# An Improved Alignment Method for the Strapdown Inertial Navigation System (SINS)

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## Abstract

**:**

## 1. Introduction

## 2. Reference Frames Definition

- $i$-frame: Earth-Centered Initially Fixed (ECIF) orthogonal reference frame.
- $e$-frame: Earth-Centered Earth-Fixed (ECEF) orthogonal reference frame.
- $t$-frame: Orthogonal reference frame aligned with East-North-Up (ENU) geographic frame.
- $b$-frame: body frame.
- $n$-frame: navigation frame.
- $p$-frame: Pseudo-Earth-Centered Earth-Fixed (PECEF) orthogonal reference frame obtained by two successive transformations from $e$-frame.
- ${t}_{p}$-frame: pseudo-orthogonal reference frame aligned with pseudo-East-North-Up $({E}_{p}{N}_{p}{U}_{p})$ pseudo-geographic frame.

## 3. The Analysis of Problem

## 4. The Principle of the Proposed Algorithm

#### 4.1. The Definition of the Pseudo-Frame

#### 4.2. The Mechanization of the SINS in Pseudo-Frame

#### 4.3. The Static Error Equations of the SINS in Pseudo-Frame

- The velocity error equations:$$\delta {\dot{V}}_{{E}_{p}}=2\Omega \mathrm{cos}{L}_{p}\mathrm{cos}{\lambda}_{p}\delta {V}_{{N}_{p}}-g{\varphi}_{{y}_{p}}+{\nabla}_{{E}_{p}}\mathrm{and}$$$$\delta {\dot{V}}_{{N}_{p}}=-2\Omega \mathrm{cos}{L}_{p}\mathrm{cos}{\lambda}_{p}\delta {V}_{{E}_{p}}+g{\varphi}_{{x}_{p}}+{\nabla}_{{N}_{p}}$$
- The attitude error equations:$${\dot{\varphi}}_{{x}_{p}}=-\Omega \mathrm{cos}{\lambda}_{p}\delta {\lambda}_{p}-\frac{\delta {V}_{{N}_{p}}}{R}+\Omega \mathrm{cos}{L}_{p}\mathrm{cos}{\lambda}_{p}{\varphi}_{{y}_{p}}+\Omega \mathrm{sin}{L}_{p}\mathrm{cos}{\lambda}_{p}{\varphi}_{{z}_{p}}+\text{}{\epsilon}_{{E}_{p}}$$$${\dot{\varphi}}_{{y}_{p}}=-\Omega \mathrm{cos}{L}_{p}\mathrm{cos}{\lambda}_{p}\delta {L}_{p}+\Omega \mathrm{sin}{L}_{p}\mathrm{sin}{\lambda}_{p}\delta {\lambda}_{p}+\frac{\delta {V}_{{E}_{p}}}{R}-\Omega \mathrm{cos}{L}_{p}\mathrm{cos}{\lambda}_{p}{\varphi}_{{x}_{p}}\text{\hspace{0.17em}}-\Omega \mathrm{sin}{\lambda}_{p}{\varphi}_{{z}_{p}}+{\epsilon}_{{N}_{p}}\mathrm{and}$$$${\dot{\varphi}}_{{z}_{p}}=-\Omega \mathrm{sin}{L}_{p}\mathrm{cos}{\lambda}_{p}\delta {L}_{p}-\Omega \mathrm{cos}{L}_{p}\mathrm{sin}{\lambda}_{p}\delta {\lambda}_{p}+\frac{\delta {V}_{{E}_{p}}}{R}\mathrm{tan}{L}_{p}-\Omega \mathrm{sin}{L}_{p}\mathrm{cos}{\lambda}_{p}{\varphi}_{{x}_{p}}\text{\hspace{0.17em}}+\Omega \mathrm{sin}{\lambda}_{p}{\varphi}_{{y}_{p}}+{\epsilon}_{{U}_{p}}$$
- The position error equations:$$\delta {\dot{L}}_{p}=\frac{\delta {V}_{{N}_{p}}}{R}\mathrm{and}$$$$\delta {\dot{\lambda}}_{p}=\frac{\delta {V}_{{E}_{p}}}{R}\mathrm{sec}{L}_{p}$$

## 5. The Filter Model of Kalman for Zero-Velocity Alignment

## 6. Simulations and Experiments

#### 6.1. The Simulations of Fine Alignment Assisted by the KF

#### 6.1.1. The Simulations Based on the Open-Loop KF

#### 6.1.2. The Simulations Based on the Closed-Loop KF

#### 6.2. The Experiments of Fine Alignment Based on the KF

#### 6.3. The Simulations of Polar Alignment

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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The Coarse Alignment Accuracy | Errors | ||
---|---|---|---|

Pitch | Roll | Heading | |

Case 1 | 0.3° | 0.3° | 1° |

Case 2 | 1° | 1° | 3° |

Case 3 | 2° | 2° | 5° |

Case 4 | 3° | 3° | 10° |

The Coarse Alignment Accuracy | Errors | ||
---|---|---|---|

Pitch | Roll | Heading | |

Case 1 | 0.2° | 0.2° | 0.5° |

Case 2 | 0.3° | 0.3° | 1° |

Case 3 | 0.5° | 0.5° | 2° |

Conditions | The Proposed Algorithm | The Traditional Algorithm | ||
---|---|---|---|---|

Mean | Variance | Mean | Variance | |

Case 1 | 1.57′ | 5 × 10^{−4} | 1.76′ | 6.25 × 10^{−4} |

Case2 | 1.74′ | 6.6 × 10^{−4} | 3.59′ | 2.6 × 10^{−3} |

Case3 | 2.53′ | 1.28 × 10^{−3} | 15.9′ | 1.28 × 10^{−5} |

Case4 | 1.82′ | 7.26 × 10^{−4} | 28.32′ | 8.35 × 10^{−3} |

Conditions | The Proposed Algorithm | The Traditional Algorithm | The Larger Noise Matrix Algorithm | |||
---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | |

Case 1 | 1.81′ | 5 × 10^{−5} | 2.80′ | 8.93 × 10^{−4} | − | − |

Case 2 | 1.33′ | 1 × 10^{−4} | × | × | 1.87′ | 4.2 × 10^{−4} |

Case 3 | −1.62′ | 1 × 10^{−3} | × | × | − | − |

^{1.}“×”denotes that alignment result diverges, and “−”denotes the experiment dose was not conducted.

The Coarse Alignment Accuracy | Errors | ||
---|---|---|---|

Pitch | Roll | Heading | |

Case 1 | 0.3° | 0.3° | 1° |

Case 2 | 0.5° | 0.5° | 2° |

Case 3 | 1° | 1° | 3° |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Liu, M.; Gao, Y.; Li, G.; Guang, X.; Li, S.
An Improved Alignment Method for the Strapdown Inertial Navigation System (SINS). *Sensors* **2016**, *16*, 621.
https://doi.org/10.3390/s16050621

**AMA Style**

Liu M, Gao Y, Li G, Guang X, Li S.
An Improved Alignment Method for the Strapdown Inertial Navigation System (SINS). *Sensors*. 2016; 16(5):621.
https://doi.org/10.3390/s16050621

**Chicago/Turabian Style**

Liu, Meng, Yanbin Gao, Guangchun Li, Xingxing Guang, and Shutong Li.
2016. "An Improved Alignment Method for the Strapdown Inertial Navigation System (SINS)" *Sensors* 16, no. 5: 621.
https://doi.org/10.3390/s16050621