# Genetic Algorithm to Solve Optimal Sensor Placement for Underwater Vehicle Localization with Range Dependent Noises

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Novelty

- Optimization method
- (a)
- Analytical (Analy)
- (b)
- Heuristics and Meta-heuristics (HM)

- Optimization criteria
- (a)
- Determinant of the Fisher information matrix (D)
- (b)
- Trace of the inverse of the Fisher information matrix (A)
- (c)
- Eigenvalue of the inverse of the Fisher information matrix (E)
- (d)
- Multi-objective (Multi)

- Measurement noise considerations
- (a)
- Parameter independent (PI)
- (b)
- Parameter dependent (PD)

- Measurements
- (a)
- Range (R)
- (b)
- Range difference (RD)
- (c)
- Time of arrival (ToA)
- (d)
- Difference time of arrival (TDoA)
- (e)
- Angle of arrival (AoA)

#### 1.2. Major Contributions

## 2. Measurement Model and Problem Description

- the range measurement model of each sensor;
- a 3D grid of points;
- the positions q that define the trajectory of a target in the 3D grid of points.

#### 2.1. Evaluation Criteria

#### 2.2. Single-Objective Optimization

#### 2.3. Multi-Objective Optimization

## 3. Genetic Algorithm for Sensor Placement

## 4. Simulations and Results

#### 4.1. Simulation Setup

^{−1}.

#### 4.2. Comparison with Known Optimal Solution

#### 4.3. Sensors in a Plane with Lawn-Mower

#### 4.4. Sensors Restricted to Half of the Plane with Lawn-Mower

#### 4.5. Sensors in a Plane with Spiral Descent

#### 4.6. Pareto Front for Lawn-Mower Maneuver

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Tables of Sensor Positions

#### Appendix A.1. Lawn-Mower

**Figure A1.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and five sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 792 | 2232 |

2 | 2542 | 1247 |

3 | 1571 | 566 |

4 | 1983 | 2319 |

5 | 501 | 1056 |

**Figure A2.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and six sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 2220 | 808 |

2 | 2535 | 1833 |

3 | 441 | 1222 |

4 | 1240 | 643 |

5 | 1677 | 2373 |

6 | 728 | 2223 |

**Figure A3.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and seven sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 2123 | 2252 |

2 | 2596 | 1447 |

3 | 525 | 2039 |

4 | 2028 | 712 |

5 | 1190 | 661 |

6 | 1310 | 2373 |

7 | 454 | 1119 |

**Figure A4.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and eight sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 448 | 1856 |

2 | 516 | 1008 |

3 | 2527 | 1057 |

4 | 1886 | 650 |

5 | 1165 | 648 |

6 | 1829 | 2321 |

7 | 1083 | 2312 |

8 | 2490 | 1928 |

#### Appendix A.2. Lawn-Mower with Sensors in Half Plane

**Figure A5.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and five sensors constrained to half of the plane. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 708 | 2589 |

2 | 2007 | 1500 |

3 | 1000 | 1500 |

4 | 2241 | 2640 |

5 | 1536 | 2884 |

**Figure A6.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and six sensors constrained to half of the plane. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 2002 | 1500 |

2 | 537 | 2453 |

3 | 1494 | 2942 |

4 | 2421 | 2509 |

5 | 1509 | 2936 |

6 | 1013 | 1500 |

**Figure A7.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and seven sensors constrained to half of the plane. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 924 | 2732 |

2 | 2056 | 2739 |

3 | 2213 | 1500 |

4 | 1497 | 1500 |

5 | 768 | 1500 |

6 | 2082 | 2726 |

7 | 933 | 2739 |

**Figure A8.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and eight sensors constrained to half of the plane. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 1012 | 2808 |

2 | 648 | 2532 |

3 | 1580 | 2925 |

4 | 1516 | 1500 |

5 | 796 | 1500 |

6 | 2226 | 1500 |

7 | 1959 | 2830 |

8 | 2314 | 2604 |

#### Appendix A.3. Spiral Descent

Sensor | X Position | Y Position |
---|---|---|

1 | 1689 | 885 |

2 | 2192 | 1698 |

3 | 998 | 1313 |

4 | 1582 | 1538 |

5 | 1300 | 2047 |

**Figure A9.**Maximum axis of uncertainty along vehicle trajectory for the spiral descent maneuver and five sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

**Figure A10.**Maximum axis of uncertainty along vehicle trajectory for the spiral descent maneuver and six sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 1405 | 921 |

2 | 1581 | 1537 |

3 | 1202 | 1992 |

4 | 2116 | 1232 |

5 | 959 | 1423 |

6 | 1917 | 2047 |

**Figure A11.**Maximum axis of uncertainty along vehicle trajectory for the spiral descent maneuver and seven sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 989 | 1297 |

2 | 1581 | 1535 |

3 | 1995 | 1045 |

4 | 2185 | 1704 |

5 | 1094 | 1819 |

6 | 1617 | 2110 |

7 | 1410 | 897 |

**Figure A12.**Maximum axis of uncertainty along vehicle trajectory for the spiral descent maneuver and eight sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

Sensor | X Position | Y Position |
---|---|---|

1 | 1437 | 855 |

2 | 1024 | 1106 |

3 | 1579 | 1539 |

4 | 902 | 1530 |

5 | 1674 | 2106 |

6 | 1915 | 1025 |

7 | 1136 | 1945 |

8 | 2178 | 1600 |

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**Figure 3.**Maximum axis of uncertainty along vehicle trajectory for lawn-mower maneuver and four sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

**Figure 4.**Maximum axis of uncertainty along vehicle trajectory for the lawn-mower maneuver and four sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

**Figure 5.**Maximum axis of uncertainty along vehicle trajectory for the spiral descent maneuver and four sensors. Red dots represent the sensors, and the green gradient line represents the maximum axis of uncertainty along the vehicle trajectory.

