# An In-Flight Alignment Method for Global Positioning System-Assisted Low Cost Strapdown Inertial Navigation System in Flight Body with Short-Endurance and High-Speed Rotation

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## Abstract

**:**

## 1. Introduction

## 2. Construction of Multi-Vector Alignment Model on Lie Group

#### 2.1. Mathematical Description of the Lie Group

#### 2.2. Alignment Principle of Multi-Vector Attitude Determination on Lie Group

## 3. Improved UKF Based on Lie Group

#### 3.1. Error Analysis

#### 3.2. Improved UKF Modeling on Lie Group

- 1)
- Initialization

- 2)
- Time Update

- (a)
- Predicting the model value at k + 1 time:$${\stackrel{\wedge}{\mathit{x}}}_{k|k-1}:=\left[{\stackrel{\wedge}{\mathit{\phi}}}_{k|k-1}^{b}{\stackrel{\wedge}{\mathit{B}}}_{k|k-1}\right],$$

Algorithm 1: Calculation of ${\stackrel{\wedge}{\mathit{\phi}}}_{k|k-1}^{b}$ on $SO(3)$. |

Input: Set of rotations $\left\{{\mathit{Z}}_{0},\dots ,{\mathit{Z}}_{12}\right\}$ in $SO(3)$ |

1.$\mathit{H}\leftarrow {\mathit{Z}}_{0}$ |

2.$\mathit{\Omega}\leftarrow {\displaystyle {\sum}_{i=0}^{12}{W}_{m}^{(i)}\mathrm{log}\left({\mathit{Z}}_{i}{\mathit{H}}^{-1}\right)}$ |

3.$\mathit{H}\leftarrow \mathrm{exp}\left(\mathit{\Omega}\right)\mathit{H}$ |

4. return $\mathit{H}$ |

- (b)
- Calculating the covariance matrix

- (c)
- Combining vectors ${\mathit{\eta}}_{i}^{\left(a\right)}$ and ${\mathit{\eta}}_{i}^{\left(b\right)}$ into vector ${\mathit{\eta}}_{i}=({\mathit{\eta}}_{i}^{\left(a\right)},{\mathit{\eta}}_{i}^{\left(b\right)})\in {\mathbf{R}}^{12}$. The predicted covariance is then calculated as:$${\mathit{P}}_{k|k-1}={\displaystyle \sum _{i=0}^{L+1}{W}_{i}^{(c)}{\mathit{\eta}}_{i}{\mathit{\eta}}_{i}^{\mathrm{T}}}+\mathit{j}({\mathit{x}}_{k-1}){\mathit{Q}}_{k-1}\mathit{j}{({\mathit{x}}_{k-1})}^{\mathrm{T}},$$

- 3)
- Measurement Update

- 4)
- Calculation of Auto-Covariance Matrix and Cross-Covariance Matrix

- 5)
- Filtering Update

- (a)
- Calculating filter gain matrix:$${\mathit{K}}_{k}={\mathit{P}}_{{\stackrel{\wedge}{x}}_{k}{\stackrel{\wedge}{z}}_{k}}{\mathit{P}}_{{\stackrel{\wedge}{z}}_{k}{\stackrel{\wedge}{z}}_{k}}^{-1},$$
- (b)
- Correcting the state prediction:$${\stackrel{\wedge}{\mathit{x}}}_{k}={\stackrel{\wedge}{\mathit{x}}}_{k|k-1}+{\mathit{K}}_{k}({\mathit{z}}_{k}-{\stackrel{\wedge}{\mathit{z}}}_{k|k-1}),$$$$\delta {\stackrel{\wedge}{\mathit{x}}}_{k}=\left[\delta {\stackrel{\wedge}{\mathit{\phi}}}_{k}^{b}\begin{array}{c},\delta {\stackrel{\wedge}{\mathit{B}}}_{k}\end{array}\right]={\mathit{K}}_{k}({\mathit{z}}_{k}-{\stackrel{\wedge}{\mathit{z}}}_{k|k-1}),$$$${\stackrel{\wedge}{\mathit{\phi}}}_{k}^{b}=\left[\mathrm{exp}(\delta {\stackrel{\wedge}{\mathit{\phi}}}_{k}^{b})\right]{\stackrel{\wedge}{\mathit{\phi}}}_{k|k-1}^{b},$$$${\stackrel{\wedge}{\mathit{B}}}_{k}=\delta {\stackrel{\wedge}{\mathit{B}}}_{k}+{\stackrel{\wedge}{\mathit{B}}}_{k|k-1},$$$${\stackrel{\wedge}{\mathit{x}}}_{k}=\left[{\stackrel{\wedge}{\mathit{\phi}}}_{k}^{b},{\stackrel{\wedge}{\mathit{B}}}_{k}\right],$$
- (c)
- Updating the covariance of the system:$${\mathit{P}}_{k}={\mathit{P}}_{k|k-1}-{\mathit{K}}_{k}{\mathit{P}}_{{\stackrel{\wedge}{z}}_{k}{\stackrel{\wedge}{z}}_{k}}{\mathit{K}}_{k}^{\mathrm{T}},$$

