# Performance Evaluation of IMU and DVL Integration in Marine Navigation

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## Abstract

**:**

## 1. Introduction

## 2. Method 1—Dead-Reckoning Navigation Using IMU and DVL

#### 2.1. Direction Estimation

#### 2.1.1. Direction Estimation by IMU with GNSS Using KF

- ${\theta}_{{G}_{k}}$: GNSS direction [°]
- ${\omega}_{{G}_{k}}$: Gyroscope angular velocity with bias taken into account as shown in Equation (14) [°/s]
- dir: Indicates that dir is a calculation of direction.
- $\mathsf{\Delta}t$ = 0.2 [s]…GNSS frequency;
- ${\sigma}_{{\theta}_{G}}=1.0[\xb0]$…GNSS error standard deviation of orientation;
- ${\sigma}_{{\omega}_{z}}=0.02\left[\xb0/\mathrm{s}\right]$…IMU error standard deviation of angular velocity.
- ${x}_{{k}_{dir}}$: State vectors
- ${\mathsf{\Phi}}_{dir}$: State − space matrix
- ${R}_{dir}$: Covariance matrix
- ${k}_{dir}$: Number of updates of KF for direction estimation

#### 2.1.2. Direction Estimation Using Only IMU

#### 2.2. Velocity Estimation

#### 2.2.1. Velocity Observation by GNSS

#### 2.2.2. Velocity Observation with DVL

- ${V}_{{x}_{k}}$: Sonar speed in X axis [m/s]
- ${V}_{{y}_{k}}$: Sonar speed in Y axis [m/s]

#### 2.2.3. Speed Estimation by KF

- $\mathsf{\Delta}{t}_{S}$ = 1 [s]…DVL observation period
- ${\sigma}_{{V}_{x}}$ = 0.11 [m/s]…DVL speed error Standard deviation in X direction
- ${\sigma}_{{V}_{y}}$ = 0.11 [m/s]…DVL speed error Standard deviation in Y direction
- ${\sigma}_{{\mathrm{a}}_{x}}$= 0.06 [m/s
^{2}]…IMU acceleration error standard deviation in X direction - ${\sigma}_{{\mathrm{a}}_{y}}$ = 0.06 [m/s
^{2}]…IMU acceleration error standard deviation in Y direction - ${k}_{speed}$: Number of updates of KF for velocity estimation

#### 2.3. Position Estimation

## 3. Method 2—INS/DVL Integrated Navigation

#### 3.1. Parameter Estimation by Allan Variance

#### 3.2. INS/DVL Integration

## 4. Experiment and Results

#### 4.1. Experiment Outline

#### 4.2. Results with First Method

#### 4.2.1. Evaluation of Estimated Direction

#### 4.2.2. Evaluation of Speed Estimation

#### 4.2.3. Position Estimation Result

#### 4.3. Results with Second Method

#### 4.3.1. Evaluation of Estimated Attitude

^{5}s. We think this is because the bias was removed by turning nearly 360°. Figure 13 shows that the difference increased exponentially when turning, and increased at a constant rate when going straight. For this difference, we can create a detailed sensor-error model, estimating the error when GNSS is available, performing the same process as DZUPT when the vessel is moving straight ahead, or estimating the error using the gyrocompass. In the future, we intend to study a method of bias estimation according to the available equipment.

#### 4.3.2. Evaluation of Speed Estimation

^{−4}(STD of 9.322 × 10

^{−2}) and −1.53 × 10

^{−4}(STD of 9.631 × 10

^{−2}), respectively. The average difference between the Y-axis INS/DVL and sonar velocities and the estimated values were 1.15 × 10

^{−1}(STD of 1.580 × 10

^{−1}) and 1.21 × 10

^{−1}(STD of 2.168 × 10

^{−1}), respectively, which were larger than that of the X-axis. As can be seen from the average, a bias-like component was detected for the Y-axis. For improving the accuracy, it might be necessary to use a better accelerometer to improve the accuracy of speed estimation of the INS, because it is difficult to replace the Doppler sonar.

