:10Use Formula (7) to update the state of the particles. Repeat steps 3, 4, 5, and 6. End for8: Output: gbest corresponds to F-ISPB. |

#### 3.3. Improved Particle Swarm Optimization Algorithm

Algorithm 2 improved particle swarm optimization to estimate F-ISPB |

1: Calculate $(\mathsf{\Delta}\nabla {N}_{ij}^{G1Ci}+ISPB)$ by the Formula (6).2: Initialize 10 particles in search space.3: If epoch>1particle 1= elite particle. End if4: Use the value of each particle to correct the value of $(\mathsf{\Delta}\nabla {N}_{ij}^{G1Ci}+ISPB)$.5: Use the LAMBDA method to fix the corrected $(\mathsf{\Delta}\nabla {N}_{ij}^{G1Ci}+ISPB)$ values.6: Calculate the value of ratio corresponding to each particle.7: Record pbest and gbest.8: For 1:10Use Formula (7) to update the state of the particles. Repeat steps 4, 5, 6, 7. End for9: If gbest in the bounds of the search space.Transform search space; Repeat steps 2, 4, 5, 6, 7, 8. End if10: Output: gbest corresponds to F-ISPB. |

## 4. Experiments of Short-Baseline Positioning

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relationship between ratio and F-ISPB for first epoch. (

**a**,

**c**) The baseline CUTB–CUTC. (

**b**,

**d**) The baseline CUTB–CUT0.

**Figure 3.**Number of GPS and BDS satellites at different cut-off elevation angles. (

**a**) 15 degrees. (

**b**) 25 degrees. (

**c**) 35 degrees. (

**d**) 45 degrees.

**Figure 4.**F-ISPB estimation results of CUTB–CUT0 baseline. (

**a**) Standard PSO. (

**b**) PSO for adaptive search space. (

**c**) PSO for elite retention strategy. (

**d**) Improved PSO with both functions.

**Figure 5.**The distribution of ratio values at different cut-off elevation angles. (

**a**) 15 degrees. (

**b**) 25 degrees. (

**c**) 35 degrees. (

**d**) 45 degrees.

**Figure 6.**Ambiguity success rate and average of ratio at different cut-off elevation angles. (

**a**) Ambiguity success rate. (

**b**) Average of ratio.

Baseline | Location | Sampling Interval | Receivers (Modules) | Duration | Length/m |
---|---|---|---|---|---|

CUTB–CUT0 | Perth, Australia | 30 s | Trimble NETR9 | 24 h | 4.27 |

CUTB–CUTC | Perth, Australia | 30 s | Trimble NETR9 | 24 h | 6.15 |

IGG01–IGG02 | Wuhan, China | 30 s | Trimble NETR9 | 60 h | 60.5 |

**Table 2.**Percentage of epochs with positioning errors less than 1 cm and 5 cm at different cut-off elevation angles.

Cut-Off Elevation Angle | Methods | <1 cm (%) | <5 cm (%) | ||||
---|---|---|---|---|---|---|---|

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

15 | Improved PSO | 100 | 100 | 95.66 | 100 | 100 | 100 |

Intra-system model | 100 | 100 | 95.45 | 100 | 100 | 100 | |

Standard PSO | 99.55 | 97.64 | 90.00 | 99.62 | 99.44 | 99.44 | |

25 | Improved PSO | 99.97 | 99.97 | 93.02 | 99.97 | 99.97 | 99.97 |

Intra-system model | 99.90 | 99.90 | 92.47 | 99.90 | 99.90 | 99.90 | |

Standard PSO | 98.47 | 96.56 | 86.32 | 98.65 | 98.72 | 98.47 | |

35 | Improved PSO | 95.14 | 98.23 | 78.61 | 95.17 | 98.51 | 98.51 |

Intra-system model | 92.05 | 96.56 | 78.78 | 92.15 | 96.84 | 97.05 | |

Standard PSO | 87.29 | 89.69 | 70.87 | 87.60 | 92.05 | 92.50 | |

45 | Improved PSO | 78.37 | 80.01 | 58.02 | 79.90 | 81.42 | 77.43 |

Intra-system model | 73.07 | 76.15 | 57.01 | 74.34 | 77.12 | 73.89 | |

Standard PSO | 66.74 | 69.31 | 53.75 | 68.33 | 71.60 | 67.53 |

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**MDPI and ACS Style**

Zhao, W.; Liu, G.; Wang, S.; Gao, M.; Lv, D.
Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm. *Remote Sens.* **2021**, *13*, 3214.
https://doi.org/10.3390/rs13163214

**AMA Style**

Zhao W, Liu G, Wang S, Gao M, Lv D.
Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm. *Remote Sensing*. 2021; 13(16):3214.
https://doi.org/10.3390/rs13163214

**Chicago/Turabian Style**

Zhao, Wenhao, Genyou Liu, Shengliang Wang, Ming Gao, and Dong Lv.
2021. "Real-Time Estimation of GPS-BDS Inter-System Biases: An Improved Particle Swarm Optimization Algorithm" *Remote Sensing* 13, no. 16: 3214.
https://doi.org/10.3390/rs13163214