# Robust Underwater Direction-of-Arrival Tracking Based on AI-Aided Variational Bayesian Extended Kalman Filter

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

#### 2.1. Kinematic Model of the DOA Tracking Process

#### 2.2. Measurement Model Based on the Received Signal of UCA

## 3. Methods

#### 3.1. EKF for DOA Tracking

#### 3.2. VB-EKF for Robust DOA Tracking

#### 3.2.1. Choice of Prior Distribution

**Σ**, and $h({x}_{k})$ is the nonlinear measurement function given by Equation (7). ${\widehat{x}}_{k|k-1}$ and ${P}_{k|k-1}$ denote the one-step prediction of state and MSEM, respectively, which are given by Equations (10) and (11).

**Ψ**[3], ${\widehat{u}}_{k|k-1}$ and ${\widehat{U}}_{k|k-1}$ are the dof and the inverse scale matrix of $p({R}_{k}|{z}_{1:k-1})$, respectively.

#### 3.2.2. Variational Approximations of Posterior PDFs

- (1)
- Update of ${x}_{k}$

- (2)
- Update of ${R}_{k}$

#### 3.3. AI-VBEKF for Robust DOA Tracking

#### 3.3.1. Input Data Processing

#### 3.3.2. Attention-Based Deep Convolutional Neural Network

#### 3.3.3. Design of the AI-VBEKF

Algorithm 1: AI-VBEKF |

Input${x}_{0}$.Calculate${\tilde{Z}}_{k}$ as Equation (50).Calculate ${e}_{V}$ as Equation (51).Initial MSEM ${\mathit{P}}_{0}$ estimation by attention-based DCNN. Inputs: ${\widehat{\mathit{x}}}_{k-1|k-1}$, ${\mathit{P}}_{k-1|k-1}$, ${\mathit{z}}_{k}$, ${\widehat{u}}_{k-1|k-1}$, ${\widehat{\mathit{U}}}_{k-1|k-1}$, ${Q}_{k-1}$, ρ, N.Time update${\widehat{\mathit{x}}}_{k|k-1}={\mathit{F}}_{k|k-1}{\widehat{\mathit{x}}}_{k-1|k-1}$. ${\mathit{P}}_{k|k-1}={\mathit{F}}_{k|k-1}{\mathit{P}}_{k-1|k-1}{\mathit{F}}_{k|k-1}^{\mathrm{T}}+{\mathit{G}}_{k-1}{\mathit{Q}}_{k-1}{\mathit{G}}_{k-1}^{\mathrm{T}}$. ${\widehat{u}}_{k\mid k-1}=\rho ({\widehat{u}}_{k-1|k-1}-n-1)+n+1$,${\widehat{\mathit{U}}}_{k|k-1}=\rho {\widehat{\mathit{U}}}_{k-1|k-1}$. Iteration measurement updateInitialization: ${\widehat{\mathit{x}}}_{k|k}^{(0)}={\widehat{\mathit{x}}}_{k|k-1}$, ${\widehat{\mathit{R}}}_{k}^{(0)}={\widehat{\mathit{U}}}_{k|k-1}/{\widehat{u}}_{k|k-1}$. For $i=0:N-1$Update ${q}^{(i+1)}({\mathit{x}}_{k})=\mathrm{N}({\mathit{x}}_{k};{\mathit{x}}_{k|k}^{(i+1)},{\mathit{P}}_{k|k}^{(i+1)})$ $h({\widehat{\mathit{x}}}_{k|k}^{(i)})$ is calculated by Equation (7), ${\mathit{H}}_{k}^{(i)}$ is calculated by Equation (13). ${\mathit{K}}_{k}^{(i+1)}={\mathit{P}}_{k|k-1}{({\mathit{H}}_{k}^{(i)})}^{\mathrm{T}}{({\mathit{H}}_{k}^{(i)}{\mathit{P}}_{k|k-1}{({\mathit{H}}_{k}^{(i)})}^{\mathrm{T}}+{\widehat{\mathit{R}}}_{k}^{(i)})}^{-1}$, ${\widehat{\mathit{x}}}_{k|k}^{(i+1)}={\widehat{\mathit{x}}}_{k|k-1}+{\mathit{K}}_{k}^{(i+1)}({z}_{k}-h({\widehat{\mathit{x}}}_{k|k}^{(i)})-{\mathit{H}}_{k}^{(i)}({\widehat{\mathit{x}}}_{k|k-1}-{\widehat{\mathit{x}}}_{k|k}^{(i)}))$, ${\mathit{P}}_{k|k}^{(i+1)}={\mathit{P}}_{k|k-1}-{\mathit{K}}_{k}^{(i+1)}{\mathit{H}}_{k}^{(i)}{\mathit{P}}_{k|k-1}$. Update ${q}^{(i+1)}({\mathit{R}}_{k})=\mathrm{IW}({\mathit{R}}_{k};{\widehat{u}}_{k|k}^{(i+1)},{\widehat{\mathit{U}}}_{k|k}^{(i+1)})$ $h({\widehat{x}}_{k|k}^{(i+1)})$ is calculated by Equation (7), ${\mathit{H}}_{k}^{(i+1)}$ is calculated by Equation (13). ${\mathit{B}}_{k}^{(i+1)}=({z}_{k}-h({\widehat{\mathit{x}}}_{k|k}^{(i+1)})){({z}_{k}-h({\widehat{\mathit{x}}}_{k|k}^{(i+1)}))}^{\mathrm{T}}+{\mathit{H}}_{k}^{(i+1)}{\mathit{P}}_{k|k}^{(i+1)}{({\mathit{H}}_{k}^{(i+1)})}^{\mathrm{T}}$, ${\widehat{\mathit{U}}}_{k|k}^{(i+1)}={\widehat{\mathit{U}}}_{k|k-1}+{\mathit{B}}_{k}^{(i+1)}$, ${\widehat{u}}_{k|k}^{(i+1)}={\widehat{u}}_{k|k-1}+1$, ${\widehat{\mathit{R}}}_{k}^{(i+1)}={\widehat{\mathit{U}}}_{k|k}^{(i+1)}/{\widehat{u}}_{k|k}^{(i+1)}$. End for${\widehat{\mathit{x}}}_{k|k}={\widehat{\mathit{x}}}_{k|k}^{(N)}$, ${\mathit{P}}_{k|k}={\mathit{P}}_{k|k}^{(N)}$, ${\widehat{u}}_{k|k}={\widehat{u}}_{k|k}^{(N)}$, ${\widehat{\mathit{U}}}_{k|k}={\widehat{\mathit{U}}}_{k|k}^{(N)}$. Outputs: ${\widehat{\mathit{x}}}_{k|k}$, ${\mathit{P}}_{k|k}$, ${\widehat{u}}_{k|k}$, ${\widehat{\mathit{U}}}_{k|k}$. |