#### 4.3.3. Evaluation of Position Estimation

^{5}[s]. Subsequently, the position estimation error increased as well as the effect of the heading error shown in Figure 13, and the final error was 579 m. In this study, we used a gyrocompass as a reference. If we use the gyrocompass values to estimate the heading bias in the circled area in Figure 17 and then correct for the heading bias, we obtain the result in Figure 18, where the final horizontal error is 170 m. This post-processing method was only applied to the results of Figure 13 found to contain bias and is, therefore, not applicable as a real-time bias detection and correction method at this time.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 12.**(

**a**) Roll angle of INS/DVL and FOG; (

**b**) Roll difference between INS/DVL and FOG; (

**c**) Pitch angle of INS/DVL and FOG; (

**d**) Pitch difference between INS/DVL and FOG.

**Figure 14.**(

**a**) INS/DVL and DVL x-axis velocities minus estimated value, respectively; (

**b**) INS/DVL and DVL y-axis velocities minus estimated value, respectively.

Static Bias (rad/s) (m/s ^{2}) | STD (rad/s) (m/s ^{2}) | Random Walk $(\mathbf{rad}/\mathbf{s}\sqrt{\mathit{H}\mathit{z}}$$\left)\right(\mathbf{m}/{\mathbf{s}}^{2}\sqrt{\mathit{H}\mathit{z}}\left)\right)$ | Bias Instability (rad/s) (m/s ^{2}) | |
---|---|---|---|---|

Gyro X | −7.825 × 10^{−}^{04} | 3.088 × 10^{−04} | 4.00 × 10^{−05} | 2.63 × 10^{−05} |

Gyro Y | 3.234 × 10^{−03} | 3.107 × 10^{−04} | 4.00 × 10^{−05} | 2.90 × 10^{−05} |

Gyro Z | 2.202 × 10^{−03} | 3.307 × 10^{−04} | 4.30 × 10^{−05} | 2.67 × 10^{−05} |

Acc X | 6.389 × 10^{−02} | 9.326 × 10^{−03} | 1.29 × 10^{−03} | 9.35 × 10^{−04} |

Acc Y | 5.178 × 10^{−01} | 1.005 × 10^{−02} | 1.69 × 10^{−03} | 1.59 × 10^{−03} |

Acc Z | −9.940 | 9.255 × 10^{−03} | 1.40 × 10^{−03} | 1.20 × 10^{−03} |

X-Axis | Y-Axis | Z-Axis | |
---|---|---|---|

${b}_{g}$ | 2.63 × 10^{−5} [rad/s] | 2.90 × 10^{−5} [rad/s] | 2.67 × 10^{−5} [rad/s] |

${b}_{f}$ | 9.34 × 10^{−4} [m/s^{2}] | 1.60 × 10^{−03} [m/s^{2}] | 1.20 × 10^{−3} [m/s^{2}] |

${S}_{rg}$ | 0.966 × 10^{−7} [rad] (Equation (14.81) of [27]) | ||

${S}_{ra}$ | 0.101 × 10^{−3} [m/s] (Equation (14.81) of [27]) | ||

${S}_{bgd}$ | 0.115 × 10^{−10} [rad] | 0.115 × 10^{−10} [rad] | 0.115 × 10^{−10} [rad] |

${S}_{bad}$ | 0.435 × 10^{−7} [m/s] | 0.261 × 10^{−7} [m/s] | 0.435 × 10^{−7} [m/s] |

${\tau}_{g}$ | 60 [s] | 60 [s] | 60 [s] |

${\tau}_{f}$ | 60 [s] | 100 [s] | 60 [s] |

${\tau}_{s}$ | 1.0 [s] |

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

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

Frequency | 5 Hz | 100 Hz | 1 Hz | |

Accuracy | Position | Gyro | Acceleration | Speed |

<0.1 [m] | ±0.01 [m/s^{2}] | ±0.00175 [rad/s] | 0.01 [knot] or 0.2% of the measured value |

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

Setting Point Error | Less than ±0.3° | Repeatability of Setting Point | Less than ±0.2° |

RMS Value | Less than 0.1° | Accuracy Under Environmental Variation | Less than ±0.5° |

Digital Output | |
---|---|

Range | ±Roll: ±180°, Pitch: ±90° |

Resolution | <0.1° |

Accuracy | <±0.15° at input <±10° <± (0.2° + 1% of input) at input = ±10°~45° |

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**MDPI and ACS Style**

Fukuda, G.; Hatta, D.; Guo, X.; Kubo, N.
Performance Evaluation of IMU and DVL Integration in Marine Navigation. *Sensors* **2021**, *21*, 1056.
https://doi.org/10.3390/s21041056

**AMA Style**

Fukuda G, Hatta D, Guo X, Kubo N.
Performance Evaluation of IMU and DVL Integration in Marine Navigation. *Sensors*. 2021; 21(4):1056.
https://doi.org/10.3390/s21041056

**Chicago/Turabian Style**

Fukuda, Gen, Daisuke Hatta, Xiaoliang Guo, and Nobuaki Kubo.
2021. "Performance Evaluation of IMU and DVL Integration in Marine Navigation" *Sensors* 21, no. 4: 1056.
https://doi.org/10.3390/s21041056