# Improving UWB-Based Localization in IoT Scenarios with Statistical Models of Distance Error

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## Abstract

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## 1. Introduction

## 2. Statistical Model of the Range Estimation Error: Experimental Derivation

## 3. Localization: General Problem Formulation and Selected Algorithms

#### 3.1. Circumference Intersection (CI) Algorithm

#### 3.2. Two-Stage Maximum-Likelihood (TSML) Algorithm

## 4. Localization: Experimental Performance Investigation

#### 4.1. “Good” Geometrical Scenario

#### 4.2. “Bad” Geometrical Scenario

#### 4.3. Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

UWB | Ultra Wide Band |

ToF | Time of Flight |

FCC | Federal Communications Commission |

WSN | Wireless Sensor Network |

RSS | Received Signal Strength |

CI | Circumference Intersection |

TSML | Two-Stage Maximum-Likelihood |

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**Figure 2.**Scenario of the experimental setup: the position of the requester (purple star) and the 10 positions of the responder (blue dots) are shown.

**Figure 4.**The values of the average range estimation error ${\overline{\nu}}_{k}$ (blue circles) and of its linear approximation ${\overline{\nu}}_{k}^{(\mathrm{LS})}$ (red pluses) are shown as functions of the true distance ${r}_{k}$ between the two sensors.

**Figure 5.**Considered geometrical scenarios: (

**a**) “good” and (

**b**) “bad.” In both cases, the true position of the TN (red star) is shown, together with the positions of three ANs (blue stars). The distances ${\{{r}_{i}\}}_{i=1}^{3}$ are also shown.

**Figure 6.**The 1000 position estimates obtained with the CI localization algorithm are shown (magenta plus), together with the 1000 position estimates obtained when taking into account the statistical model for range estimates (green circles) and the TN (blue star).

**Figure 7.**CDFs of the distance errors obtained with the CI localization algorithm in Figure 6 without (solid line) and with (dashed line) the application of the statistical model on the distance estimation error.

**Figure 8.**The 1000 position estimates obtained with the TSML localization algorithm are shown (magenta plus), together with the 1000 position estimates obtained when taking into account the statistical model for range estimates (green circles) and the TN (blue star).

**Figure 9.**CDFs of the distance errors obtained with the TSML localization algorithm in Figure 8 without (solid line) and with (dashed line) the statistical model.

**Figure 10.**The 1000 position estimates obtained with the CI localization algorithm are shown (magenta plus), together with the 1000 position estimates obtained when taking into account the statistical model for range estimates (green circles) and the TN (blue star).

**Figure 11.**CDFs of the distance errors obtained with the CI localization algorithm in Figure 10 without (solid line) and with (dashed line) the statistical model.

**Figure 12.**The 1000 position estimates obtained with the TSML localization algorithm are shown (magenta plus), together with the 1000 position estimates obtained when taking into account the statistical model for range estimates (green circles) and the TN (blue star).

**Figure 13.**CDFs of the distance errors obtained with the TSML localization algorithm in Figure 12 without (solid line) and with (dashed line) the statistical model.

**Table 1.**This table shows the values of ${\overline{\nu}}_{k}$ (second column), their linear approximation ${\overline{\nu}}_{k}^{(LS)}$ (third column), and the absolute value $\Delta {\overline{\nu}}_{k}$ of their difference (fourth column), as a function of the true distances ${r}_{k}$.

${\mathit{r}}_{\mathit{k}}$ (m) | ${\overline{\mathit{\nu}}}_{\mathit{k}}$ (mm) | ${\overline{\mathit{\nu}}}_{\mathit{k}}^{(\mathit{LS})}$ (mm) | $\Delta {\overline{\mathit{\nu}}}_{\mathit{k}}$ (mm) |
---|---|---|---|

1 | −144 | −122 | 22 |

2 | −70 | −105 | 35 |

3 | −89 | −89 | 0 |

4 | −112 | −72 | 40 |

5 | −51 | −55 | 4 |

6 | −17 | −38 | 21 |

7 | 12 | −22 | 34 |

8 | −13 | −5 | 8 |

9 | −6 | 12 | 18 |

10 | 24 | 28 | 4 |

**Table 2.**The average and the maximum distance between the true TN position and its estimates with (${\stackrel{\u02c7}{d}}_{\mathrm{avg}}$, ${\stackrel{\u02c7}{d}}_{\mathrm{max}}$) and without (${\widehat{d}}_{\mathrm{avg}}$, ${\widehat{d}}_{\mathrm{max}}$) the use of the statistical model, when using the CI algorithm and the TSML algorithm, are shown.