## 4. Simulation and Experimental Results

#### 4.1. Simulation Results

#### 4.2. Experimental Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Li, N.; Guan, L.; Gao, Y.; Du, S.; Wu, M.; Guang, X.; Cong, X. Indoor and Outdoor Low-Cost Seamless Integrated Navigation System Based on the Integration of INS/GNSS/LIDAR System. Remote Sens.
**2020**, 12, 3271. [Google Scholar] [CrossRef] - Li, T.; Zhang, H.; Gao, Z.; Niu, X.; El-sheimy, N. Tight Fusion of a Monocular Camera, MEMS-IMU, and Single-Frequency Multi-GNSS RTK for Precise Navigation in GNSS-Challenged Environments. Remote Sens.
**2019**, 11, 610. [Google Scholar] [CrossRef] [Green Version] - Sun, Y. Autonomous Integrity Monitoring for Relative Navigation of Multiple Unmanned Aerial Vehicles. Remote Sens.
**2021**, 13, 1483. [Google Scholar] [CrossRef] - Ge, B.; Zhang, H.; Fu, W.; Yang, J. Enhanced Redundant Measurement-Based Kalman Filter for Measurement Noise Covariance Estimation in INS/GNSS Integration. Remote Sens.
**2020**, 12, 3500. [Google Scholar] [CrossRef] - Cheng, Q.; Chen, P.; Sun, R.; Wang, J.; Mao, Y.; Ochieng, W.Y. A New Faulty GNSS Measurement Detection and Exclusion Algorithm for Urban Vehicle Positioning. Remote Sens.
**2021**, 13, 2117. [Google Scholar] [CrossRef] - Kim, Y.; An, J.; Lee, J. Robust Navigational System for a Transporter Using GPS/INS Fusion. IEEE Trans. Ind. Electron.
**2017**, 65, 3346–3354. [Google Scholar] [CrossRef] - Chang, L.; Qin, F.; Jiang, S. Strapdown Inertial Navigation System Initial Alignment Based on Modified Process Model. IEEE Sens. J.
**2019**, 19, 6381–6391. [Google Scholar] [CrossRef] - Dingjie, W.; Hanfeng, L.; Jie, W. In-flight initial alignment for small UAV MEMS-based navigation via adaptive unscented Kalman filtering approach. Aerosp. Sci. Technol.
**2017**, 61, 73–84. [Google Scholar] - Dmitriyev, S.P.; Stepanov, O.A.; Shepel, S.V. Nonlinear filtering methods application in INS alignment. IEEE Trans. Aerosp. Electron. Syst.
**1997**, 33, 260–272. [Google Scholar] [CrossRef] - Cao, S.; Guo, L. Multi-objective robust initial alignment algorithm for Inertial Navigation System with multiple disturbances. Aerosp. Sci. Technol.
**2012**, 21, 1–6. [Google Scholar] [CrossRef] - Li, J.; Xu, J.; Chang, L.; Zha, F. An Improved Optimal Method For Initial Alignment. J. Navig.
**2014**, 67, 727–736. [Google Scholar] [CrossRef] - Wu, J.; Zhou, Z.; Fourati, H.; Li, R.; Liu, M. Generalized Linear Quaternion Complementary Filter for Attitude Estimation From Multisensor Observations: An Optimization Approach. IEEE Trans. Autom. Sci. Eng.
**2019**, 16, 1330–1343. [Google Scholar] [CrossRef] - Liu, J.; Zhao, T. In-flight alignment method of navigation system based on microelectromechanical systems sensor measurement. Int. J. Distrib. Sens. Netw.
**2019**, 15, 155014771984492. [Google Scholar] [CrossRef] - Wu, M.; Wu, Y.; Hu, X.; Hu, D. Optimization-based alignment for inertial navigation systems: Theory and algorithm. Aerosp. Sci. Technol.
