# A Fuzzy Cooperative Localisation Framework for Underwater Robotic Swarms

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Underwater Localisation Methods

#### 2.1. Time of Flight (ToF) Acoustic Navigation

#### 2.2. Inertial Navigation System

#### 2.3. Least-Squares Trilateration

## 3. Fuzzy-Based Localisation

`IF Operational Depth is Shallow AND Battery Level is High THEN Localisation Method is L.`

## 4. Simulation

#### 4.1. Simulation Platform

**dB**re $\mathsf{\mu}$Pa

**@**1 m would be ${\mathbf{S}}_{\mathbf{L}}-{\mathbf{T}}_{\mathbf{L}}-{\mathbf{N}}_{\mathbf{L}}$.

#### 4.2. Implementation

#### 4.3. Simulation Scenario and Settings

#### 4.4. Results and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1.

- IF $\mathcal{D}$ is Shallow AND $\mathcal{R}$ is Short THEN $\mathcal{Y}$ is ${L}_{1}$.
- IF $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Not Enough THEN $\mathcal{Y}$ is ${L}_{1}$.
- IF $\mathcal{B}$ is Low AND $\mathcal{U}$ is Available AND $\mathcal{G}$ is Not Enough AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is Low AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{U}$ is Available AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{G}$ is Not Enough AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is High AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{D}$ is Shallow AND $\mathcal{B}$ is High AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{D}$ is Shallow AND $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{U}$ is Available AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is Low AND $\mathcal{U}$ is Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{D}$ is Deep AND $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{D}$ is Deep AND $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{D}$ is Shallow AND $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is High AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{D}$ is Deep AND $\mathcal{B}$ is High AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{1}$.
- IF $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is High AND $\mathcal{U}$ is Available AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{B}$ is High AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Long THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Mid THEN $\mathcal{Y}$ is ${L}_{3}$.
- IF $\mathcal{D}$ is Deep AND $\mathcal{U}$ is Available THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{D}$ is Deep AND $\mathcal{U}$ is Available THEN $\mathcal{Y}$ is ${L}_{2}$.
- IF $\mathcal{D}$ is Deep AND $\mathcal{U}$ is Not Available AND $\mathcal{G}$ is Enough AND $\mathcal{R}$ is Short THEN $\mathcal{Y}$ is ${L}_{1}$.

#### Appendix A.2.

**Hypothesis**

**A1 (HA1).**

**Hypothesis**

**A2 (HA2).**

Swarm Size | H0 REJECTED | p-Value | Degree of Freedom | t-Statistics | Critical Value |
---|---|---|---|---|---|

50 | No | 1.0000 | 75.45 | −4.35 | 1.6653 |

100 | Yes | 0.0252 | 194.48 | 1.96 | 1.6527 |

150 | Yes | 0.0324 | 255.45 | 1.85 | 1.6508 |

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**Figure 1.**Underwater wireless network of Autonomous Underwater Vehicles (yellow vessels with blue fins) and anchored sensors (red).

**Figure 3.**Two-dimensional trilateration problem of determining an object (red solid circle) location $\mathbf{X}$ given the location of three stations (green diamonds) ${A}_{i}$ and the range measurements/distances ${d}_{i}$ ($i=1,2,3$).

**Figure 4.**Histograms of mean multilateration error of around 15,000 multilateration process carried out in 100 walkers (nodes); the vertical line in each histogram represents the mean error of the entire simulation.

**Figure 5.**An example of input-output mapping for the localisation problem: “Given three variable inputs what the localisation plan should be?”

**Figure 7.**Underwater Webots simulation scene of 50 AUVs deployment, the USBL transceiver is hull-mounted on the deployment vessel.

**Figure 8.**The implemented simulation platform used to validate the proposed navigation algorithm. Webots robotic simulator is employed for physics simulation and UnetStack is employed for simulating the underwater acoustic communication properties.

**Figure 9.**Angular velocity, acceleration, and local magnetic field of the earth are generated as ground truth readings at each time instant of each IMU that is modelled in an underwater environment in Webots simulator. Given the ground truth readings and the IMU properties, a realistic nine-axis IMU is modelled on MATLAB Navigation toolbox.

**Figure 10.**Fuzzy inference variable inputs of each AUV in a swarm of AUVs based on its on-board acoustic communication modems and sensors.