Scenario 1 | CI Algorithm | TSML Algorithm |
---|---|---|

${\widehat{d}}_{\mathrm{avg}}$ (mm) | 90 | 40 |

${\stackrel{\u02c7}{d}}_{\mathrm{avg}}$ (mm) | 28 | 26 |

${\widehat{d}}_{\mathrm{max}}$ (mm) | 166 | 104 |

${\stackrel{\u02c7}{d}}_{\mathrm{max}}$ (mm) | 129 | 101 |

**Table 3.**The average and the maximum distance between the true TN position and its estimates with (${\stackrel{\u02c7}{d}}_{\mathrm{avg}}$, ${\stackrel{\u02c7}{d}}_{\mathrm{max}}$) and without (${\widehat{d}}_{\mathrm{avg}}$, ${\widehat{d}}_{\mathrm{max}}$) the use of the statistical model, when using the CI algorithm and the TSML algorithm, are shown.

Scenario 2 | CI Algorithm | TSML Algorithm |
---|---|---|

${\widehat{d}}_{\mathrm{avg}}$ (mm) | 153 | 150 |

${\stackrel{\u02c7}{d}}_{\mathrm{avg}}$ (mm) | 62 | 68 |

${\widehat{d}}_{\mathrm{max}}$ (mm) | 229 | 210 |

${\stackrel{\u02c7}{d}}_{\mathrm{max}}$ (mm) | 130 | 112 |

**Table 4.**Values of ${{[\widehat{d}]}_{j}}^{\mathrm{th}}$, expressed in mm, such that $\mathbb{P}({[\widehat{d}]}_{j}<{{[\widehat{d}]}_{j}}^{\mathrm{th}})=0.9$ (columns 1, 4, 7, 10); values of ${{[\stackrel{\u02c7}{d}]}_{j}}^{\mathrm{th}}$, expressed in mm, such that $\mathbb{P}({[\stackrel{\u02c7}{d}]}_{j}<{{[\stackrel{\u02c7}{d}]}_{j}}^{\mathrm{th}})=0.9$ (columns 2, 5, 8, 11); and their differences $\Delta \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$ (columns 3, 6, 9, 12) are shown for different scenarios and algorithms.

Good Scenario | Bad Scenario | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

CI | TSML | CI | TSML | ||||||||

${{[\widehat{d}]}_{j}}^{\mathrm{th}}$ | ${{[\stackrel{\u02c7}{d}]}_{j}}^{\mathrm{th}}$ | $\Delta $ | ${{[\widehat{d}]}_{j}}^{\mathrm{th}}$ | ${{[\stackrel{\u02c7}{d}]}_{j}}^{\mathrm{th}}$ | $\Delta $ | ${{[\widehat{d}]}_{j}}^{\mathrm{th}}$ | ${{[\stackrel{\u02c7}{d}]}_{j}}^{\mathrm{th}}$ | $\Delta $ | ${{[\widehat{d}]}_{j}}^{\mathrm{th}}$ | ${{[\stackrel{\u02c7}{d}]}_{j}}^{\mathrm{th}}$ | $\Delta $ |

118 | 39 | 79 | 54 | 35 | 19 | 168 | 74 | 94 | 164 | 81 | 83 |

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**MDPI and ACS Style**

Monica, S.; Ferrari, G. Improving UWB-Based Localization in IoT Scenarios with Statistical Models of Distance Error. *Sensors* **2018**, *18*, 1592.
https://doi.org/10.3390/s18051592

**AMA Style**

Monica S, Ferrari G. Improving UWB-Based Localization in IoT Scenarios with Statistical Models of Distance Error. *Sensors*. 2018; 18(5):1592.
https://doi.org/10.3390/s18051592

**Chicago/Turabian Style**

Monica, Stefania, and Gianluigi Ferrari. 2018. "Improving UWB-Based Localization in IoT Scenarios with Statistical Models of Distance Error" *Sensors* 18, no. 5: 1592.
https://doi.org/10.3390/s18051592