**2011**, 15, 1–17. [Google Scholar] [CrossRef] - Chang, L.; Li, J.; Li, K. Optimization-based Alignment for Strapdown Inertial Navigation System Comparison and Extension. IEEE Trans. Aero. Elec. Sys.
**2016**, 52, 1697–1713. [Google Scholar] [CrossRef] - Shuster, M.D. A Survey of Attitude Representations. J. Astronaut. Sci.
**1993**, 41, 439–517. [Google Scholar] - Pan, C.; Qian, N.; Li, Z.; Gao, J.; Liu, Z.; Shao, K. A Robust Adaptive Cubature Kalman Filter Based on SVD for Dual-Antenna GNSS/MIMU Tightly Coupled Integration. Remote Sens.
**2021**, 13, 1943. [Google Scholar] [CrossRef] - Lin, Y.; Miao, L.; Zhou, Z. An Improved MCMC-based Particle Filter for GPS-aided SINS In-motion Initial Alignment. IEEE Trans. Instrum. Meas.
**2020**, 69, 7895–7905. [Google Scholar] [CrossRef] - Shao, H.; Miao, L.; Gao, W.; Shen, J. Ensemble Particle Filter Based on KLD and Its Application to Initial Alignment of the SINS in Large Misalignment Angles. IEEE Trans. Ind. Electron.
**2018**, 65, 8946–8955. [Google Scholar] [CrossRef] - Atia, M.M.; Liu, S.; Nematallah, H.; Karamat, T.B.; Noureldin, A. Integrated Indoor Navigation System for Ground Vehicles With Automatic 3-D Alignment and Position Initialization. IEEE Trans. Veh. Technol.
**2015**, 64, 1279–1292. [Google Scholar] [CrossRef] - Zhang, P.; Zhao, Y.; Lin, H.; Zou, J.; Wang, X.; Yang, F. A Novel GNSS Attitude Determination Method Based on Primary Baseline Switching for A Multi-Antenna Platform. Remote Sens.
**2020**, 12, 747. [Google Scholar] [CrossRef] [Green Version] - Yang, Y.; Liu, X.; Zhang, W.; Liu, X.; Guo, Y. A Nonlinear Double Model for Multisensor-Integrated Navigation Using the Federated EKF Algorithm for Small UAVs. Sensors
**2020**, 20, 2974. [Google Scholar] [CrossRef] [PubMed] - Wang, M.; Wu, W.; Zhou, P.; He, X. State transformation extended Kalman filter for GPS/SINS tightly coupled integration. GPS Solut.
**2018**, 22, 112. [Google Scholar] [CrossRef] - Pei, F.; Zhu, D.; Yin, S. An In-Motion Initial Alignment Algorithm for SINS Using Lie Group Matrix Kalman Filter. In Proceedings of the 2019 Chinese Automation Congress (CAC), Hangzhou, China, 22–24 November 2019. [Google Scholar] [CrossRef]
- Yunfeng, W.; Chirikjian, G.S. Error propagation on the Euclidean group with applications to manipulator kinematics. IEEE Trans. Robot.
**2006**, 22, 591–602. [Google Scholar] [CrossRef] - Joukov, V.; Ćesić, J.; Westermann, K.; Marković, I.; Petrović, I.; Kulić, D. Estimation and Observability Analysis of Human Motion on Lie Groups. IEEE Trans. Cybern.
**2020**, 50, 1321–1332. [Google Scholar] [CrossRef] - Barrau, A.; Bonnabel, S. Intrinsic filtering on Lie groups with applications to attitude estimation. IEEE Trans. Automat. Contr.
**2014**, 60, 436–449. [Google Scholar] [CrossRef] [Green Version] - Zhang, C.; Taghvaei, A.; Mehta, P.G. Feedback Particle Filter on Riemannian Manifolds and Matrix Lie Groups. IEEE Trans. Automat. Contr.