**Figure 11.**Fuzzy-based underwater localisation approach (Decision-making) with five variable inputs & their fuzzy/crisp sets and three underwater location estimators. The aggregated fuzzy output shows an example of a final localisation plan of $(\frac{2}{3}{L}_{1}+\frac{1}{3}{L}_{3})$.

**Figure 12.**An Example of round-robin scheduling for USBL navigation aid (represented by the green bar) in a swarm of AUVs. The USBL, in this example, can only navigationally aid five AUVs in a single TDMA frame of $\Delta T$.

**Figure 13.**The proposed fuzzy-based localisation framework harnesses round-robin scheduling for USBL navigation aid assuming that all AUVs requested USBL navigation aid. The USBL can only navigatioanlly aid five AUVs in a single TDMA frame of $\Delta T$ utilising low-frequency ACOMMS (in black arrows). NBs (in blue) broadcast localisation aid to their neighbouring AUV utilising high-frequency ACOMMS (in red arrows).

**Figure 14.**The entire swarm mean and standard deviation localisation error in both the proposed Fuzzy-based USBL/trilateration aided DR navigation (in blue) and round-robin EKF-based USBL/trilateration aided DR navigation (in red). The error bar around the mean point represents $2\sigma $ standard deviation.

**Figure 15.**Histogram of the mean localisation error and standard deviation of each AUV of the entire swarm of 150 AUVs. The vertical lines represent the entire swarm mean localisation error, which is 49.76 m and 59.62 m in fuzzy-based and round-robin EKF-based localisation, respectively.

**Figure 16.**Instantaneous localisation performance of an AUV in a swarm of 150 when the proposed fuzzy-based, round-robin EKF-based and DR-only methods are adopted. A time window of 500 s shows the fused external navigation aid.

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

Accelerometer Resolution | 60.958 $\mathsf{\mu}$g |

Accelerometer Constant Bias | 14 $\mathsf{\mu}$g |

Accelerometer Noise Density | 57 $\mathsf{\mu}$g/$\sqrt{Hz}$ |

Gyroscope Resolution | 0.0625${}^{\circ}$ |

Gyroscope Constant Bias | 7${}^{\circ}$/hour |

Gyroscope Noise Density | 0.15${}^{\circ}$/$\sqrt{\mathrm{hour}}$ |

Magnetometer Resolution | 1 mGauss |

Magnetometer Constant Bias | 1.5 mGauss |

Magnetometer Noise Density | 3 mGauss |

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

Swarm Size | 50; 100; 150 AUVs |

Simulation Time Step | 100 ms |

Clock-synchronisation error | 1.2 ms 1-$\sigma $ |

Seabed Depth | 1000 m |

Depth Sensor | 2 Hz, 0.1 m 1-$\sigma $ error |

USBL Transponder Communication Range | 6000 m |

USBL Localisation Accuracy in 1000 m | 2.7 m 1-$\sigma $ error |

Number of AUVs positioned by the USBL in a single TDMA frame | 10 AUVs |

USBL TDMA Frame length | 1 s |

USBL update rate | 4 s |

Number of NBs | 10 AUVs |

NBs broadcasting period | 1 s |

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

Communication modem Frequency band | 160 kHz |

Communication data rate | 50 kbit/s |

Navigation aid length and duration | 20 bytes; 3.2 ms |

Navigation aid allocated TDMA time-slot length | 20 ms |

Noise level | 60 dB |

Water salinity | 35 ppt |

Water temperature | 10 ${}^{\circ}$C |

Rician fading parameter | 10 |

Fast fading | enabled |

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Sabra, A.; Fung, W.-K.
A Fuzzy Cooperative Localisation Framework for Underwater Robotic Swarms. *Sensors* **2020**, *20*, 5496.
https://doi.org/10.3390/s20195496

**AMA Style**

Sabra A, Fung W-K.
A Fuzzy Cooperative Localisation Framework for Underwater Robotic Swarms. *Sensors*. 2020; 20(19):5496.
https://doi.org/10.3390/s20195496

**Chicago/Turabian Style**

Sabra, Adham, and Wai-Keung Fung.
2020. "A Fuzzy Cooperative Localisation Framework for Underwater Robotic Swarms" *Sensors* 20, no. 19: 5496.
https://doi.org/10.3390/s20195496