# Application of Initial Bias Estimation Method for Inertial Navigation System (INS)/Doppler Velocity Log (DVL) and INS/DVL/Gyrocompass Using Micro-Electro-Mechanical System Sensors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Using Initial Bias Estimation by Inversion of Inertial Navigation Calculations

#### 2.1. Initial Bias Estimation

#### 2.2. IGG and IDG Integration

#### 2.3. Acceleration and Angular Velocity Estimation Using Inverted Form of INS Calculation

## 3. Experiment Outline and Results

#### 3.1. Initial Bias Estimation

#### 3.2. Estimation Results from Inverse Inertial Navigation Calculations

#### 3.2.1. Comparison between IGG and Reference Estimations

#### 3.2.2. Comparison of the TG and IGG Estimates

#### 3.2.3. Estimates of Angular Velocity and Acceleration (Specific Force)

^{2}, respectively, and the average differences are 0.47, 0.70, and 0.13 m/s

^{2}, respectively.

#### 3.3. Comparison of Estimates Based on the Interval of Data Used to Estimate the Initial Bias

^{−2}, 2.30 × 10

^{−1}, and 1.23 × 10

^{−1}°/s for the TG, and −1.75 × 10

^{−2}, 2.37 × 10

^{−1}, and 1.59 × 10

^{−1}°/s for the KF. For acceleration, when the data interval used for bias estimation is more than 400 s, the averages for the x, y, and z axes are 4.74 × 10

^{−1}, 7.09 × 10

^{−1}, and −1.32 × 10

^{−1}m/s

^{2}for the TG, and 4.62 × 10

^{−1}, 8.23 × 10

^{−1}, and −1.23 × 10

^{−1}m/s

^{2}for the KF.

#### 3.3.1. Comparison of Results with Initial Bias Estimation Using the TG and KF

^{−2}°, and based on the accuracy of FOG in Table 1, we concluded that the estimation results were the same in this case.

#### 3.3.2. Initial Bias Estimation and Position Estimation Results Using TG and KF

^{−3}°/s (horizontal error of 491.03 m) and 2.175 × 10

^{−3}°/s (horizontal error of 486.20 m).

## 4. Discussion

^{−5}°/s. Therefore, for the accuracy of this IMU, it is necessary to estimate the initial bias with an accuracy of approximately 2.00 × 10

^{−5}°/s or better.

^{−2}m/s

^{2}is observed between the TG and KF for the x-axis acceleration. For the y-axis, the TG is stable at 1.15 × 10

^{−1}m/s

^{2}, whereas the KF shows greater variation. For the z-axis, there is an average difference of 8.26 × 10

^{−3}m/s

^{2}. The acceleration calculated by the TG is an estimate that can satisfy the average difference of 0.04 m from the position estimated by the IGG when used in pure INS, as shown in Figure 5. Although it can be inferred that TG is superior to KF, to verify this conclusion, a highly accurate system, which is not available in our laboratory, is required to obtain the reference values of velocity in each axis direction.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References and Notes

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**Figure 5.**(

**a**) Roll angle of IGG and FOG; (

**b**) roll difference (RMS) between IGG and FOG; (

**c**) pitch angle of IGG and FOG; (

**d**) pitch difference (RMS) between IGG and FOG.

**Figure 7.**(

**a**) Angular velocity along x-axis angular velocity of the IMU (orange) and TG (blue); (

**b**) y-axis angular velocity of the IMU (orange) and TG (blue); (

**c**) z-axis angular velocity of the IMU (orange) and TG (blue); (

**d**) difference in the x-axis angular velocity between the IMU and TG; (

**e**) difference in the y-axis angular velocity between the IMU and TG; (

**f**) difference in the z-axis angular velocity between the IMU and TG.

**Figure 8.**(

**a**) Acceleration along x-axis acceleration of the IMU (orange) and TG (blue); (

**b**) y-axis acceleration of the IMU (orange) and TG (blue); (

**c**) z-axis acceleration of the IMU (orange) and TG (blue); (

**d**) difference in x-axis acceleration between the IMU and TG; (

**e**) difference in y-axis acceleration between the IMU and TG; (

**f**) difference in z-axis angular velocity between the IMU and TG.

**Figure 9.**Initial bias estimates and data interval relationships obtained by the TG and KF for angular velocity; (

**a**) x-axis angular velocity bias by TG; (

**b**) y-axis angular velocity bias by TG; (

**c**) z-axis angular velocity bias by TG; (

**d**) x-axis angular velocity bias by KF; (

**e**) y-axis angular velocity bias by KF; (

**f**) z-axis angular velocity bias by KF.

**Figure 10.**Initial bias estimates and data interval relationships obtained by the TG and KF for acceleration. (

**a**) x-axis acceleration bias by TG; (

**b**) y-axis acceleration bias by TG; (

**c**) z-axis acceleration bias by TG; (

**d**) x-axis acceleration bias by KF; (

**e**) y-axis acceleration bias by KF; (

**f**) z-axis acceleration bias by KF.

**Figure 11.**RMS difference between estimates by the INS/DVL and FOG during a 1 h voyage and the initial bias estimates from the TG and KF; (

**a**) roll difference when initial bias is estimated by TG; (

**b**) roll difference when initial bias is estimated by KF; (

**c**) pitch difference when initial bias is estimated by TG; (

**d**) pitch difference when initial bias is estimated by KF.

**Figure 12.**RMS difference between estimates by the IDG and FOG during a 1 h voyage and the initial bias estimates from the TG and KF. (