**2018**, 63, 2465–2480. [Google Scholar] [CrossRef] - Celledoni, E.; Owren, B. Lie group methods for rigid body dynamics and time integration on manifolds. Comput. Methods Appl. Mech. Eng.
**2003**, 192, 421–438. [Google Scholar] [CrossRef] - Kang, D.; Jang, C.; Park, F.C. Unscented Kalman Filtering for Simultaneous Estimation of Attitude and Gyroscope Bias. IEEE/ASME Trans. Mech.
**2019**, 24, 350–360. [Google Scholar] [CrossRef] - Brossard, M.; Condomines, J.P. Unscented Kalman Filtering on Lie Groups. In Proceedings of the 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vancouver, BC, Canada, 24–28 September 2017. [Google Scholar]
- Phogat, K.S.; Chang, D.E. Invariant extended Kalman filter on matrix Lie groups. Automatica
**2020**, 114, 108812. [Google Scholar] [CrossRef] [Green Version] - Chang, L.; Li, J.; Chen, S. Initial Alignment by Attitude Estimation for Strapdown Inertial Navigation Systems. IEEE Trans. Instrum. Meas.
**2015**, 64, 784–794. [Google Scholar] [CrossRef] - Huang, Y.; Zhang, Z.; Du, S.; Li, Y.; Zhang, Y. A high-accuracy GPS-aided coarse alignment method for MEMS-based SINS. IEEE Trans. Instrum. Meas.
**2020**, 69, 7914–7932. [Google Scholar] [CrossRef] - Wu, Y.; Pan, X. Velocity/Position Integration Formula Part I: Application to In-Flight Coarse Alignment. IEEE Trans. Aero. Electron. Sys.
**2011**, 49, 1006–1023. [Google Scholar] [CrossRef] [Green Version] - Mortari, D. ESOQ-2 Single-Point Algorithm for Fast Optimal Spacecraft Attitude Determination. Am. Soc. Mech. Eng.
**1997**, 95, 817–826. [Google Scholar] - Markley, F.L. Attitude Determination Using Vector Observations and the Singular Value Decomposition. J. Astronaut. Sci.
**1987**, 38, 245–258. [Google Scholar] - Wu, Y.; Pan, X. Velocity/Position Integration Formula Part II: Application to Strapdown Inertial Navigation Computation. IEEE Trans. Aero. Elec. Sys.
**2013**, 49, 1024–1034. [Google Scholar] [CrossRef] - Wahba, G. A Least Squares Estimate of Satellite Attitude. Siam Rev.
**2006**, 7, 409. [Google Scholar] [CrossRef] - Mortari, D. Euler-q Algorithm for Attitude Determination from Vector Observations. J. Guid. Control. Dyn.
**1998**, 7, 328–334. [Google Scholar] [CrossRef] - Aboutaleb, A.; El-Wakeel, A.S.; Elghamrawy, H.; Noureldin, A. LiDAR/RISS/GNSS Dynamic Integration for Land Vehicle Robust Positioning in Challenging GNSS Environments. Remote Sens.
**2020**, 12, 2323. [Google Scholar] [CrossRef] - Julier, S.J. Unscented filtering and nonlinear estimation. Proc. IEEE
**2004**, 92, 401–422. [Google Scholar] [CrossRef] [Green Version] - Turner, D.; Lucieer, A.; Wallace, L. Direct Georeferencing of Ultrahigh-Resolution UAV Imagery. IEEE Trans. Geosci. Remote Sens.
**2014**, 5, 2738–2745. [Google Scholar] [CrossRef] - Wu, Q.; Li, K.; Song, T.X. The calibration for inner and outer lever-arm errors based on velocity differences of two RINSs. Mech. Syst. Signal Process.
**2021**, 160, 107868. [Google Scholar] [CrossRef]

