# Local Wireless Sensor Networks Positioning Reliability Under Sensor Failure

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## Abstract

**:**

## 1. Introduction

## 2. Taylor-Based Positioning Algorithm in Time Difference of Arrival (TDOA) Systems

## 3. Cramer Rao Lower Bound (CRLB) Modeling in TDOA Systems

## 4. Genetic Algorithm (GA) Optimization

## 5. Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AGVs | Automatic Ground Vehicles |

CRLB | Cramer Rao Lower Bound |

FIM | Fisher Information Matrix |

GA | Genetic Algorithm |

GNSS | Global Navigation Satellite Systems |

LPS | Local Positioning Systems |

NLOS | Non-Line-of-Sight |

PDOP | Position Dilution of Precision |

RMSE | Root Mean Square Error |

TDOA | Time Difference of Arrival |

TOA | Time of Arrival |

TS | Target Sensor |

UAVs | Unmanned Aerial Vehicles |

WGN | White Gaussian Noise |

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**Figure 2.**The scenario of simulations. The reference surface is depicted is grey tones. Node Location Environment (NLE) and Target Location Environment (TLE) regions are respectively shown in orange and purple colors. The discretized points of the TLE zone are the points employed for the optimization of the Time Difference of Arrival (TDOA) architecture performance. In the case of the NLE area, the points shown are only a representation of the area where every sensor can be located.

**Figure 3.**Accuracy analysis in terms of Cramer Rao Lower Bound (CRLB) in meters for a random sensor distribution of five sensors, under the assumption of one randomly malfunction sensor. Black spheres indicate the location of active sensors and red spheres highlights the sensor which is not available. Red tones in the color bar indicate bad accuracy evaluations, while green tones imply acceptable accuracy values.

**Figure 4.**Convergence radius analysis in meters for a random sensor distribution of 5 sensors, under the assumption of one randomly malfunction sensor. The convergence radius represents the maximum radius of the sphere of convergence in which every inside point used as initial iterating point of the positioning algorithm guarantees the unequivocal position determination by using the four available sensors. It represents the same operating condition than Figure 2. Red tones in the color bar indicate bad convergence radius values, while green tones imply acceptable convergence magnitudes.

**Figure 5.**Accuracy analysis in terms of CRLB in meters for the optimized distribution of 5 sensors under possible failure. The condition represented corresponds with the Case I - Sensor Fail 1 of Table 3. Red tones in the color bar indicate badly accuracy evaluations, while green tones imply acceptable accuracy values.

**Figure 6.**Convergence radius analysis in meters for the optimized distribution of 5 sensors under possible failure. The condition represented corresponds with the Case I - Sensor Fail 5 of Table 3. Red tones in the color bar indicate badly convergence radius values, while green tones imply acceptable convergence magnitudes.

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

Transmission power | 100 W |

Mean noise power | −94 dBm |

Frequency of emission | 1090 MHz |

Bandwidth | 100 MHz |

Path loss exponent | 2.05 |

Antennae gains | Unity |

Time-Frequency product | 1 |

GA | Selection |
---|---|

Population size | 90 |

Selection technique | Tournament 2 |

% Elitism | 5 |

Crossover technique | Single-point |

% Mutation | 3 |

Convergence criteria | 80% individuals equals |

Parameter Considered | Case I | Case II |
---|---|---|

Nominal Operating Conditions (5 sensors distribution) | ✓ | ✓ |

Failure Conditions (4 sensors distributions) | ✓ | X |

Convergence Maximization | ✓ | X |

Sensor Distributions | Sensor Fail | CRLB Evaluation TDOA (meters) | Convergence Evaluation (meters) | ||||
---|---|---|---|---|---|---|---|

Max | Mean | Min | Max | Mean | Min | ||

Case I | Sensor 1 | 62.408 | 0.651 | 0.233 | 300 | 138.684 | 35 |

Sensor 2 | 133.556 | 0.875 | 0.216 | 240 | 125.786 | 40 | |

Sensor 3 | 117.304 | 0.627 | 0.223 | 280 | 154.237 | 40 | |

Sensor 4 | 191.480 | 2.005 | 0.196 | 300 | 138.851 | 35 | |

Sensor 5 | 188.676 | 7.425 | 0.237 | 220 | 129.149 | 4 | |

None | 0.795 | 0.326 | 0.154 | 300 | 140.229 | 40 | |

Case II | Sensor 1 | 206.049 | 1.340 | 0.225 | 240 | 103.711 | 2 |

Sensor 2 | 159.772 | 1.512 | 0.149 | 280 | 84.650 | 2 | |

Sensor 3 | 65.487 | 1.688 | 0.169 | 220 | 102.037 | 4 | |

Sensor 4 | 199.168 | 0.629 | 0.182 | 260 | 113.604 | 2 | |

Sensor 5 | 2340.42 | 9.674 | 0.181 | 240 | 70.850 | 2 | |

None | 0.872 | 0.312 | 0.143 | 300 | 128.306 | 10 |

**Table 5.**Comparative between the optimizations of Case I and II. Values presented show the comparison in relative terms of the failure consideration distribution regarding the optimization for normal operation of the system.

Performance Analysis | Case I | Case II | Sensor Distribution: Case I vs Case II | |
---|---|---|---|---|

Mean CRLB Evaluation TDOA (meters) | Failure conditions | 2.316 | 2.969 | −22.0 % |

Non-Failure conditions | 0.326 | 0.312 | +4.3 % | |

Mean Convergence Evaluation (meters) | Failure conditions | 137.341 | 94.970 | +30.9 % |

Non-Failure conditions | 140.229 | 128.306 | +8.5 % |

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**MDPI and ACS Style**

Díez-González, J.; Álvarez, R.; Prieto-Fernández, N.; Perez, H. Local Wireless Sensor Networks Positioning Reliability Under Sensor Failure. *Sensors* **2020**, *20*, 1426.
https://doi.org/10.3390/s20051426

**AMA Style**

Díez-González J, Álvarez R, Prieto-Fernández N, Perez H. Local Wireless Sensor Networks Positioning Reliability Under Sensor Failure. *Sensors*. 2020; 20(5):1426.
https://doi.org/10.3390/s20051426

**Chicago/Turabian Style**

Díez-González, Javier, Rubén Álvarez, Natalia Prieto-Fernández, and Hilde Perez. 2020. "Local Wireless Sensor Networks Positioning Reliability Under Sensor Failure" *Sensors* 20, no. 5: 1426.
https://doi.org/10.3390/s20051426