**Figure 6.**Pareto front considering average maximum eigenvalue and determinant of inverse of FIM with respective positioning of sensors.

**Figure 7.**Pareto front considering average maximum eigenvalue and trace of inverse of FIM with respective positioning of sensors for the lawn-mower path.

**Table 1.**Differences between our work and some of the relevant literature. The checkmarks represent the points covered by a particular paper.

Ref | Method | Criteria | Noise | Measurement | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Analy | HM | D | A | E | Multi | PD | PI | R | RD | ToA | TDoA | AoA | |

Our work | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||

[26] | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||

[20] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||

[2] | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||

[6] | ✓ | ✓ | ✓ | ✓ | |||||||||

[7] | ✓ | ✓ | ✓ | ✓ | ✓ | ||||||||

[16] | ✓ | ✓ | |||||||||||

[9] | ✓ | ✓ | ✓ | ✓ |

# Sensors | Mean Deviation from Optimal (%) | Mean Computational Time (s) |
---|---|---|

4 | 0.073 | 7.43 |

5 | 0.056 | 8.11 |

6 | 0.071 | 8.67 |

7 | 0.039 | 9.28 |

8 | 0.045 | 9.28 |

Sensor | X Position | Y Position | Z Position |
---|---|---|---|

1 | 712 | 2126 | 0 |

2 | 827 | 746 | 0 |

3 | 2283 | 871 | 0 |

4 | 2168 | 2253 | 0 |

**Table 4.**Distance and measurements standard deviation to the center of the place for the solution with 4 sensors and the lawn-mower maneuver.

Sensor | Distance to Center (m) | ${\mathit{\sigma}}_{\mathit{c}}$ (m) |
---|---|---|

1 | 1350 | 10.25 |

2 | 1353 | 10.28 |

3 | 1348 | 10.24 |

4 | 1350 | 10.25 |

mean | 1350 | 10.25 |

# Sensors | Worst Axis of Uncertainty (m) | ${\mathit{\sigma}}_{\mathit{c}}$ (m) | Computational Time (s) |
---|---|---|---|

4 | 8.15 | 10.25 | 1521 |

5 | 7.08 | 10.30 | 1698 |

6 | 6.22 | 10.27 | 1804 |

7 | 5.76 | 10.25 | 1859 |

8 | 5.32 | 10.26 | 1977 |

Sensor | X Position | Y Position | Z Position |
---|---|---|---|

1 | 1989 | 1500 | 0 |

2 | 2030 | 2689 | 0 |

3 | 966 | 2693 | 0 |

4 | 996 | 1500 | 0 |

# Sensors | Worst Axis of Uncertainty (m) | ${\mathit{\sigma}}_{\mathit{c}}$ (m) | Computational Time (s) |
---|---|---|---|

4 | 11.41 | 9.94 | 1740 |

5 | 10.02 | 10.55 | 1834 |

6 | 9.32 | 10.97 | 1746 |

7 | 8.52 | 10.53 | 1771 |

8 | 8.00 | 10.84 | 1929 |

Sensor | X Position | Y Position | Z Position |
---|---|---|---|

1 | 1624 | 2181 | 0 |

2 | 1947 | 1075 | 0 |

3 | 1580 | 1547 | 0 |

4 | 958 | 1334 | 0 |

# Sensors | Worst Axis of Uncertainty (m) | Computational Time (s) |
---|---|---|

4 | 9.33 | 2390 |

5 | 7.53 | 2309 |

6 | 6.70 | 2575 |

7 | 611 | 2652 |

8 | 5.64 | 2875 |

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## Share and Cite

**MDPI and ACS Style**

Villa, M.; Ferreira, B.; Cruz, N.
Genetic Algorithm to Solve Optimal Sensor Placement for Underwater Vehicle Localization with Range Dependent Noises. *Sensors* **2022**, *22*, 7205.
https://doi.org/10.3390/s22197205

**AMA Style**

Villa M, Ferreira B, Cruz N.
Genetic Algorithm to Solve Optimal Sensor Placement for Underwater Vehicle Localization with Range Dependent Noises. *Sensors*. 2022; 22(19):7205.
https://doi.org/10.3390/s22197205

**Chicago/Turabian Style**

Villa, Murillo, Bruno Ferreira, and Nuno Cruz.
2022. "Genetic Algorithm to Solve Optimal Sensor Placement for Underwater Vehicle Localization with Range Dependent Noises" *Sensors* 22, no. 19: 7205.
https://doi.org/10.3390/s22197205