# A Self-Calibrating Localization Solution for Sport Applications with UWB Technology

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## Abstract

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## 1. Introduction

#### 1.1. Related Works

#### 1.2. Study Contributions

- Time of Flight (TOF) measurements among APs are used to reconstruct the network geometry in the SL procedure. An algorithm based on GN is used to handle the errors due to antenna delays in the TOF estimate. The algorithm relies on an iterative procedure for estimating the antenna delay by minimizing the difference between the true and estimated TOF at each UWB node. Compared with other methods such as [34], the proposed method converges to the optimal solution and can be applied to conventional ranging schemes, avoiding the definition of custom messages as in [37].
- Outlier identification and removal is here addressed by proposing a modified IFA. Compared with the standard IFA [45], the proposed algorithm can tackle a very-low-dimensional set of measurements while achieving high accuracy in detecting outliers. This method was designed due to the low number of TOF measurements available for anchors’ SL.
- For anchors’ SL, we developed an iterative algorithm that exploits the filtered TOF measurements to reconstruct the system geometry. The algorithm, inspired by [26], iteratively searches for an AP configuration that minimizes the residual error between the TOF measurements provided by the UWB system and the distances extracted from the positioned anchors. Rather than relying on complex optimization procedures, such as the ones proposed in [31], the developed algorithm relies on GN algorithm which is computationally efficient.
- Tag localization is achieved by multilateration of TDOA measurements. To compensate for the TDOA antenna delay at the anchors, we statistically modeled the TDOA measurements and extracted the corresponding delay through an inversion operation. Compared with the other approaches available in the literature, the proposed approach does not require any training procedure, as opposed to [40] or complex optimization procedures as in [39].

## 2. System and Measurement Models

#### 2.1. Measurements for AP Network Localization

#### 2.2. Measurements for Tag Localization

## 3. Antenna Delay Calibration

#### 3.1. ADS-TWR Antenna Delay Calibration

#### 3.2. TDOA Antenna Delay Calibration

## 4. Modified 1D IFA for Outlier Removal

**Isolation Tree (iTree)**: It is a binary tree, where each node has either zero or two child nodes. Nodes can be either external or internal depending on their position into the tree. An internal node is denoted as $\mathsf{intNode}({C}_{L},{C}_{R},\alpha )$, where ${C}_{L}$ and ${C}_{R}$ are the left and right child nodes, respectively; and $\alpha $ is the split value that defines the separation between ${C}_{L}$ data and ${C}_{R}$ data. An external node is denoted as $\mathsf{extNode}(set,size)$, being thus defined on the set of data points belonging to the extNode and its cardinality.**Path length**is denoted as $P\left({s}_{m}\right)$; it measures the depth of the data point ${s}_{m}$ in the iTree. Outliers typically have shorter path lengths because they are more likely to be isolated.**Isolation forest**: It is a set composed by a fixed number ${N}_{F}$ of iTrees that are generated on the same set of data $\mathcal{S}$.

Algorithm 1: iTree($\mathcal{S}$, n, ${L}_{\mathrm{MAX}}$) |

## 5. Localization Methods

#### 5.1. Anchors Self-Localization

Algorithm 2:Autolocalization($\overline{\mathcal{D}},{\u03f5}_{min},{I}_{max},\delta $) |

#### 5.2. Tag Localization

## 6. Experimental Setup and Performance Metrics

## 7. Results and Discussion

## 8. Conclusions and Future Studies

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Abbreviation | Definition |

ADC | Antenna Delay Calibration |

ADS-TWR | Asymmetric Double-Sided Two-Way Ranging |

AOA | Angle of Arrival |

AP | Access Point |

CEP | Circular Error Probable |

CDF | Cumulative Distribution Function |

GN | Gauss–Newton |

GNSS | Global Navigation Satellite System |

IFA | Isolation Forest Algorithm |

IoT | Internet of Things |

LM | Levenberg–Marquardt |

LS | Least Square |

PSO | Particle Swarm Optimization |

RMSE | Root Mean Square Error |

RSS | Received Signal Strength |

RTT | Round Trip Time |

SL | Self-Localization |

TDMA | Time Division Multiple Access |

TDOA | Time Difference of Arrival |

TOA | Time of Arrival |

TOF | Time of Flight |

TWR | Two-Way Ranging |

UWB | Ultrawide-Band |

WSN | Wireless Sensor Network |

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**Figure 5.**Outlier filtering by the proposed 1D IFA: (

