# A Study on Graph Optimization Method for GNSS/IMU Integrated Navigation System Based on Virtual Constraints

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## Abstract

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## 1. Introduction

- A method to establish virtual pseudoranges from previous satellite positions is proposed, and a corresponding GO model is developed when these virtual constraints are injected.
- A comparative analysis between the proposed model and GNSS/IMU and SLAM models is conducted. Additionally, the differences in the marginalization processes among these models are explored.
- Real-world GNSS/IMU data were used to conduct performance tests in both instantaneous and continuous scenarios. The pseudoranges were artificially replaced with virtual constraints created using the proposed method. These results were subsequently compared against the original GO results to demonstrate the feasibility and effectiveness of the method.

## 2. Methodology

#### 2.1. Virtual Pseudorange Establishment

#### 2.2. Graph Model Construction with Virtual Constraints

#### 2.2.1. IMU Factor

#### 2.2.2. GNSS Pseudorange Factor

#### 2.3. Graph Model Analysis Akin to SLAM Model

#### 2.4. Marginalization

## 3. Experiments and Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ai, Q.; Zhang, B.; Yuan, Y.; Xu, T.; Chen, Y.; Tan, B. Evaluation and mitigation of the influence of pseudorange biases on GNSS satellite clock offset estimation. Measurement
**2022**, 193, 111015. [Google Scholar] - Wen, W.; Hsu, L.T. Towards robust GNSS positioning and real-time kinematic using factor graph optimization. In Proceedings of the 2021 IEEE International Conference on Robotics and Automation (ICRA), Xi’an China, 30 May–5 June 2021; pp. 5884–5890. [Google Scholar]
- Zhang, X.; Zhang, Y.; Zhu, F. Factor Graph Optimization Method of GNSS positioning in complex urban Scenarios and its resistance Analysis. Geomat. Inf. Sci. Wuhan Univ.
**2023**, 48, 1050–1057. [Google Scholar] [CrossRef] - Tian, Y.; Liu, F.; Liu, H.; Liu, Y.; Suwoyo, H.; Tao, J.; Long, L.; Wang, J. A Real-Time and Fast LiDAR–IMU–GNSS SLAM System with Point Cloud Semantic Graph Descriptor Loop-Closure Detection. Adv. Intell. Syst.
**2023**, 5, 2300138. [Google Scholar] [CrossRef] - Navisa, C.S.; Cahyadi, M.N.; Asfihani, T. Analysis of GNSS and IMU Sensor Data Fusion Using the Unscented Kalman Filter Method on Medical Drones in Open Air. IOP Conf. Ser. Earth Environ. Sci.
**2023**, 1250, 012019. [Google Scholar] [CrossRef] - Kaczmarek, A.; Rohm, W.; Klingbeil, L.; Janusz, T. Experimental 2D extended Kalman filter sensor fusion for low-cost GNSS/IMU/Odometers precise positioning system. Measurement
**2022**, 193, 110963. [Google Scholar] [CrossRef] - McGrath, T.; Stirling, L. Body-Worn IMU Human Skeletal Pose Estimation Using a Factor Graph-Based Optimization Framework. Sensors
**2020**, 20, 6887. [Google Scholar] [CrossRef] [PubMed] - Chang, L.; Niu, X.; Liu, T.; Tang, J.; Qian, C. GNSS/INS/LiDAR-SLAM integrated navigation system based on graph optimization. Remote Sens.
**2019**, 11, 1009. [Google Scholar] [CrossRef] - Wen, W.; Pfeifer, T.; Bai, X.; Tang, J.; Qian, C. Factor graph optimization for GNSS/INS integration: A comparison with the extended Kalman filter. Navig. J. Inst. Navig.
**2021**, 68, 315–331. [Google Scholar] [CrossRef] - Suzuki, T. Time-relative RTK-GNSS: GNSS loop closure in pose graph optimization. IEEE Robot. Autom. Lett.
**2020**, 5, 4735–4742. [Google Scholar] [CrossRef] - Zhang, G.; Ng, H.F.; Wen, W.; Hsu, L.T. 3D mapping database aided GNSS based collaborative positioning using factor graph optimization. IEEE Trans. Intell. Transp. Syst.
**2020**, 22, 6175–6187. [Google Scholar] [CrossRef] - Li, W.; Cui, X.; Lu, M. A robust graph optimization realization of tightly coupled GNSS/INS integrated navigation system for urban vehicles. Tsinghua Sci. Technol.
**2018**, 23, 724–732. [Google Scholar] [CrossRef] - Wen, W.; Zhang, G.; Hsu, L.T. GNSS outlier mitigation via graduated non-convexity factor graph optimization. IEEE Trans. Veh. Technol.
**2021**, 71, 297–310. [Google Scholar] [CrossRef] - Jiang, C.; Chen, Y.; Xu, B.; Jia, J.; Sun, H.; Chen, C.; Duan, Z.; Bo, Y.; Hyyppä, J. Vector tracking based on factor graph optimization for GNSS NLOS bias estimation and correction. IEEE Internet Things J.
**2022**, 9, 16209–16221. [Google Scholar] [CrossRef] - Das, A.; Elfring, J.; Dubbelman, G. Real-time vehicle positioning and mapping using graph optimization. Sensors
**2021**, 21, 2815. [Google Scholar] [CrossRef] [PubMed] - Fu, W.; Hu, S.; Luo, C.; Li, Y.; Guo, S.; Zhang, J. Development and Test of GNSS/IMU-Based Speed Measurement Device for Agricultural Machinery. In Proceedings of the International Conference on Intelligent Agriculture 2017 (ICIA2017) Part I, Macau, China, 18–20 July 2017; pp. 455–466. [Google Scholar]
- Jiang, C.; Chen, S.; Chen, Y.; Liu, D.; Bo, Y. GNSS vector tracking method using graph optimization. IEEE Trans. Circuits Syst. II Express Briefs
**2020**, 68, 1313–1317. [Google Scholar] [CrossRef] - Wen, W.; Kan, Y.C.; Hsu, L.T. Performance comparison of GNSS/INS integrations based on EKF and factor graph optimization. In Proceedings of the 32nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2019), Miami, FL, USA, 16–20 September 2019; pp. 3019–3032. [Google Scholar]
- Kaess, M.; Ranganathan, A.; Dellaert, F. iSAM: Incremental smoothing and mapping. IEEE Trans. Robot.
**2008**, 24, 1365–1378. [Google Scholar] [CrossRef] - Dellaert, F. Factor graphs and GTSAM: A hands-on introduction. Ga. Inst. Technol. Tech. Rep.
**2012**, 2, 4. [Google Scholar] - Groves, P.D. Principles of GNSS, inertial, and multisensor integrated navigation systems. IEEE Aerosp. Electron. Syst. Mag.
**2015**, 30, 26–27. Available online: https://github.com/benzenemo/TightlyCoupledINSGNSS (accessed on 14 December 2018). [CrossRef]

