# Satellite Launcher Navigation with One Versus Three IMUs: Sensor Positioning and Data Fusion Model Analysis

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## Abstract

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## 1. Introduction

- compare the performances of three IMUs against that of one better quality IMU;
- investigate the effect of collocating IMUs versus distributing them along the launcher structure;
- test three multi-IMU navigation solutions:
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- fusion of all IMUs in one INS,
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- fusion of multiple INSs, and
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- fusion of multiple INSs with geometrical constraints.

## 2. Data Fusion Architectures

#### 2.1. Fusion with One IMU

#### 2.2. Fusion of Multiple IMUs within One INS

#### 2.3. Fusion of Multiple INSs in a Stacked Filter

#### 2.3.1. Geometrical Constraint Equations

#### 2.3.2. Fusion of all INSs

#### 2.3.3. Individual INS Error State Models

## 3. Test Parameters

## 4. Results Analysis

#### 4.1. Tests with All Sensors in the Launcher Head

#### 4.1.1. Test with GPS Receiver

#### 4.1.2. Test without GPS Receiver

#### 4.1.3. Conclusion on Sensors Installed in the Launcher Head

#### 4.2. Tests with IMUs Distributed Along the Launcher Structure

## 5. Conclusions

- evaluating the impact of using more than three IMUs in the fusion of multiple INSs, either with or without geometrical constraints;
- testing the fusion of multiple INSs with geometrical constraints, when independent GPS receivers and attitude reference sensors are used for each INS;
- investigating the reasons which lead the Kalman filter to diverge when the GPS receiver and attitude reference sensor measurement correlations among INSs are considered in the observation covariance matrix;
- comparing the impact of including the knowledge of the launcher dynamics in the fusion of multiple IMUs in one INS to the one obtained by the authors in a previous work with a single IMU.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The attitude estimation error standard deviation obtained with the navigation solutions, when all sensor measurements are present.

**Figure 5.**The velocity estimation error standard deviation obtained with the navigation solutions, when all sensor measurements are present.

**Figure 6.**The position estimation error standard deviation obtained with the navigation solutions, when all sensor measurements are present.

**Figure 7.**The position estimation error standard deviation obtained with the first INS of the multi-INS fusion without constraints versus the one obtained with the single IMU solution.

**Figure 8.**The velocity standard deviation obtained with a common GPS bias estimation versus one GPS bias estimated for each INS in the multi-INS fusion without constraint, when all sensor measurements are present.

**Figure 9.**The position standard deviation obtained with a common GPS bias estimation versus one GPS bias estimated for each INS in the multi-INS fusion without constraint, when all sensor measurements are present.

**Figure 10.**The attitude estimation error standard deviation obtained with the navigation solutions, when the GPS receiver is not present.

**Figure 11.**The velocity estimation error standard deviation obtained with the navigation solutions, when the GPS receiver is not present.

**Figure 12.**The position estimation error standard deviation obtained with the navigation solutions, when the GPS receiver is not present.

**Figure 13.**The velocity estimation error standard deviation computed by the navigation filter and the one obtained from Monte Carlo simulation, when the GPS receiver is not present.

**Figure 14.**The position estimation error standard deviation computed by the navigation filter and the one obtained from Monte Carlo simulation, when the GPS receiver is not present.

**Figure 15.**Graphical representation of the correlation between the states of the navigation filter at the 30th second of the simulation, when all geometrical constraints are present.

**Figure 16.**Graphical represetation of the correlation between the states of the navigation filter at the 30th second of the simulation, when only the relative attitude constraint is present.

**Figure 17.**Graphical represetation of the correlation between the states of the navigation filter at the 674th second of simulation, when only the relative attitude constraint is present.

**Figure 18.**The attitude estimation error standard deviation obtained with the multi-INS fusion when all geometrical constraints are applied versus when only the relative attitude constraint is used.

**Figure 19.**The attitude estimation error standard deviation obtained with the navigation solutions, when the attitude reference sensor and GPS receiver are present.

