# Processing Chain for Localization of Magnetoelectric Sensors in Real Time

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Magnetoelectric Sensor

## 3. Forward Problem

## 4. Localization Processing Chain

#### 4.1. Signal Generation and Equalizer

#### 4.2. Matched Filter

#### 4.3. Localization Algorithm

#### 4.4. Postprocessing

## 5. Measurements and Results

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TDMA | Time Division Multiple Access |

FDMA | Frequency Division Multiple Access |

CDMA | Code Division Multiple Access |

SNR | Signal-to-Noise Ratio |

## Appendix A. Localization Errors

**Figure A1.**Exemplary cost function. The sensor is located at point A. Due to the relatively coarse grid (black lines), there will be a localization error of at least the distance between the point B and point A. However, due to the shape of the cost function, the minimum that is crossing the grid lines—and thus the localization outcome—is at point C.

## Appendix B. Initialization of the Kalman Matrices

- State transition matrix$$\mathit{F}=\left[\begin{array}{ccc}{\mathit{I}}_{{N}_{\mathrm{m}}}& \Delta t{\mathit{I}}_{{N}_{\mathrm{m}}}& \frac{1}{2}\Delta {t}^{2}{\mathit{I}}_{{N}_{\mathrm{m}}}\\ {\mathbf{0}}_{{N}_{\mathrm{m}}\times {N}_{\mathrm{m}}}& {\mathit{I}}_{{N}_{\mathrm{m}}}& \Delta t{\mathit{I}}_{{N}_{\mathrm{m}}}\\ {\mathbf{0}}_{{N}_{\mathrm{m}}\times {N}_{\mathrm{m}}}& {\mathbf{0}}_{{N}_{\mathrm{m}}\times {N}_{\mathrm{m}}}& {\mathit{I}}_{{N}_{\mathrm{m}}}\end{array}\right]$$
- Measurement matrix$$\mathit{H}=\left[{\mathit{I}}_{{N}_{\mathrm{m}}},{\mathbf{0}}_{{N}_{\mathrm{m}}\times 2{N}_{\mathrm{m}}}\right]$$
- Covariance matrix of the process noise$$\mathit{Q}=\mathrm{E}\left\{{\mathit{n}}_{\mathrm{p}}\phantom{\rule{0.166667em}{0ex}}{\mathit{n}}_{\mathrm{p}}^{\mathrm{T}}\right\}=\left[\begin{array}{ccc}\frac{\Delta {t}^{4}}{4}{\mathit{I}}_{{N}_{\mathrm{m}}}& \frac{\Delta {t}^{3}}{2}{\mathit{I}}_{{N}_{\mathrm{m}}}& \frac{\Delta {t}^{2}}{2}{\mathit{I}}_{{N}_{\mathrm{m}}}\\ \frac{\Delta {t}^{3}}{2}{\mathit{I}}_{{N}_{\mathrm{m}}}& \frac{\Delta {t}^{2}}{2}{\mathit{I}}_{{N}_{\mathrm{m}}}& \Delta t{\mathit{I}}_{{N}_{\mathrm{m}}}\\ \frac{\Delta {t}^{2}}{2}{\mathit{I}}_{{N}_{\mathrm{m}}}& \Delta t{\mathit{I}}_{{N}_{\mathrm{m}}}& {\mathit{I}}_{{N}_{\mathrm{m}}}\end{array}\right]{\widehat{\sigma}}_{p}^{2}$$
- Covariance matrix of the measurement noise$$\mathit{R}=\mathrm{E}\left\{{\mathit{n}}_{\mathrm{m}}\phantom{\rule{0.166667em}{0ex}}{\mathit{n}}_{\mathrm{m}}^{\mathrm{T}}\right\}={\mathit{I}}_{{N}_{\mathrm{m}}}{\widehat{\sigma}}_{m}^{2}$$
- State covariance matrix$$\mathit{S}\left(0\right)={\mathit{I}}_{3{N}_{\mathrm{m}}}{\widehat{\sigma}}_{\mathrm{s}}^{2}$$

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**Figure 1.**General overview of a medical system operating with magnetic sensors. The measurements are performed simultaneously with localizing the sensors. After transforming the signals into the digital domain, the signals are processed and analyzed. Since the analysis of the measured magnetic signals is not in the focus of this contribution, the corresponding box is depicted in gray.

