# A Modified Magnetic Gradient Contraction Based Method for Ferromagnetic Target Localization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. STAR Concept and Asphericity Errors

## 3. Asphericity Errors Compensation Algorithm

- Step 1:
- The initial value of position vector $\stackrel{\rightharpoonup}{R}$ is estimated by STAR method.
- Step 2:
- Substituting position vector $\stackrel{\rightharpoonup}{R}$ into Equation (3), the magnetic moment vector $\stackrel{\rightharpoonup}{M}$ is estimated by least square method.
- Step 3:
- The parameters ${C}_{1}$ and ${C}_{2}$ in Equation (16) are calculated with the estimated values of $\stackrel{\rightharpoonup}{R}$ and $\stackrel{\rightharpoonup}{M}$.
- Step 4:
- The new unit bearing vector is calculated by the following expression.$$\stackrel{\rightharpoonup}{{r}_{\mathrm{new}}}=\frac{\sqrt{{C}_{1}^{2}+{C}_{2}^{2}+2{C}_{1}{C}_{2}\mathrm{cos}\theta}\xb7\stackrel{\rightharpoonup}{V}-{C}_{2}\stackrel{\rightharpoonup}{{m}_{0}}}{{C}_{1}}$$

## 4. Simulation Study

^{2}. In the first simulation experiment, the root mean square (RMS) of magnetometers noise was 0.05 nT. The simulation results were shown in Figure 2.

^{2}, shown as Figure 3b. The estimated values of the modified STAR method were consistent with the true values.

## 5. Experiment Result

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**The estimated positions of two methods. (

**a**) The estimated positions on the XY plane; (

**b**) The estimated positions on the Z-axis.

**Figure 3.**The estimated magnetic moment of the two methods. (

**a**) X-component; (

**b**) Y-component; (

**c**) Z-component.

**Figure 4.**The error distributions of these two methods: (

**a**) The error distributions of these two methods on XY plane; (

**b**) The error distributions of these two methods in Z-axis.

**Figure 5.**The maximum relative error between the estimated parameters and the true values versus SNR using synthetic data with additive Gaussian noise.

Hardware | Manufacturer | Model Number |
---|---|---|

Magnetometer | Bartington | Mag-03MSL100 |

Data acquisition | National Instruments | NI PXIe-4497 and NI PXI-4462 |

Localization Method | X-Component (cm) | Y-Component (cm) | Z-Component (cm) |
---|---|---|---|

Modified STAR | 3.25 | 2.08 | 2.03 |

STAR | 5.56 | 6.79 | 10.67 |

Methods | Computation Times |
---|---|

The traditional STAR method | 3.13 (ms) |

One iteration of the proposed method | 0.78 (ms) |

The proposed method | 6.25 (ms) |

Localization Method | X-Magnetic Moment (Am^{2}) | Y-Magnetic Moment (Am^{2}) | Z-Magnetic Moment (Am^{2}) | |||
---|---|---|---|---|---|---|

Mean Value | RMS | Mean Value | RMS | Mean Value | RMS | |

Modified STAR | 0.08 | 0.23 | 0.35 | 0.18 | −1.54 | 0.16 |

STAR | 0.35 | 0.38 | −0.04 | 0.48 | −1.26 | 0.38 |

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**MDPI and ACS Style**

Wang, C.; Zhang, X.; Qu, X.; Pan, X.; Fang, G.; Chen, L. A Modified Magnetic Gradient Contraction Based Method for Ferromagnetic Target Localization. *Sensors* **2016**, *16*, 2168.
https://doi.org/10.3390/s16122168

**AMA Style**

Wang C, Zhang X, Qu X, Pan X, Fang G, Chen L. A Modified Magnetic Gradient Contraction Based Method for Ferromagnetic Target Localization. *Sensors*. 2016; 16(12):2168.
https://doi.org/10.3390/s16122168

**Chicago/Turabian Style**

Wang, Chen, Xiaojuan Zhang, Xiaodong Qu, Xiao Pan, Guangyou Fang, and Luzhao Chen. 2016. "A Modified Magnetic Gradient Contraction Based Method for Ferromagnetic Target Localization" *Sensors* 16, no. 12: 2168.
https://doi.org/10.3390/s16122168