# 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

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**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