# Subsea Cable Tracking by Autonomous Underwater Vehicle with Magnetic Sensing Guidance

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## Abstract

**:**

## 1. Introduction

## 2. Magnetic Sensing and Cable Localization

#### 2.1. Passive Magnetic Sensing

#### 2.2. Cable Location

#### 2.2.1. Heading Deviation

#### 2.2.2. Buried Depth

#### 2.2.3. Horizontal Offset

## 3. Magnetic Guidance and Tracking Control

#### 3.1. AUV Modeling

#### 3.2. Guidance and Control Design

#### 3.2.1. Magnetic Guidance

#### 3.2.2. Cable Tracking Control

## 4. Numerical Simulation

#### 4.1. Magnetic Field Simulation

#### 4.2. Cable Tracking Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

ROV | Remotely-operated vehicle |

3-DOF | Three-degrees-of-freedom |

AUV | Autonomous underwater vehicle |

AC | Alternating current |

AE | AQUA EXPLORER |

LOS | Line-of-sight |

ILOS | Integral line-of-sight |

USV | Unmanned surface vessel |

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**Figure 1.**The yellow AUV equipped with two tri-axial magnetometers is tracking the orange cable. The induced electromotive forces in the x-axis, y-axis and z-axis of the tri-axial Magnetometer 1 and Magnetometer 2 are $({A}_{1},{B}_{1},{C}_{1})$ and $({A}_{2},{B}_{2},{C}_{2})$, respectively. The distance between those two magnetometers is ${O}_{1}{O}_{2}=L$, and $\{I\}$ is the inertial frame. In addition, ${\psi}_{Br}$, Y and Z denote the heading deviation, horizontal offset and vertical distance between the AUV and the cable.

**Figure 2.**The yellow AUV equipped with two tri-axial magnetometers and the blue-black cable are located in Plane 1 and Plane 2, respectively. In Plane 1, the red dashed line denotes the projection line of the cable; the green dashed line denotes the AUV heading; and the purple line is the connection line between two tri-axial magnetometers, represented by ${O}_{1}{x}_{1}{y}_{1}{z}_{1}$ and ${O}_{2}{x}_{2}{y}_{2}{z}_{2}$, respectively. Note that their x-axes are parallel to the longitudinal axis directed from aft to fore of the AUV and the z-axes are directed from top to bottom. Constructing the auxiliary lines as ${O}_{1}{E}_{1}\perp {E}_{1}{E}_{2}$, ${E}_{1}{F}_{1}\perp {O}_{1}{F}_{1}$, ${O}_{2}{E}_{2}\perp {E}_{1}{E}_{2}$, ${E}_{2}{F}_{2}\perp {O}_{2}{F}_{2}$ and ${O}_{2}{D}_{1}\perp {O}_{1}{D}_{1}$ gives the following geometrical relationships for locating the cable: (1) ${F}_{1}{E}_{1}={F}_{2}{E}_{2}=-Z$; (2) $\u25b5{O}_{1}D{F}_{1}\sim \u25b5{O}_{2}D{F}_{2}$; (3) ${F}_{1}{D}_{1}={O}_{2}{F}_{2}$; (4) $\angle {O}_{2}{O}_{1}{D}_{1}={\psi}_{Br}$.

**Figure 3.**For the tri-axial Magnetometer 1, the induced electromotive force projected in the horizontal plane is perpendicular to the cable. Obviously, $tan{\gamma}_{1}=\frac{{A}_{1}}{{B}_{1}}$ and ${\gamma}_{1}={\psi}_{Br}$.

**Figure 4.**Assume that the cable is located in the middle of two tri-axial magnetometers; there is ${C}_{1}<0$ and ${C}_{2}>0$. Hence, $tan({\delta}_{1})=\frac{\sqrt{{A}_{1}^{2}+{B}_{1}^{2}}}{-{C}_{1}}$ and $tan({\delta}_{2})=\frac{\sqrt{{A}_{2}^{2}+{B}_{2}^{2}}}{{C}_{2}}$. In addition, ${\alpha}_{1}={\delta}_{1}$ and ${\alpha}_{2}={\delta}_{2}$.

**Figure 5.**Let the position and course angle of the under-actuated AUV be denoted by $O={(x,y,{\psi}_{w})}^{\top}$ in the inertial frame $\{I\}$, and let the position and orientation of the projection point of the AUV on the path be denoted by $P={({x}_{r},{y}_{r},{\psi}_{r})}^{\top}$ in the inertial frame $\{I\}$. The side-slip angle $\beta =arctan(\frac{v}{u})$. In the Serret–Frenet frame $\{F\}$ with the origin at P, the LOS guidance angle is defined as $arctan(\frac{-Y}{\Delta})$, where Y is the horizontal offset of the AUV relative to the cable, and the look ahead distance Δ is constant. The choice of Δ is instrumental to shape the AUV moving towards the straight-line path.