#### 3.3.4. Performance Metrics

## 4. Results and Discussions

#### 4.1. Simulations

#### 4.1.1. Simulation Scenario and Data Set Generation of the Attention-Based DCNN

#### 4.1.2. Data Set Generation and Training of the Attention-Based DCNN

#### 4.1.3. Simulation Results

#### 4.2. Experimental Scenario

#### 4.2.1. Experimental Scenario

#### 4.2.2. Verification of the AI-VBEKF for DOA Tracking

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The basic structure of the attention-based DCNN [35].

**Figure 3.**The structure of the attention-based DCNN blocks combined with the residual model [35].

**Figure 5.**Simulation results of EKF, SH-EKF and AI-VBEKF. (

**a**) Trajectories tracking results under ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$, (

**b**) BEEs obtained under ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$, (

**c**) trajectories tracking results under ${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$, (

**d**) BEEs obtained under ${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$, (

**e**) trajectories tracking results under ${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$, and (

**f**) BEEs obtained under ${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$.

**Figure 6.**Configuration of the underwater buoy system: (

**a**) photo of the underwater buoy system; (

**b**) configuration of the UCA and the compass system.

**Figure 7.**Experimental results of CBF-based DOA estimation, EKF, SH-EKF and AI-VBEKF on raw data. (

**a**) Trajectories obtained using the raw data under ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$, (

**b**) BEEs obtained using the raw data under ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$, (

**c**) trajectories obtained using the raw data under ${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$, (

**d**) BEEs obtained using the raw data under ${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$, (

**e**) trajectories obtained using the raw data under ${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$, and (

**f**) BEEs obtained using the raw data under ${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$.

**Figure 8.**Experimental results of CBF-based DOA estimation, EKF, SH-EKF and AI-VBEKF on data with noise. (

**a**) Trajectories obtained using the data with added noise under ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$, (