**a**) range measurements before outlier filtering and (

**b**) after outlier filtering.

**Figure 7.**Experimental area: (

**a**) picture of the area; (

**b**) ground-truth positions of tags and anchors.

**Figure 8.**Performance of outlier detection algorithm for ranging measurements: (

**a**) without outlier detection; (

**b**) with outlier detection.

**Figure 9.**CDF of the anchors’ SL position error considering different levels of correction. Dashed lines highlight the CEP 95 values.

**Figure 10.**CDF of the tag localization error ${e}_{u}$ for GM and LM algorithms with and without TDOA bias correction. Dashed lines indicate the CEP 95 values.

**Figure 11.**CDF of the localization error ${e}_{u}$ for GN and LM algorithms considering different corrections.

RMSE (m) | CEP 95 (m) | Mean Error (m) | |
---|---|---|---|

SL | 0.6154 | 1.3068 | 0.4708 |

SL + IFA | 0.3365 | 0.5223 | 0.3056 |

SL + ADS-TWR ADC + IFA | 0.1626 | 0.2482 | 0.1463 |

**Table 2.**Performance metrics of tag localization for GN and LM algorithms with real AP positions and with/without TDOA antenna delay correction.

RMSE (m) | CEP 95 (m) | Mean Error (m) | |
---|---|---|---|

GN with true AP positions | 0.3509 | 0.6887 | 0.2817 |

GN with true AP position + TDOA ADC | 0.2817 | 0.4875 | 0.2181 |

LM with true AP positions | 0.3654 | 0.7208 | 0.2959 |

LM with true AP positions + TDOA ADC | 0.2617 | 0.4965 | 0.2176 |

**Table 3.**Comparison of GN and LM algorithms for tag localization with/without outlier and antenna delay corrections.

RMSE (m) | CEP 95 (m) | Mean Error (m) | |
---|---|---|---|

GN with true AP positions | 0.2817 | 0.4875 | 0.2181 |

GN with SL | 0.6316 | 1.4089 | 0.4842 |

GN with SL + IFA | 0.3447 | 0.6381 | 0.2660 |

GN with SL + ADS-TWR ADC + IFA | 0.3072 | 0.5584 | 0.2487 |

LM with true AP positions | 0.2617 | 0.4965 | 0.2176 |

LM with SL | 0.6606 | 1.3991 | 0.4635 |

LM with SL + IFA | 0.3447 | 0.6887 | 0.2770 |

LM with SL + ADS-TWR ADC + IFA | 0.2993 | 0.5335 | 0.2494 |

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**MDPI and ACS Style**

Piavanini, M.; Barbieri, L.; Brambilla, M.; Cerutti, M.; Ercoli, S.; Agili, A.; Nicoli, M. A Self-Calibrating Localization Solution for Sport Applications with UWB Technology. *Sensors* **2022**, *22*, 9363.
https://doi.org/10.3390/s22239363

**AMA Style**

Piavanini M, Barbieri L, Brambilla M, Cerutti M, Ercoli S, Agili A, Nicoli M. A Self-Calibrating Localization Solution for Sport Applications with UWB Technology. *Sensors*. 2022; 22(23):9363.
https://doi.org/10.3390/s22239363

**Chicago/Turabian Style**

Piavanini, Marco, Luca Barbieri, Mattia Brambilla, Mattia Cerutti, Simone Ercoli, Andrea Agili, and Monica Nicoli. 2022. "A Self-Calibrating Localization Solution for Sport Applications with UWB Technology" *Sensors* 22, no. 23: 9363.
https://doi.org/10.3390/s22239363