**Figure 1.**Schematic representation of the mathematical relation between the Lie algebra and the Lie group.

**Figure 8.**Simulation results of pitch angle alignment with different alignment methods. (

**a**) Pitch angle alignment results of different alignment methods; (

**b**) absolute value of pitch alignment error of different alignment methods.

**Figure 9.**Simulation results of roll angle alignment with different alignment methods. (

**a**) Roll angle alignment results of different alignment methods; (

**b**) absolute value of roll alignment error of different alignment methods.

**Figure 10.**Simulation results of yaw angle alignment with different alignment methods. (

**a**) Yaw angle alignment results of different alignment methods; (

**b**) absolute value of yaw alignment error of different alignment methods.

**Figure 19.**Radar map for statistical data (RMSE) of alignment attitude errors and alignment time of three alignment methods.

**Figure 20.**Comparison of alignment results obtained from three alignment methods in 20-times repeated experiments. (

**a**) Violin plot for RMSE of pitch angle errors obtained from three alignment methods in 20-times field experiments; (

**b**) violin plot for RMSE of roll angle errors obtained from three alignment methods in 20-times repeated field experiments; (

**c**) violin plot for RMSE of yaw angle errors obtained from three alignment methods in 20-times repeated field experiments; (

**d**) Violin plot for alignment time of three alignment methods in 20-times repeated field experiments.

Device | Parameter | Value | Frequency |
---|---|---|---|

SINS | Gyro bias | 50°/h | 100 Hz |

Gyro random walk | $0.5\xb0/\sqrt{\mathrm{h}}$ | ||

Accelerometer bias | 5 mg | ||

GPS (CNS50) | Position accuracy | 2 m | 5 Hz |

Velocity accuracy | 0.2 m/s | ||

Time accuracy | 50 ns | ||

Reference system (NovAtel SPAN-LCI) | Position accuracy | 1 cm ± 1 ppm | 100 Hz |

Velocity accuracy | 0.03 m/s | ||

Time accuracy | 20 ns | ||

Vehicle turntable | Roll angle accuracy | 0.05° | - |

**Table 2.**Std and RMSE of alignment attitude errors of different IFA methods after convergence in the experiment.

Alignment Error | Method | Std | RMSE |
---|---|---|---|

Pitch angle error (°) | OBA | 0.04 | 0.52 |

EKF | 0.02 | 0.32 | |

Proposed method | 0.01 | 0.13 | |

Roll angle error (°) | OBA | 1.02 | 4.56 |

EKF | 0.56 | 2.20 | |

Proposed method | 0.08 | 1.16 | |

Yaw angle error (°) | OBA | 0.18 | 2.07 |

EKF | 0.09 | 0.90 | |

Proposed method | 0.02 | 0.23 |

Method | OBA | EKF | Proposed Method |
---|---|---|---|

Alignment time (s) | 33.79 | 29.56 | 13.71 |

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## Share and Cite

**MDPI and ACS Style**

Wei, X.; Li, J.; Han, D.; Wang, J.; Zhan, Y.; Wang, X.; Feng, K.
An In-Flight Alignment Method for Global Positioning System-Assisted Low Cost Strapdown Inertial Navigation System in Flight Body with Short-Endurance and High-Speed Rotation. *Remote Sens.* **2023**, *15*, 711.
https://doi.org/10.3390/rs15030711

**AMA Style**

Wei X, Li J, Han D, Wang J, Zhan Y, Wang X, Feng K.
An In-Flight Alignment Method for Global Positioning System-Assisted Low Cost Strapdown Inertial Navigation System in Flight Body with Short-Endurance and High-Speed Rotation. *Remote Sensing*. 2023; 15(3):711.
https://doi.org/10.3390/rs15030711

**Chicago/Turabian Style**

Wei, Xiaokai, Jie Li, Ding Han, Junlin Wang, Ying Zhan, Xin Wang, and Kaiqiang Feng.
2023. "An In-Flight Alignment Method for Global Positioning System-Assisted Low Cost Strapdown Inertial Navigation System in Flight Body with Short-Endurance and High-Speed Rotation" *Remote Sensing* 15, no. 3: 711.
https://doi.org/10.3390/rs15030711