# Research on Dynamic Inertial Estimation Technology for Deck Deformation of Large Ships

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## Abstract

**:**

## 1. Introduction

## 2. Principle of Dynamic Inertial Measurement Method

## 3. Estimation Models

#### 3.1. Ship Benchmark Model

**D**is given as below

**F**

_{L}is a 12 × 12 dimensional state transition matrix, the 12-dimensional state noise vector is

#### 3.2. Sliding Estimation Model

_{1}, β

_{2}, and β

_{3}are the inverse correlation times of the corresponding stochastic processes, and λ

_{x}, λ

_{y}, and λ

_{z}are three-axis attitude angles of the SINS. The attitude angles are obtained by the attitude matrix

**R**

_{S}. The coordinate transformation matrix of the SINS carrier coordinate system to the local horizontal coordinate system is given as [17]

_{x}, λ

_{y}, and λ

_{z}are obtained by the following equation

**F**

_{S}is a state transition matrix, and the state noise vector is a 12-dimensional zero-mean white noise vector, which is

## 4. Dynamic Filtering Algorithm

- Prediction: $\widehat{\mathit{X}}(\left.k+1\right|k)=\mathit{\Phi}(\left.k+1\right|k)\widehat{\mathit{X}}(\left.k\right|k)$
- Correction: $\widehat{\mathit{X}}(\left.k+1\right|k+1)=\widehat{\mathit{X}}(\left.k+1\right|k)+\mathit{K}(k+1){\left[\mathit{Z}(k+1)-\mathit{H}(k+1)\widehat{\mathit{X}}(\left.k+1\right|k)\right]}^{}$
- Kalman gain matrix: $\mathit{K}(k+1)=\mathit{P}(\left.k+1\right|k){\mathit{H}}^{T}(k+1){\left[\mathit{H}(k+1)\mathit{P}(\left.k+1\right|k){\mathit{H}}^{T}(k+1)+\mathit{R}(k+1)\right]}^{-1}$
- Prediction error variance matrix: $\mathit{P}(\left.k+1\right|k)=\mathit{\Phi}(\left.k+1\right|k)\mathit{P}(\left.k\right|k){\mathit{\Phi}}^{\mathrm{T}}(\left.k+1\right|k)+\mathit{Q}(k)$
- Correction error variance matrix: $\mathit{P}(\left.k+1\right|k+1)=\mathit{P}(\left.k+1\right|k)-\mathit{K}(k+1)\mathit{H}(k+1)\mathit{P}(\left.k+1\right|k)$

**Z**

_{m}is smooth approximation and

**Z**

_{d}is detail signal.

## 5. Deck Deformation Measurement Simulation

#### 5.1. Parameter Setting

_{1}, β

_{2}, β

_{3}were 0.15, 0.12, and 0.10 respectively. And the SINS moved uniformly along the estimation trajectory at a velocity of 5 m/s, the estimation trajectory was 100 m long, the estimation time was 20 s.

#### 5.2. Results and Discussion

#### 5.2.1. Simulation Results of the Ship Benchmark Model

#### 5.2.2. Simulation Results of the Sliding Estimation Model

#### 5.2.3. Curvature and Torsion of the Deck

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Comparison of angular velocity filtering results of the inertial navigation systems (INS). (

**a**) ${\omega}_{r}$ of the INS; (

**b**) ${\omega}_{p}$ of the INS; (

**c**) ${\omega}_{a}$ of the INS.

**Figure 4.**Comparison of angular velocity filtering results of the sub-inertial navigation systems (SINS). (

**a**) Noiseless ${\omega}_{x}$ of the SINS; (

**b**) Noisy ${\omega}_{x}$ of the SINS; (

**c**) Noiseless ${\omega}_{y}$ of the SINS; (

**d**) Noiseless ${\omega}_{y}$ of the SINS; (

**e**) Noisy ${\omega}_{z}$ of the SINS; (

**f**) Noiseless ${\omega}_{z}$ of the SINS.

**Figure 5.**Spectrum analysis before and after filtering of the SINS angular velocity. (

**a**) Spectrogram of observation values of the ${\omega}_{x}$; (

**b**) Spectrogram of filter values of the ${\omega}_{x}$; (

**c**) Spectrogram of observation values of the ${\omega}_{y}$; (

**d**) Spectrogram of filter values of the ${\omega}_{y}$; (

**e**) Spectrogram of observation values of the ${\omega}_{z}$; (

**f**) Spectrogram of filter values of the ${\omega}_{z}$.

**Figure 6.**Curvature and torsion of the estimation trajectory: (

**a**) Vertical curvature of the deck; (

**b**) Horizontal curvature of the deck; (

**c**) Torsion of the deck.

**Figure 7.**Sliding position of the estimation trajectory: (

**a**) Original trajectory and deformation trajectory; (

**b**) Partial enlargement of the xoy plane (Potential danger zone).

Axial | Parameter | Original Data | Filter Data (Wavelet Combined with Kalman Filter) |
---|---|---|---|

x | Mean | 0.9180 | 0.3812 |

RMS | 1.0157 | 0.4036 | |

y | Mean | 0.3271 | 0.1460 |

RMS | 0.5328 | 0.1487 | |

z | Mean | 0.3874 | 0.1974 |

RMS | 0.5833 | 0.1981 |

Parameter | Original Data | Filter Data (Wavelet) |
---|---|---|

Mean | 1.76 | 0.64 |

RMS | 1.98 | 0.76 |

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**MDPI and ACS Style**

Ren, B.; Li, T.; Li, X.
Research on Dynamic Inertial Estimation Technology for Deck Deformation of Large Ships. *Sensors* **2019**, *19*, 4167.
https://doi.org/10.3390/s19194167

**AMA Style**

Ren B, Li T, Li X.
Research on Dynamic Inertial Estimation Technology for Deck Deformation of Large Ships. *Sensors*. 2019; 19(19):4167.
https://doi.org/10.3390/s19194167

**Chicago/Turabian Style**

Ren, Bo, Tianjiao Li, and Xiang Li.
2019. "Research on Dynamic Inertial Estimation Technology for Deck Deformation of Large Ships" *Sensors* 19, no. 19: 4167.
https://doi.org/10.3390/s19194167