**Figure 20.**The velocity estimation error standard deviation obtained with the navigation solutions, when the attitude reference sensor and GPS receiver are present.

**Figure 21.**The position estimation error standard deviation obtained with the navigation solutions, when the attitude reference sensor and GPS receiver are present.

**Figure 22.**The raw gyroscope measurement standard deviation and the standard deviation of the angular velocity estimated in the fusion of all IMU into one INS, when the attitude reference sensor and GPS receiver are present.

**Figure 23.**The raw accelerometer measurement standard deviation and the standard deviation of the acceleration estimated in the fusion of all IMU into one INS, when the attitude reference sensor and GPS receiver are present.

GPS Receiver | C/A Code with Wide Correlator |
---|---|

Attitude reference sensor noise standard deviation | ${1}^{\xb0}$ |

Gyroscope random walk (multi-IMUs) | ${0.72}^{\xb0}/h/\sqrt{Hz}$ |

Gyroscope bias stability (multi-IMUs) | ${0.05}^{\xb0}/h$ |

Accelerometer random walk (multi-IMUs) | 117 $\mathsf{\mu}$g/$\sqrt{Hz}$ |

Accelerometer bias stability (multi-IMUs) | 7500 $\mathsf{\mu}$g |

Gyroscope random walk (single-IMU) | ${0.416}^{\xb0}/h/\sqrt{Hz}$ |

Gyroscope bias stability (single-IMU) | ${0.029}^{\xb0}/h$ |

Accelerometer random walk (single-IMU) | 68 $\mathsf{\mu}$g/$\sqrt{Hz}$ |

Accelerometer bias stability (single-IMU) | 4330 $\mathsf{\mu}$g |

**Table 2.**Reduction of the estimation error standard deviations provided by the fusion of multiple INSs at the satellite injection (values in parentheses represent the maximum improvement when it does not occur at the satellite injection).

With GPS Receiver | Without GPS Receiver | ||||
---|---|---|---|---|---|

w/o Constraint | All Constraints | w/o Constraint | Attitude Constraint | ||

Attitude | roll | $16\%$ | $31\%$ | $17\%$ | $32\%$ |

pitch | $16\%$ | $28\%$ | $17\%$ | $32\%$ | |

yaw | $17\%$ | $29\%$ | $17\%$ | $32\%$ | |

Velocity | x | $39\%$ | $51\%$ | $11\%$ | $11\%$ |

y | $35\%$$(40\%)$ | $48\%$$(52\%)$ | $12\%$ | $12\%$ | |

z | $35\%$$(41\%)$ | $49\%$$(52\%)$ | $11\%$ | $12\%$ | |

Position | x | $50\%$$(54\%)$ | $54\%$$(59\%)$ | $18\%$ | $18\%$ |

y | $50\%$$(54\%)$ | $53\%$$(59\%)$ | $19\%$ | $19\%$ | |

z | $50\%$$(54\%)$ | $54\%$$(59\%)$ | $18\%$ | $18\%$ |

© 2018 by Her Majesty the Queen in Right of Canada, Department of National Defence. 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**

Beaudoin, Y.; Desbiens, A.; Gagnon, E.; Landry, R., Jr. Satellite Launcher Navigation with One Versus Three IMUs: Sensor Positioning and Data Fusion Model Analysis. *Sensors* **2018**, *18*, 1872.
https://doi.org/10.3390/s18061872

**AMA Style**

Beaudoin Y, Desbiens A, Gagnon E, Landry R Jr. Satellite Launcher Navigation with One Versus Three IMUs: Sensor Positioning and Data Fusion Model Analysis. *Sensors*. 2018; 18(6):1872.
https://doi.org/10.3390/s18061872

**Chicago/Turabian Style**

Beaudoin, Yanick, André Desbiens, Eric Gagnon, and René Landry, Jr. 2018. "Satellite Launcher Navigation with One Versus Three IMUs: Sensor Positioning and Data Fusion Model Analysis" *Sensors* 18, no. 6: 1872.
https://doi.org/10.3390/s18061872