Hardware Parameters | Value |
---|---|

GNSS Signal Frequency (Hz) | 1 |

IMU Frequency (Hz) | 125 |

Gyroscope Bias (rad/s) | 0.0005 |

$Accelerometer\mathrm{Bias}(\mathsf{\mu}\mathrm{g}$) | 80 |

Carrier-to-Noise Ratio Threshold | 30 |

Operation Time (s) | 292 |

Methods | X-Direction Error | Y-Direction Error | Z-Direction Error |
---|---|---|---|

Kalman filter | 2.83 m | 4.16 m | 0.61 m |

GO | 1.96 m | 2.64 m | 0.45 m |

Error reduction | 30.1% | 36.5% | 25.8% |

Methods | X-Direction Error | Y-Direction Error | Z-Direction Error |
---|---|---|---|

VC GO | 1.99 m | 2.68 m | 0.48 m |

GO | 1.96 m | 2.64 m | 0.45 m |

Error variation rate | 1.667% | 1.623% | 4.891% |

**Table 4.**The RMSE in three directions for VC GO and GO compared with the reference trajectory over 5 and 10 s in a continuous performance test scenario.

Period | Methods | X-Direction Error | Y-Direction Error | Z-Direction Error |
---|---|---|---|---|

5 s | VC GO | 2.45 m | 2.99 m | 0.53 m |

GO | 3.10 m | 3.85 m | 0.55 m | |

Error reduction | 21.02% | 22.32% | 3.2% | |

10 s | VC GO | 3.11 m | 3.54 m | 0.69 m |

GO | 4.43 m | 5.15 m | 0.74 m | |

Error reduction | 29.73% | 31.26% | 6.92% |

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**MDPI and ACS Style**

Qiu, H.; Zhao, Y.; Wang, H.; Wang, L.
A Study on Graph Optimization Method for GNSS/IMU Integrated Navigation System Based on Virtual Constraints. *Sensors* **2024**, *24*, 4419.
https://doi.org/10.3390/s24134419

**AMA Style**

Qiu H, Zhao Y, Wang H, Wang L.
A Study on Graph Optimization Method for GNSS/IMU Integrated Navigation System Based on Virtual Constraints. *Sensors*. 2024; 24(13):4419.
https://doi.org/10.3390/s24134419

**Chicago/Turabian Style**

Qiu, Haiyang, Yun Zhao, Hui Wang, and Lei Wang.
2024. "A Study on Graph Optimization Method for GNSS/IMU Integrated Navigation System Based on Virtual Constraints" *Sensors* 24, no. 13: 4419.
https://doi.org/10.3390/s24134419