**a**) Roll difference when initial bias is estimated by TG; (

**b**) Roll difference when initial bias is estimated by KF; (

**c**) Pitch difference when initial bias is estimated by TG; (

**d**) Pitch difference when initial bias is estimated by KF.

**Figure 13.**Horizontal position errors after 1 h estimated by INS/DVL and IDG with initial bias estimation by TG and KF; (

**a**) horizontal error by INS/DVL with initial bias estimation by TG; (

**b**) horizontal error by INS/DVL with initial bias estimation by KF; (

**c**) horizontal error by IDG with initial bias estimation by TG; (

**d**) horizontal error by IDG with initial bias estimation by KF.

**Figure 14.**(

**a**) Relationship between z-axis angular velocity initial bias and horizontal error after 1 h in INS/DVL; (

**b**) relationship between z-axis angular velocity initial bias and horizontal error after 1 h in IDG.

Abbreviation | Full Term |
---|---|

TG | Trajectory Generator |

INS | Inertial Navigation System |

DVL | Doppler Velocity Log |

IDG | INS/DVL/Gyrocompass |

IGG | INS/GPS/Gyrocompass |

KF | Kalman Filter |

GNSS | IMU | DVL | FOG | |||
---|---|---|---|---|---|---|

Name | Trimble SPS855 | CSM-MG100 | ATLAS DOLOG SYSTEM | JCS7402-A | ||

Freq. | 5 Hz | 100 Hz | 1 Hz | 1 Hz | ||

Accuracy | Position | Gyro | Acc. | Position | Speed | Roll and Pitch |

<0.1 [m] | ±0.00175 [rad/s] | ±0.01 [m/s ^{2}] | 3.0 m RMS or Less | 0.01 [knot] or 0.2% of the measured value | ≤±0.15° at input ≤±10° ≤±(0.2° + 1% of input) at input = ±10°–45° |

Setting Time | Within 2 h | Accuracy on Scorsby Table | ≤±0.5° |

Setting Point Error | ≤±0.3° | Repeatability of Setting Point | ≤±0.2° |

RMS Value | ≤0.1° | Accuracy Under Environmental Variation | ≤±0.5° |

**Table 4.**Maximum and minimum differences in the RMS roll and pitch when using the initial bias estimates from the TG and KF for INS/DVL.

Roll by ISN/DVL (°) | Pitch by INS/DVL (°) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Total | Over 400 s | Total | Over 400 s | |||||||

Max. | Min. | Max. | Min. | Ave. | Max. | Min. | Max. | Min. | Ave. | |

By TG | 1.33 | 1.10 | 1.15 | 1.12 | 1.13 | 0.31 | 0.22 | 0.24 | 0.23 | 0.24 |

By KF | 1.86 | 1.60 | 1.83 | 1.69 | 1.78 | 0.37 | 0.16 | 0.20 | 0.16 | 0.18 |

**Table 5.**Maximum and minimum differences in the RMS roll and pitch when using the initial bias estimates from the TG and KF for IDG.

Roll by IDG (Degree) | Pitch by IDG (Degree) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Total | Over 400 s | Total | Over 400 s | |||||||

Max. | Min. | Max. | Min. | Ave. | Max. | Min. | Max. | Min. | Ave. | |

By TG | 1.33 | 1.11 | 1.14 | 1.12 | 1.13 | 0.31 | 0.23 | 0.24 | 0.23 | 0.23 |

By KF | 1.86 | 1.57 | 1.85 | 1.72 | 1.80 | 0.39 | 0.15 | 0.20 | 0.15 | 0.18 |

**Table 6.**Maximum and minimum differences in the horizontal positions for the INS/DVL for the TG and KF.

Err. by INS/DVL after 1 h (m) | Err. by IDG after 1 h (m) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Total | Over 400 s | Total | Over 400 s | |||||||

Max. | Min. | Max. | Min. | Ave. | Max. | Min. | Max. | Min. | Ave. | |

By TG | 1.76 × 10^{4} | 340 | 1569 | 340 | 921 | 515 | 466 | 494 | 492 | 493 |

By KF | 2.08 × 10^{4} | 217 | 2.06 × 10^{4} | 345 | 1.31 × 10^{4} | 544 | 442 | 522.00 | 495 | 507 |

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**MDPI and ACS Style**

Fukuda, G.; Kubo, N.
Application of Initial Bias Estimation Method for Inertial Navigation System (INS)/Doppler Velocity Log (DVL) and INS/DVL/Gyrocompass Using Micro-Electro-Mechanical System Sensors. *Sensors* **2022**, *22*, 5334.
https://doi.org/10.3390/s22145334

**AMA Style**

Fukuda G, Kubo N.
Application of Initial Bias Estimation Method for Inertial Navigation System (INS)/Doppler Velocity Log (DVL) and INS/DVL/Gyrocompass Using Micro-Electro-Mechanical System Sensors. *Sensors*. 2022; 22(14):5334.
https://doi.org/10.3390/s22145334

**Chicago/Turabian Style**

Fukuda, Gen, and Nobuaki Kubo.
2022. "Application of Initial Bias Estimation Method for Inertial Navigation System (INS)/Doppler Velocity Log (DVL) and INS/DVL/Gyrocompass Using Micro-Electro-Mechanical System Sensors" *Sensors* 22, no. 14: 5334.
https://doi.org/10.3390/s22145334