**Figure 6.**(

**a**) Three-dimensional view of the magnetic sensing; (

**b**) Two-dimensional vertical view; and (

**c**) Two-dimensional horizontal view. There are $\sqrt{{A}_{1}^{2}+{B}_{1}^{2}}=\frac{{k}_{\epsilon}}{{R}_{1}}sin({\delta}_{1})$ with ${\delta}_{1}={\alpha}_{1}=arcsin(\frac{{z}_{r}-z}{{R}_{1}})$, ${A}_{1}=\sqrt{{A}_{1}^{2}+{B}_{1}^{2}}sin({\gamma}_{1})$ with ${\gamma}_{1}={\psi}_{Br}={\psi}_{B}-{\psi}_{r}$, ${B}_{1}=\sqrt{{A}_{1}^{2}+{B}_{1}^{2}}cos({\gamma}_{1})$, and ${C}_{1}=\sqrt{\frac{{k}_{\epsilon}^{2}}{{R}_{1}^{2}}-{A}_{1}^{2}-{B}_{1}^{2}}=\frac{{k}_{\epsilon}}{{R}_{1}}\sqrt{(1-\frac{{({z}_{r}-z)}^{2}}{{R}_{1}^{2}})}$.

**Figure 7.**(

**a**) Three-dimensional view of the cable tracking; and (

**b**) Two-dimensional projection in the X-Yplane. These two pictures intuitively describe the cable tracking process, where the green line represents the underwater cable, the red dashed line represents the desired tracking path and the blue line represents the actual path of the under-actuated AUV.

**Figure 8.**(

**a**) The induced electromotive forces measured by the port tri-axial magnetometer; and (

**b**) The induced electromotive forces measured by the starboard tri-axial magnetometer in the case of ${k}_{\epsilon}=1$. The red dashed line, the green dash dotted line and the blue line represent the force components along the x-axis, y-axis and z-axis of the tri-axial magnetometer, respectively.

**Figure 9.**(

**a**) The angle ${\psi}_{Br}$ between the AUV heading and the direction of the cable; (

**b**) The horizontal offset Y between the AUV and the cable; and (

**c**) The vertical distance Z between the AUV and the cable.

**Figure 10.**The red dashed line, the blue line and the green line represent the desired yaw angle, the actual yaw angle and the course angle of the AUV, respectively, and the pink dash dotted line represents the orientation angle of the cable.

**Figure 11.**(

**a**) Surge linear velocity u; (

**b**) Sway linear velocity v; (

**c**) Yaw angular velocity r; and (

**d**) Resultant velocity ${v}_{t}$.

**Figure 14.**(

**a**) Three-dimensional view of the cable tracking; and (

**b**) Two-dimensional projection in the X-Y plane. These two pictures intuitively describe the cable tracking process, in which the green line represents the underwater cable, the red dashed line represents the desired tracking path, the magenta dot-dashed line represents the actual path of the under-actuated AUV without sensor noise and the blue line represents the actual path of the under-actuated AUV with sensor noise.

**Figure 15.**(

**a**) The induced electromotive forces measured by the port tri-axial magnetometer; and (

**b**) The induced electromotive forces measured by the starboard tri-axial magnetometer in the case of ${k}_{\epsilon}=1$. The red dashed line, the green dash dotted line and the blue line represent the force components along the x-axis, y-axis and z-axis of the tri-axial magnetometer, respectively.

**Figure 16.**(

**a**) The angle ${\psi}_{Br}$ between the AUV heading and the direction of the cable; (

**b**) The horizontal offset Y between the AUV and the cable; and (

**c**) The vertical distance Z between the AUV and the cable. Two simulation cases are plotted in the same figure, in which the solid line and dot-dashed line represent the error without noise and the error with noise, respectively.

**Table 1.**The hydrodynamic parameters of the under-actuated AUV moving in the horizontal plane are obtained from Chapter 12 in [57].

Parameter | Value | Unit |
---|---|---|

L | 1.5 | m |

${m}_{11}$ | 1116 | kg |

${m}_{22}$ | 2133 | kg |

${m}_{33}$ | 4061 | kg·m^{2} |

${d}_{11}$ | 25.5 | kg·s^{−1} |

${d}_{22}$ | 138 | kg·s^{−1} |

${d}_{33}$ | 490 | kg·m^{2}·s^{−1} |

${d}_{u2}$ | 0 | kg·m^{−1} |

${d}_{u3}$ | 0 | kg·m^{−2}·s |

${d}_{v2}$ | 920.1 | kg·m^{−1} |

${d}_{v3}$ | 750 | kg·m^{−2}·s |

${d}_{r2}$ | 0 | kg·m^{2} |

${d}_{r3}$ | 0 | kg·m^{2}·s |

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## Share and Cite

**MDPI and ACS Style**

Xiang, X.; Yu, C.; Niu, Z.; Zhang, Q. Subsea Cable Tracking by Autonomous Underwater Vehicle with Magnetic Sensing Guidance. *Sensors* **2016**, *16*, 1335.
https://doi.org/10.3390/s16081335

**AMA Style**

Xiang X, Yu C, Niu Z, Zhang Q. Subsea Cable Tracking by Autonomous Underwater Vehicle with Magnetic Sensing Guidance. *Sensors*. 2016; 16(8):1335.
https://doi.org/10.3390/s16081335

**Chicago/Turabian Style**

Xiang, Xianbo, Caoyang Yu, Zemin Niu, and Qin Zhang. 2016. "Subsea Cable Tracking by Autonomous Underwater Vehicle with Magnetic Sensing Guidance" *Sensors* 16, no. 8: 1335.
https://doi.org/10.3390/s16081335