**Figure 2.**(

**a**) Exchange bias ΔE-effect sensor used in this study. The sensor is based on a cantilever of size 3 mm × 1 mm. The cantilever is placed on a printed circuit board and connected to a low-noise JFET charge amplifier [19]. The sensor is encapsulated by a brass cylinder for electrical shielding and mechanical protection. (

**b**) Visualization of the relationship between the sensitive and the long axis of the sensor. (

**c**) Magnitude and phase response of the sensor in the first bending mode applying a magnetic field of b

_{ac}= 1 μT. The sensor has a resonance frequency of f

_{r}= 7.712 kHz and a bandwidth of bw

_{−3 dB}= 10.2 Hz.

**Figure 3.**Real (

**a**) and schematic (

**b**) measurement setup for the localization of magnetoelectric sensors. The coils are placed outside of the localization area and transmit orthogonal signals, which are measured by the sensor. The localization area (box bounded by white stripes in (

**a**)) is of size 50 cm × 50 cm.

**Figure 4.**Coil impedances separated into magnitude and phase. In (

**a**) the whole spectrum from 100 Hz up to 1 MHz is shown, so that the resonance of the coils can be seen. In (

**b**) the frequency range is scaled to the frequency range of the excitation signals.

**Figure 5.**Processing chain for the localization of magnetoelectric sensors in real time. The input signal of the sensor is matched filtered with the equalized coil signals. The matched filter outputs at time lag zero are compared with the lead-field matrix entries. A first order Kalman filter smooths the estimated position-orientation-pairs over time to mitigate possible outliers.

**Figure 7.**Flow chart of the iterative localization approach for one time step k. As long as neither the maximum number of iterations nor the desired resolution is reached, the algorithm keeps refining the localization grid.

**Figure 8.**Graphical user interface of the real-time system used for localizing the magnetic sensors. The estimated position and orientation of the sensor is shown graphically in the 3D view and as text in the lower left corner. The number of iterations ${N}_{\mathrm{it}}$, the number of considered position-orientation pairs for refining the localization grid ${N}_{\mathrm{b}}$ and the desired resolution in position and orientation are adjustable during runtime.

**Figure 10.**Real and estimated position and orientation of ${\mathit{p}}_{\mathrm{s},3}$ over time. The variances in the localization result are due to the presence of noise and cross talk in the measurement hardware.

**Figure 11.**Mean localization errors for all tested position-orientation pairs. The index j depicts the respective position-orientation pair ${\mathit{p}}_{\mathrm{s},j}$ of the sensor as defined in Figure 9.

**Table 1.**Parameter of the coils and amplifier channels used in this study. The conversion factors of the coils describing the relationship between current and magnetic field are described by the column conversion. The gains of the coil amplifier channels are normalized to the maximum value (channel 6). Both values are determined at the resonance frequency of the sensor.

Number | Conversion (mT/A) (@7712 Hz) | Amplifier Gain (Relative, @7712 Hz) |
---|---|---|

1 | 13.2 | 0.9632 |

2 | 12.4 | 0.9823 |

3 | 12.8 | 0.9592 |

4 | 12.8 | 0.9751 |

5 | 12.2 | 0.9549 |

6 | 12.5 | 1.0000 |

Parameter | ${\mathit{N}}_{\mathbf{c}}$ | ${\mathit{L}}_{\mathbf{sig}}$ | ${\mathit{L}}_{\mathbf{mf}}$ | ${\mathit{f}}_{\mathbf{s}}$ (kHz) | ${\mathit{N}}_{\mathbf{it}}$ | ${\mathit{N}}_{\mathbf{b}}$ | ${\mathit{N}}_{\mathbf{p}}$ |
---|---|---|---|---|---|---|---|

Value | 6 | 2048 | 28,672 | 192 | 10 | 10 | 49,419 |

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Bald, C.; Schmidt, G.
Processing Chain for Localization of Magnetoelectric Sensors in Real Time. *Sensors* **2021**, *21*, 5675.
https://doi.org/10.3390/s21165675

**AMA Style**

Bald C, Schmidt G.
Processing Chain for Localization of Magnetoelectric Sensors in Real Time. *Sensors*. 2021; 21(16):5675.
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**Chicago/Turabian Style**

Bald, Christin, and Gerhard Schmidt.
2021. "Processing Chain for Localization of Magnetoelectric Sensors in Real Time" *Sensors* 21, no. 16: 5675.
https://doi.org/10.3390/s21165675