**b**) BEEs obtained using the data with added noise under ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$, (

**c**) trajectories obtained using the data with added noise under ${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$, (

**d**) BEEs obtained using the data with added noise under ${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$, (

**e**) trajectories obtained using the data with added noise under ${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$, and (

**f**) BEEs obtained using the data with added noise under ${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$.

Layer Name | [Kernel Size, Filters] $\times $ No. of Blocks |
---|---|

SE-Comv1 | $[5\times 5,64]\times 1$ |

SE-Conv2 | $\left[\begin{array}{cc}1\times 1& 64\\ 3\times 3& 64\\ 1\times 1& 256\\ full\text{\hspace{0.33em}}connection& \left[16,256\right]\end{array}\right]\times 3$ |

SE-Conv3 | $\left[\begin{array}{cc}1\times 1& 128\\ 3\times 3& 128\\ 1\times 1& 512\\ full\text{\hspace{0.33em}}connection& \left[32,512\right]\end{array}\right]\times 4$ |

SE-Conv4 | $\left[\begin{array}{cc}1\times 1& 256\\ 3\times 3& 256\\ 1\times 1& 1024\\ full\text{\hspace{0.33em}}connection& \left[64,1024\right]\end{array}\right]\times 6$ |

SE-Conv5 | $\left[\begin{array}{cc}1\times 1& 512\\ 3\times 3& 512\\ 1\times 1& 2048\\ full\text{\hspace{0.33em}}connection& \left[128,2048\right]\end{array}\right]\times 3$ |

SE-Global | Global average pool, fully-connected layer with ReLU |

${\mathit{P}}_{0}$ | EKF | SH-EKF | VB-AEKF | |
---|---|---|---|---|

$\overline{RMS{E}_{\theta}}$ (°) | ${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$ | 7.4 | 2.8 | 2.7 |

${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$ | 13.2 | 7.4 | 2.7 | |

${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$ | 61.7 | 7.3 | 2.7 |

Data | ABEE (°) | ||||
---|---|---|---|---|---|

CBF | EKF | SH-EKF | AI-VBEKF | ||

${P}_{0}=\mathrm{diag}\left([25,{10}^{-5}]\right)$ | Raw data | 13.2 | 8.2 | 7.1 | 6.3 |

With noise | 20.1 | 12.2 | 7.3 | 6.5 | |

${P}_{0}=\mathrm{diag}\left([50,{10}^{-5}]\right)$ | Raw data | 13.2 | 11.3 | 11.6 | 6.8 |

With noise | 20.1 | 12.9 | 11.8 | 7.2 | |

${P}_{0}=\mathrm{diag}\left([100,{10}^{-5}]\right)$ | Raw data | 13.2 | 18.3 | 14.2 | 6.4 |

With noise | 20.1 | 18.5 | 14.9 | 6.6 |

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## Share and Cite

**MDPI and ACS Style**

Hou, X.; Qiao, Y.; Zhang, B.; Yang, Y.
Robust Underwater Direction-of-Arrival Tracking Based on AI-Aided Variational Bayesian Extended Kalman Filter. *Remote Sens.* **2023**, *15*, 420.
https://doi.org/10.3390/rs15020420

**AMA Style**

Hou X, Qiao Y, Zhang B, Yang Y.
Robust Underwater Direction-of-Arrival Tracking Based on AI-Aided Variational Bayesian Extended Kalman Filter. *Remote Sensing*. 2023; 15(2):420.
https://doi.org/10.3390/rs15020420

**Chicago/Turabian Style**

Hou, Xianghao, Yueyi Qiao, Boxuan Zhang, and Yixin Yang.
2023. "Robust Underwater Direction-of-Arrival Tracking Based on AI-Aided Variational Bayesian Extended Kalman Filter" *Remote Sensing* 15, no. 2: 420.
https://doi.org/10.3390/rs15020420